Radiation Heat Transfer Analysis in Two-Phase Mixture Associated with Liquid

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

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

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].

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.

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

3-4-3 Application of Current Model to Sodium Experiments ...... 78 3-4-4 Bubble Collapse Time ...... 84 3-4-5 Boundary Work and Fuel-Gas Bubble ...... 91 CHAPTER IV SUMMARY AND CONCLUSIONS ...... 93 4-1 Rayleigh, Mie, and Expansion Work Models ...... 93 4-2 Recommendations ...... 97 REFERENCES ...... 98 APPENDIX A Matzler’s Functions ...... 104 A-1 Computes a Matrix of Mie Coefficients ...... 104 A-2 Computation of Mie Efficencies ...... 105 A-3 Computation and Plot of Mie Power Scattering ...... 107 A-4 Computation and Plot of Mie Power Scattering and Diffraction Functions ...... 108 APPENDIX B MATLAB Codes ...... 114 B-1 Mie Scattering for FAST Tests ...... 114 B-2 Bubble Collapse Time for Fast Tests ...... 116 B-3 The Absorption Coefficient for Small Particles ...... 127 APPENDIX C Mie Theory ...... 129 C-1 Introduction ...... 129 C-2 Rayleigh, Mie, and Optical Scattering ...... 129 C-3 Mie Efficiencies and Cross Sections ...... 132 C-4 Angular Scattering ...... 133 C-5 Phase Function ...... 134 C-6 Absorption and Scattering Coefficients ...... 135 C-7 Nucleation Rate ...... 136 C-8 Angular Scattering Application ...... 137 C-9 Rayleigh Scattering Model ...... 141 C-10 Sodium Data ...... 142 C-11 Absorption and Scattering Efficiencies In Rayleigh Scattering Model ...... 142 C-12 The Absorption Coefficient In Rayleigh Model ...... 143

vii

LIST OF FIGURES

Figure 1. Pool type sodium-cooled fast reactor (SFR) [66]...... 3 Figure 2. Aerosol release scenario [3]...... 4 Figure 3. Summary of sequence of events in the disassembly phase for SRF accident [48]...... 5 Figure 4. Reactor geometry (adapted from WTT) [74]...... 5 Figure 5. Example of bubble and reactor [3]...... 7 Figure 6. The fog thickness...... 26 Figure 7. Temperature distribution for two components and a two-phase mixture...... 29 Figure 8. Physical model of a bubble contains UO2 fuel and fission gas...... 34 Figure 9. Physical model at T1 ≠ 0, T2 ≠ 0, and T ≠ 0 ...... 46 Figure 10. Transient temperature profile for semi-infinite mixture [56]...... 53 Figure 11. Fog formation under water tests at ORNL [3, 63]...... 54 Figure 12. Transient temperature profile in spherical coordinates at t=1.85 ms, k=10 m-1 and R=100 mm...... 55 Figure 13. Transient temperature profile in spherical coordinates at t=2.5 ms, k=10 m-1 and R=100 mm...... 56 Figure 14. Transient temperature profile in spherical coordinates at t=3.5 ms, k=10 m-1 and R=100 mm...... 57 Figure 15. Transient temperature profile for slab at 3.5 ms, τ = 1 and k = 10 m-1...... 58 Figure 16. Transient heat flux for semi-infinite condensable mixture [56]...... 59 Figure 17. Transient wall heat flux for bubble at different sizes...... 60 Figure 18. Bubble position relative to pool [63]...... 61 Figure 19. Transient temperature profile and fog penetration depth at bubble collapse time. ... 63 Figure 20. Comparison of temperature distribution at K = 10 m-1 and k = 30 m-1...... 65 Figure 21. Physical model of a bubble containing UO2 fuel and fission gas...... 67 Figure 22. The variation of dimensionless incident radiation for case1 [28]...... 68 Figure 23. The variation of dimensionless radiative flux for case1 [28]...... 69 Figure 24. Dimensionless incident radiation variation for case2 [28]...... 70 Figure 25. Dimensionless radiative flux variation for cas2 [28]...... 70 Figure 26. The effect of dimensionless incident radiation for case3 [28]...... 72 Figure 27. The effect of dimensionless radiative flux for case3 [28]...... 72 Figure 28. Dimensionless irradiance behavior under sodium experiments ...... 74 Figure 29. Radiative Heat Flux behavior under sodium experiments ...... 75 Figure 30. Effect of τ1/τ2 ratio on radiative heat transfer under sodium tests...... 76 Figure 31. Dimensionless irradiance behavior under sodium experiments...... 77 Figure 32. Dimensionless radiative heat flux behavior under sodium experiments...... 77 Figure 33. Effect of τ1/τ2 ratio on radiative heat transfer under sodium tests...... 78 Figure 34. Radiative heat flux variation under sodium tests for case1...... 79 Figure 35. The variation of dimensionless irradiance under sodium tests for case1...... 80 Figure 36. Radiative heat flux variation under sodium tests for case2...... 81 Figure 37. The variation of dimensionless irradiance under sodium tests for case2...... 81 Figure 38. Dimensionless heat flux variation under sodium tests for case3...... 82

viii

Figure 39. The variation of dimensionless irradiance for sodium tests for case3...... 83 Figure 40. Physical model drawing and coordinates...... 84 Figure 41. The variation of the volumetric heat generation inside the system ...... 85 Figure 42. The attenuation of incident radiation in participating medium [30]...... 90 Figure 43. Work transfer trends (Na Tests)...... 92 Figure 44. Angular Mie scattering diagram [39]...... 138 Figure 45. The scattering function [39]...... 139 Figure 46. The phase function at θ' = 0°...... 139 Figure 47. The phase function at θ' = 180°...... 140 Figure 48. Angular scattering diagram...... 141

LIST OF TABLES

Table 1. The bubble rise time...... 62 Table 2. The bubble collapse time at k = 10 m-1 ...... 64 Table 3. The bubble collapse time at k = 2.2 m-1 ...... 65 Table 4. Dimensionless heat fluxes at the outer surface...... 76 Table 5. The bubble collapse time at k = 0.082 m-1 ...... 87 Table 6. The bubble collapse time at k = 1.1 m-1 ...... 89 Table 7. Asymmetry factors of Legendre polynomials for phase function...... 137 Table 8. Mie model inputs...... 141 Table 9. Mie model parameters...... 141 Table 10. Rayleigh model inputs...... 141 Table 11. Rayleigh model parameters...... 142 Table 12. Sodium data...... 142 Table 13. Absorption and scattering efficiencies...... 143 Table 14. The absorption coefficient at different bubble radius...... 144

LIST OF ABBREVIATIONS AND NOTATIONS

Nomenclature Description

Upper case

2 A1 and A2 The inner and the outer surface area, m

Ai, I = 1,2,3,4,5,6 Constants used in Cramer’s rule.

B Surface radiosity, (W- m2/m)

3 C Molar concentration, (C1+C2) (mol/m )

Cp Molar specific heat at constant pressure, (J/mol-K)

D Molar diffusion coefficient, (m2/s)

D Diameter, (m)

A, B, D, C, n Coefficients in the general Bessel equation.

Ei Exponential integral of ith kind.

E1 and E2 Parameters used to simplify the scattering model.

Ein Energy input, (J)

Eout Energy output, (J)

Eg Energy generation, (J)

∆Est Energy stored, (J)

F1 and F2 Parameters used to simplify the scattering model.

G(τ) The irradiance, (W/m2)

H Pool level relative to the vaporizer, (m)

I Radiation intensity, (W/m2-sr)

퐼1 The modified Bessel functions of the first kind of order 퐼1 . 2 2

2 퐼푣푏(푇) The intensity of radiation by black body, (W/m -sr).

푘1 The modified Bessel functions of the second kind of order 푘1 2 2 M Molecular weight, (g/mol)

N Molar flux, (mol/m2-s)

P Pressure (Pa)

P (μ, μ’) The scattering phase function.

Pn (μ) Legendre polynomials of scattering radiation.

Pn (μ') Legendre polynomials of incident radiation.

R Bubble radius, (m)

R Molar fog production rate/volume, (mol/s-m3)

R Universal gas constant, (J/mol-K)

RE, RF Internal and external radius, (m)

S Saturation ratio.

T Temperature, (K)

Ti Initial temperature level of coolant, (K)

V Volume, (m3)

W Work transfer, (J)

X The mole fraction.

Z Physical coordinate normal to the wall, (m)

Lower Case c Molar concentration, (mol/m3)

2 ebλ Spectral black body emissive power, (W-m /m) g Gravitational acceleration, (m/s2) hfg Latent heat of condensation, (J/mol-K) k Thermal conductivity, (W/m-K) k Absorption coefficient, m-1 n Number of moles, (mol) n Refractive index q(τ) The radiative heat flux, (W/m2) r Radius of the sphere, (m)

3 r1 and r3 The amount of vapor/fog rate per unit volume, (mol/s-m ) t Bubble collapse time, (s) v* Molar average velocity, (m) xi ith component molar fraction

xii

Greek

Α Thermal diffusivity, (m2/s)

β Special function, (s-1)

-1 βλ Spectral extinction coefficient, (m )

γ Specific heat ratio

ρ Density, (kg/m3)

δkj Kronecker delta; = 1 when j = k; = 0 when j ≠ k

ϵλ Spectral emissivity

θ Polar angle measured from normal of surface, (rad)

μ The cosine direction of θ, (rad)

ρ Reflectivity

-1 σλ Spectral scattering coefficient, (m )

σ Stefan Boltzmann

τλ Optical coordinate

2 Ψ0 (τ) The incident radiation, (W/m )

2 Ψ1 (τ) The radiant heat flux, (W/m )

∅ Azimuthal angle, (rad)

ωλ Albedo of scattering

γ Special function

ϵλ Spectral emissivity

η(T) Special function

Subscripts cg Covergas quantity max Maximum value sat Saturation property

∞ Far-field quantity

0 Initial condition

1 Condensable vapor

2 Non-condensable gas

xiii

3 Fog condensates

F Fog boundary w wall sat saturation

λ Wavelength, (m)

τ Varying with τ- direction

xiv

CHAPTER I

INTRODUCTION

A Fast Neutron Reactor (FNR) is a class of nuclear reactors in which the fission chain is maintained by fast neutrons. The nuclear fuel employed must be rich in fissile material compared to that used in thermal nuclear reactors. Because of this constraint, they can use fuel that is discarded as waste in other classes of reactors. So for FNRs, a neutron moderator is not required, resulting in a smaller less complex reactor. FNR’s can provide substantial energy in the present and foreseeable future [68].

Currently all FNR’s employ liquid metal as the coolant and, therefore, FNR’s are frequently called Liquid Metal Coolant Reactors (LMR’s). The commonly used liquid metals are mercury, sodium, lead, and tin. As compared to a water coolant, the liquid metal coolant removes heat much faster. This has proven to be more efficient and allows a much higher power density than reactors using alternatives methods. The higher power densities and efficiency of LMR’s is further amplified by the fact that they can use spent fuel from ordinary pressurized water reactors

(PWRs). This prolongs their life cycle significantly and reduces their long-term costs.

Concerns about their safety must also be addressed to a level that is comparable to other types of nuclear power generation reactors. Research is on-going in several countries to reduce cost and mitigate safety concerns [68].

Other reasons for adopting LMR’s besides their efficiency and power generation capabilities are:

• A LMR core is smaller than a conventional reactor, and it has higher power output of

560 – 1500 MW.

• The liquid metal coolant reactor has a higher fission/absorption ratio (the number of

neutrons released after the chain reaction) than the pressurized water reactor

(except the U-235 the pressurized water reactor is higher) [69].

• The pressure in a LMR reactor core is smaller compared to conventional reactors

[69].

• The LMR’s uses spent fuel from conventional reactors. Fast reactors use Uranium

238 as well as Uranium 235. Liquid metal breeding reactors (LMR’s), convert

Uranium 238 to Plutonium 239 (Plutonium production). LMR’s can consume long-

lived actinides (U, Np, Pu, and Am). The fast reactor can be used to convert nuclear

weapon grade plutonium to a less harmful state [68].

• The reduction in decay time of a long-lived actinide is a chief advantage to employ

LMR’s as they are less harmful to the environment. Moreover, a cost saving is

realized in the initial Uranium and repository costs. This is achieved from recycling

and converting of the actinides which lower the thermal heating load inside the

repository as well as resulting in shorter decay times [71].

The typical LMR consists of a stainless-steel vessel containing the core, coolant, control rods, cover gas, and the heat exchanger. An example of a LMR is shown in Figure 1. The vessels of these reactors have a cylindrical shape with a spherical base. Often it is placed in a cylindrical underground hole or under a mountain for safety concerns. The core of the reactor contains the

UO2 fuel rod and the coolant channels for flow of the liquid metal coolant (here, Sodium).

The coolant is pumped from the cold region in the spherical base to the center of the core where the chain reaction is continually generating an enormous amount of heat. The upward moving coolant between the fuel rods removes heat from the core which is transferred to the heat exchanger. The cooled sodium flows out of the heat exchanger shell bottom by gravity [4].

The secondary sodium loop is located close to the power plant for generating steam and cools the reactor side to keep the reactor core temperature within design operating conditions.

Figure 1. Pool type sodium-cooled fast reactor (SFR) [66].

Superheated pressurized steam flows through the turbine to generate electricity. In the top of the reactor, a cover gas, from inert gases such as Argon, is maintained at specific pressure.

The role of the cover gas is to keep the sodium in a safe condition away from air and water. Sodium is extremely flammable in air and explosively reacts on contact with water. Graphite control rods are located in the top of the reactor and have the ability to absorb neutrons. This allows the controllability of the fission chain reaction inside the reactor core. The chain reaction can be

3 accelerated by raising the control rods [4,68]. If the cooling pump, valves or control rods are damaged, the reactor core could overheat with the possibility of an explosion. Accident prevention is addressed during the design phase of LMRs. Care is taken to ensure that leakage of radioactive material into the core and subsequent release to the coolant pool is avoided. The goal is containment of the radioactive material and not to permit environmental release as shown in

Figure 2. The mechanism of this transfer could occur due to multiphase bubbles which contain

“fuel vapor, fission gas, and UO2 particles” [3]. LMR’s are becoming an increasingly important in the field of nuclear engineering. More efforts should be encouraged to study, analyze, and to improve the performance of fast neutron reactors. Safety issues should especially be addressed so that possible negative impacts can be mitigated.

Figure 2. Aerosol release scenario [3].

The hypothetical core disruptive accident (HCDA) will occur when there is no coolant flow inside the cooling channels among the fuel rods. Conditions such as this may include for example the failure of circulation pumps, control room systems, and control rods. When the reactor is at

4 full load, and the cooling system is in a failure state, the fuel in the core starts to heat up or boil and the vapor can leave the reactor as shown in Figure 3 [55].

Figure 3. Summary of sequence of events in the disassembly phase for SRF accident [48].

As the fuel starts vaporizing, UO2 vapor and non-condensable Xe gas are produced from the fission chain reaction. The bubbles expand radially and axially at high temperature and pressure, leaving the core and proceeding to the coolant pool as indicated in Figure 4 [55].

Figure 4. Reactor geometry (adapted from WTT) [74].

The bubble rises under the effect of gravity, and it takes a certain time to contact the top of the reactor [44,52]. During this journey, the condensation of UO2 vapor takes place due to cooling from the boundary. The condensation process of the UO2 vapor may cause the bubble to collapse before it hits the reactor head. Another scenario is that the radioactive material inside the bubble will release outside the reactor to the surrounding environment as shown in Figures 2 and 3 due to the slowing of the condensation process [64]. From the preceding paragraphs, it is realized that it is important to study the effect of the fuel bubble’s geometry on radiative heat flux and transient temperature distribution in a participating medium during a hypothetical core disruptive accident (HCDA) for liquid metal cooled reactor (LMR). In addition, it’s interesting and to study the vapor bubble behaviour, in particular the bubble collapse is of interest. Figures 5 shows a stationary, spherical bubble containing fuel vapor (UO2), fission gas (Xe) at high temperature (4500 K) surrounded by Sodium coolant (811 K) [3, 63].

The system as described is under local thermodynamic equilibrium, and radiation dominates the heat transfer. Furthermore, the thermodynamic properties of UO2 vapor and Xe gas are assumed constant, and the vapor and gas are modelled as a perfect gas [11, 56]. Under these conditions, it was also assumed that the effects of other types of heat and mass transfer can be neglected such as convection, diffusion and conduction as compared with the radiation heat transfer where it is radiating at high temperatures [56]. Mie theory will be employed to ascertain the amount of radiative energy that is lost due to the presence of UO2 particles within the medium in the model. The magnitude of the energy reduction can be determined by computing the scattering and absorbing coefficients of the participating medium.

Figure 5. Example of bubble and reactor [3].

There are two types of scattering that are of interest in the participating medium, Rayleigh

1 scattering and Mie scattering. Rayleigh scattering occurs when the particle size (r) is less than λ 10

1 of the photon wavelength. The Mie scattering region is in the range from λ of the particle size 10 to less than λ [30] as described in Appendix C. Both models (Rayleigh and Mie) were considered in this study to analyze and interpret the data associated with severe accident assessments. (e.g.,

Oak Ridge Experiments) as indicated in the next chapter. This assessment is based on a comparison of the bubble rise time, computed using the Taylor formula, against the elapsed time required for energy transfer by radiation from the bubble [44, 63].

The importance of the present study is to provide information about the tolerances and the safety profile of LMR, so that limits can be established to prevent catastrophic failure involved with melting the reactor core and to avoid radioactive material release into the coolant pool and to the environment.

In the past, a safety analysis of a liquid metal fast breeder reactor (LMFBR) was studied by

Michael et al. (1979) using an experimental model describing two-phase liquid flow motion with coolant vaporization phenomena and without external thermal radiation effects. Michael et al.

(1979) theorized that the accident was initiated by the loss of the flow of the coolant. In the case of this severe accident, the strength of the accident impact was equal to the size of the expansion work, and because of the pressure of non-condensable gas inside the bubble, the condensate fuel around the coolant appears as fog [49]. However, they neglected the absorption/scattering effects on the thermal radiation. Kress et al. (1977) presented a significant examination of nuclear reactor safety research regarding the study of important parameters affecting two phase flow generated during a hypothetical core disruptive accident (HCDA). These parameters include the internal circulation velocity and properties of the non-condensable gas. The problem was examined by solving the involved turbulent transient heat and mass transfer. The important conclusion of Kress et al. (1977) is that the heat transfer coefficient for condensation of UO2 can be determined at different concentration values of a non-condensable gas. However, they simulated the bubble as a flat plate [2]. The intent of the study by Tobias et al. (1979) was to study the heat and mass transfer arising from a Uranium dioxide (UO2) bubble for a sodium experiment in the Fuel Aerosol Simulant Test (FAST) facility. In this assessment, a spherical cap shape related to the bubble size was employed in the model. A concentric spheres formula was used for the emissivity of the gas-sodium, but they ignored the absorption/scattering effects on thermal radiation [5, 6].

A. B. Reynolds et al. (1979) illustrated bubble behavior in experiments. The bubbles oscillated and moved either upward or downward depending on the pressure difference between the tank and bubble. The bubble rose if the bubble pressure was greater than the tank pressure; otherwise, it moved down. Some parameters affecting the bubble motion were studied in this model such

8 as buoyancy force, free surface, and rigid surface boundary. In contrast, this study neglected the effect of non-condensable gases and the thermal radiation effects [51].

The expansion of two phase UO2 in a tank of inviscid and incompressible coolant has been analyzed by B. Reynolds et al. (1987) using the UVABUBL computer model. This code consists of two distinct parts: a hydrodynamics model that uses two-dimensional cylindrical geometry, and a thermal model that uses one-dimensional spherical geometry. An agreement has been achieved between the experimental and calculated results for period, bubble radius, and pressure for all

FAST-underwater experiments [55]. However, there are some omissions in this model such as condensation, the presence of non-condensable gas, fog formation, and particle scattering. H.

Chan et al. (1979) investigated the condensation phenomena and heat transfer from a very hot mixture of gases at 4050 K to a low temperature field boundary consisting of a sodium coolant at

823 K. An analytical solution was obtained for two-component, two-phase mixtures in this study.

This analysis showed that the fog formation was due to rapid radiative cooling from the mixture to the coolant. Thus, fog formation is a significant parameter that must be considered in computing the heat transfer from a very high-temperature condensable mixture [56]. In contrast, they neglected the effect of scattering, and they used unrealistic geometry. R. L. Webb et al.

(1985) have analyzed the heat and mass transfers at ORNL for UO2 bubble under sodium FAST tests by using the UVABUBL II Package. This code was designed based on a one-dimensional radiative heat transfer model, and the coolant motion was simulated in a two-dimensional (r, z) hydrodynamics model. A transient finite difference technique was used at each node to solve the equations. A comparison between the theoretical study and measured data was made. This showed no occurrences of surface vaporization during the sodium tests. because the sodium has higher thermal conductivity and lower emissivity than water. The purpose of this paper was to calculate the bubble behavior, which involves the bubble period and size [53]. In contrast,

9 condensation, and the presence of non-condensable gas, along with fog formation and scattering were not included in this research. In full scale reactor experiment, Fauske et al. (1973) have shown that the effect of non-condensable gas, seriously retards the condensation process. The bubble expansion of the fuel-coolant is isentropic. Fauske et al. (1973) found that the measurements indicate an interaction zone covering 40 % of Fast Flux Test Facility (FFTF) core a

750 ft3 bubble at 2270 °F and 185 psia is obtained. Fauske et al. (1973) found that the bubble consists of 116 lb. of sodium vapor, 174 lb. of liquid sodium, 680 lb. of UO2 vapor, and 35 lb. of xenon gas. The bubble had a large diameter is 11.3 ft and rising from the bottom of the Sodium pool at 8.5 - 8.8 ft/s, and it reached the upper surface of the Sodium pool in approximately 2 s

[53].

1-1 Objectives

The results extracted from the following three mathematical models can be used by others to analyze and interpret the phenomenology and existing database associated with severe accident assessments. (e.g., Oak Ridge Experiments and Reactor-Scale Events):

1- Rayleigh model (non-scattering model) of radiation heat transfer

The Rayleigh model will use the work already done by H. Chan et ai. (1979) who

investigated fog phenomena and heat transfer from a very hot mixture of gases to a low

temperature field boundary at the Argonne National Laboratory.

2- Mie model (scattering model) of radiation heat transfer

To analyze and interpret the results observed at (CDV) experiments conducted in the

FAST facility at Oak Ridge National Laboratory and with Fast Flux Test Facility (FFTF)

accident in Argonne National Laboratory the Mie scattering model will be used. For the

interpretation of the results, work done by W. Li et al. (1990) who investigated the

radiative heat transfer in an emitting, absorbing and scattering media between two

concentric a spherical surface will be considered [28].

3- Boundary work and fuel-gas bubble

Michael et al, (1978) studied and investigated the experimental model describing a two-

phase liquid flow motion with coolant vaporization phenomena and without external

thermal radiation effects. That study theorized that the strength of impact in a disruptive

accident was equal to the size of the work done by the expansion. The present work will

quantify the work done by the bubble expansion.

1-2 New Contributions

In this research, the effect of the fuel bubble’s geometry on the radiative heat flux and the transient temperature distribution in the participating medium during a hypothetical core disruptive accident (HCDA) for a liquid metal cooled reactor (LMR) will be investigated.

Furthermore, the influence of the extinction coefficient on the rate of radiative heat transfer energy in spherical coordinates containing UO2 particles during (HCDA) for (LMR) will be researched. Theoretical and numerical results, which will include bubble collapse times, will be compared with the capacitor discharge vaporization (CDV) experiments conducted in the FAST facility at Oak Ridge National Laboratory and with Fast Flux Test Facility (FFTF) accident in Argonne

National Laboratory.

Chapter two of this dissertation is concerned with developing expression for the Rayleigh model, the Mie model and the boundary work involved. Chapter three is devoted to analyzing and reporting the results. Chapter four is concerned with drawing conclusions based on the results and the analysis done on Chapter three

CHAPTER II

MATHEMATICAL MODELS

While continuing the study of safety profiles for fast breeder reactors, this chapter examines the problems associated with the transient heat flux at the wall. Also, a model of the fuel-gas bubble with transient temperature distributions in spherical coordinates is presented.

Another objective of this chapter is to investigate the differences between the results of realistic sphere bubble geometry and an unrealistic slab bubble geometry considered in past work. From there, this study will prove and determine the influences of the bubble’s geometry on the heat transfer and the wall temperature are through the fuel-gas bubble moving up inside the sodium coolant. This will be a new contribution to this field.

The physical problem and objectives for this work have been described. The mathematical models of interest are now presented. The first model is non-scattering model, the second model is the scattering model, and the last model is boundary work of fuel-gas bubble.

2-1 Non-Scattering Model

In this section, the radiative heat equation will be simplified and solved using spherical coordinates. Scattering effects are neglected in the presence one-dimensional thermal radiation transfer in r-direction. Then by substituting equations for radiant heat transfer into the energy equation, they are solved together along with the continuity equations of the mixture species under local thermodynamic equilibrium.

The spherical coordinate system for the equation of radiative heat transfer (RTE) has the following form [26, 30].

∂퐼 (r,μ) 1−μ2 ∂퐼 (r,μ) 1 μ 휆 + 휆 + 훽 퐼 (r, μ) = 푘 퐼 + 1 휎 ∫ 푝(휇, μ′)퐼(r, μ′) dμ′ (2.1) ∂r r ∂μ 휆 휆 휆 휆푏 2 휆 −1

The first and the second terms of the above equation represents the change of 퐼휆 (r, μ) intensity with respect to r and μ directions, where μ is the direction cosine of propagating beam radiation. The third term represents the reduction of the radiation due to the absorption and out scattering effect on the thermal radiation that is propagating through the medium. Both terms on the right-hand side of Equation (2.1) represent the augmentation of 퐼휆 (r, μ) intensity. The first term on right hand side is the stimulated emission which is always in the same direction of the

푛2휎푇4(휏) radiation causing the stimulation where 퐼 = . In scattering is the second term, which is 휆푏 휋 the amount of intensity scattered in the direction of interest, and its magnitude equals to the phase function times the average radiation scattered in all directions [30].

By neglecting the scattering effects and assuming azimuthal symmetry ∅ (θ, ∅) = ∅(θ) [30] in Equation (2.1), the effect of the bubble’s geometry on the radiative heat flux and the transient temperature distribution in the participating medium during a hypothetical core disruptive accident (HCDA) for a liquid metal cooled reactor can be simulated.

Based on these assumptions, Equation (2.1) can be simplified to

d퐼 (r,μ) μ 휆 + 훽 퐼 (r, μ) = 푘 퐼 (2.2) dr 휆 휆 휆 휆푏

-1 Where βλ and kλ are the extinction and the absorption coefficients with m units.

푟 Let 휏휆 = ∫0 훽휆 푑푟 (2.3)

푅 휏0휆 = ∫0 훽휆 푑푟 (2.4)

휇 = 푐표푠휃 (2.5)

푑휏 푑푟 = 휆 (2.6) 훽휆

Back substituting Equation (2.6) into Equation (2.2) reduces to [63]

d퐼휆 (r,μ) 푘휆푒휆푏(τ) μ + 훽휆퐼휆 (r, μ) = (2.7) 푑휏휆⁄훽휆 휋

where 푒휆푏(τ) is the black – body monochromatic emissive power.

d퐼휆 (r,μ) 푘휆푒휆푏(τ) μ훽휆 + 훽휆퐼휆 (r, μ) = (2.8) 푑휏휆 휋

Upon dividing Equation (2.8) by 훽휆 gives

d퐼휆 (r,μ) 푘휆푒휆푏(τ) μ + 퐼휆 (r, μ) = (2.9) 푑휏휆 휋훽휆

It is known that

휎휆 휔휆 = (2.10) 훽휆

푘휆 and (1 − 휔휆) = (2.11) 훽휆

Equation (2.11) is a non–homogenous ordinary differential equation that can be solved by using integrating factor (IF) method or the variation of parameters method. Here the integrating factor method is used to solve Equation (2.11). This equation is in standard form with

(1−휔 ) 1 the quantities of Q and IF as 휆 푒 (τ), 푃 = again Equation (2.11) can be re-written as 휋휇 휆푏 휇

푑퐼 (r,μ) + P퐼휆 (r, μ) = 푄 (2.12) 푑휏휆

(1−휔 ) 1 Where 푄 = 휆 푒 (τ), and 푃 = To solve this equation, the integrating factor can be 휋휇 휆푏 휇

1 ∫ 푑휏 computed by using the relationship 퐼퐹 = 푒 휇 휆 (2.13)

휏휆 =≫ 퐼퐹 = 푒 휇 (2.14)

To check the solution of Equation (2.12), the integrating factor should satisfy the following equation.

푑 (퐼퐹 × 퐼휆) = (푄 × 퐼퐹) (2.15) 푑휏휆

휏 휏 휏 휆 푑퐼 1 휆 휆 휇 휆 휇 휇 푒 + 푒 퐼휆 = 푄푒 (2.16) 푑휏휆 휇

휏휆 By cancelling out 푒 휇 from both sides of Equation (2.16), Equation (2.16) is satisfied and resulting in Equation (2.12).

푑퐼휆 1 + 퐼휆 = 푄 (2.17) 푑휏휆 휇

Now, integrating Equation (2.17) from (0 → 휏휆) gives the general solution,

푑 휏휆 ∫ 퐼퐹 퐼휆 푑휏 = ∫ 푄 퐼퐹 d휏휆 (2.18) 푑휏휆 0

휏 휏 휆 휏 휆 휇 휆 휇 푒 퐼휆 = ∫0 푄 푒 d휏휆 + 푐1 (2.19)

The boundary conditions of Equation (2.19) are: [62]

+ + 퐼휆 (휏휆, 휇) = 퐼휆 (0, 휇) (2.20)

− − 퐼휆 (휏휆, 휇) = 퐼휆 (휏0휆, −휇) (2.21)

+ − Where 퐼휆 (휏휆, 휇)represents the 퐼휆intensity in positive μ direction, but 퐼휆 (휏휆, 휇)represents

퐼휆 in negative μ direction. By applying the boundary condition (2.20) into (2.19) the integration constant 푐1can be calculated by setting 휏휆 = 0 into Equation (2.19).

0 0 0 휇 휇 +( ) 푒 퐼휆 = ∫0 푄 푒 d휏휆 + 푐1 =≫ 푐1 = 퐼휆 0, 휇 back substitution 푐1into Equation (2.19) yields

휏 푡 휆 휏 푘 휇 + 휆 휇 + 휆 푒 퐼휆 = ∫ 푄(푡) 푒 dt + 퐼휆 (0, 휇), 푄 = 푒휆푏(t) (2.22) 0 휇 훽휆

휏 푡 휆 휏 푘 휇 + 휆 휆 휇 + 푒 퐼휆 = ∫ 푒푏휆(t) 푒 dt + 퐼휆 (0, 휇) (2.23) 0 휇휋 훽휆

(휏 −푡) −휏 1 휏 푘 − 휆 푑푡 휆 + 휆 휆 휇 + 휇 퐼휆 = ∫ [ 푒푏휆(t) 푒 ] + 퐼휆 (0, 휇) 푒 (2.24) 휋 0 훽휆 휇

By applying the boundary condition (2.21) into Equation (2.19), the constant 푐2 can be calculated at 휏휆 = 휏0휆

휏 휏 휏 0휆 휏 0휆 0휆 휇 − 0휆 휇 − 휇 푒 퐼 = ∫ 푄 푒 푑휏휆 + 푐2 =≫ 푐2 = 퐼 (휏0휆, 휇) 푒 again, back substitution 휆 휏0휆 휆

푐2into Equation (2.19) yields

휏 휏 휏 휆 휏 0휆 0휆 휇 − 0휆 휇 − 휇 푒 퐼 = ∫ 푄 푒 d휏휆 + 퐼 (휏0휆, 휇) 푒 (2.25) 휆 휏휆 휆

휏휆 − 휇 Solving for 퐼휆 by dividing Equation (2.25) by 푒 gives

(휏 −푡) (휏 −휏 ) −1 휏 푘 − 휆 푑푡 0휆 휆 − 0휆 휆 휇 − 휇 퐼휆 = ∫ [ 푒푏휆(t) ] 푒 + 퐼휆 (휏0휆, 휇) 푒 (2.26) 휋 휏휆 훽휆 휇

The hemispherical monochromatic radiation flux can be calculated from the following formula [63].

′′ ( ) ( ) 푞푅휆 휏휆 = ∫4휋 퐼휆 휏휆, 휇 푐표푠휃 푑휔 (2.27)

2휋 휋⁄2 The solid angle ω can be expressed by using ∫ℎ 푑휔 = ∫0 ∫0 푠푖푛휃 푑휃 푑∅ (2.28)

From Equations (2.27) and (2.28) we find, [27]

⁄ ′′( ) 2휋 휋 2 ( ) 푞푅 휆 = ∫0 ∫0 퐼휆,푒 휆, 휃, ∅ 푐표푠휃푠푖푛휃 푑휃 푑∅ (2.29)

′′( ) 1 ( ) 푞푅 휆 = 2휋 ∫−1 퐼휆 휏휆, 휇 휇 푑휇 (2.30)

+ − + − Equation (2.30) [26] can be written in terms of 퐼휆 and 퐼휆 according to 휇 and 휇

′′( ) 1 + −1 − 푞푅 휆 = 2휋 ∫0 퐼휆 휇 푑휇 − 2휋 ∫0 퐼휆 휇 푑휇 (2.31)

Upon substituting Equations (2.24) and (2.26) into Equation (2.31) gives

(휏 −푡) −휏 1 휏 푘 − 휆 1 휆 ′′ 휆 휆 휇 + 휇 푞푅 (휆) = 2 ∫ ∫ 푒푏휆(t) 푒 dµ dt + 2휋 ∫ 퐼휆 (0, 휇) 푒 휇 푑휇 + 0 0 훽휆 0

(휏 −푡) −(휏 −휏 ) −1 휏 푘 − 휆 −1 0휆 휆 휆0 휆 휇 − 휇 2 ∫ ∫ 푒푏휆(t) 푒 dµ dt − 2휋 ∫ 퐼휆 (휏0휆, 휇) 푒 휇 푑휇 (2.32) 0 휏휆 훽휆 0

To simplify Equation (2.32), it is necessary to introduce the definition of exponential integrals as shown below. [63]

1 −푡 ( ) 푛−2 휇 퐸푛 푡 = ∫0 휇 푒 푑휇 (2.33)

−푡 1 ( ) 휇 When n = 2 =≥ 퐸2 푡 = ∫0 푒 푑휇 (2.34)

By substituting Equation (2.34) into Equation (2.32) the radiative heat flux becomes

−휏 −(휏 −휏 ) 1 휆 1 0휆 휆 ′′( ) +( ) 휇 −( ) 휇 푞푅 휆 = 2휋 ∫0 퐼휆 0, 휇 푒 휇 푑휇 − 2휋 ∫0 퐼휆 휏0휆, −휇 푒 휇 푑휇 +

휏휆 푘휆 휏0휆 푘휆 2 ∫ 푒푏휆(t) 퐸2(휏휆 − 푡)푑푡 − 2 ∫ 푒푏휆(t) 퐸2(푡 − 휏휆)푑푡 (2.35) 0 훽휆 휏휆 훽휆

Now, by differentiating Equation (2.35) with respect to τ, it will be substituted it into the energy equation in the next section.

∞ ∞ 1 휕 ∞ 푑푞′′(휆) ∞ 2 ∫ 푞′′(휆)푑휆 = ∫ (푟2푞′′(휆))푑휆 = ∫ 푅 푑휆 + ∫ 푞′′(휆) 푑휆 (2.36) 0 푅 0 푟2 휕푟 푅 0 푑푟 0 푟 푅

푟 휏휆 = ∫0 퐵휆푑푟 (2.37)

푑휏 = 훽 (2.38) 푑푟

In order to couple the heat flux with the energy equation, the divergence of radiative flux is needed. Thus, the divergence of radiative flux becomes:

′′ ∞ ′′ ∞ 1 휕 2 ′′ ∞ 퐵휆 푑푞푅 (휆) ∞ 2퐵휆 ′′ ∫ 푞푅 (휆)푑휆 = ∫ 2 (푟 푞푅 (휆))푑휆 = ∫ 푑휆 + ∫ 푞푅 (휆) 푑휆 (2.39) 0 0 푟 휕푟 0 푑휏휆 0 휏휆

Starting with the first term on the right-hand side of Equation (2.39) and using the Leibniz integral rule and then examining the second term of Equation (2.39).

−휏 −(휏 −휏 ) 푑푞′′(휆) 푑 1 휆 푑 1 0휆 휆 푅 + 휇 − 휇 = 2휋 ∫ [ 퐼휆 (0, 휇) 푒 휇 ]푑휇 − 2휋 ∫ [퐼휆 (휏0휆, −휇) 푒 휇] 푑휇 + 푑휏휆 푑휏휆 0 푑휏휆 0

푑 휏휆 푘휆 푑 휏0휆 푘휆 2 ∫ 푒푏휆(t) 퐸2(휏휆 − 푡)푑푡 − 2 ∫ 푒푏휆(t) 퐸2(푡 − 휏휆)푑푡 (2.40) 푑휏휆 0 훽휆 푑휏휆 휏휆 훽휆

−휏 푑푞′′(휆) 1 휆 푅 + 휇 = −2휋 ∫ 퐼휆 (0, 휇) 푒 푑휇 − 푑휏휆 0

−(휏 −휏 ) 1 0휆 휆 휏 푘 휏 푘 − 휇 휆 휆 0휆 휆 2휋 ∫ 퐼휆 (휏0휆, −휇) 푒 푑휇 − 2 ∫ 푒푏휆(t) 퐸1(휏휆 − 푡)푑푡 − 2 ∫ 푒푏휆(t) 퐸1(푡 − 0 0 훽휆 휏휆 훽휆

푘휆 푑휏휆 푑[0] 푘휆 휏휆) 푑푡 + 2 { 푒푏휆(휏휆)} [퐸2(휏휆 − 휏휆) − 퐸2(휏휆 − 0) ] − 2 { 푒푏휆(휏휆)} [퐸2(휏0휆 − 훽휆 푑휏휆 푑휏휆 훽휆

푑휏0휆 푑휏휆 휏휆) − 퐸2(휏휆 − 휏휆) ] (2.41) 푑휏휆 푑휏휆

The values of 퐸2 at 휏휆 ,and 휏휆 = ∞ are (퐸2(0) = 1, 퐸2 (∞) = 0). Upon substituting these values in Equation (2.41) produces: [63]

−휏 −(휏 −휏 ) 푑푞′′(휆) 1 휆 1 0휆 휆 푅 + 휇 − 휇 = −2휋 ∫ 퐼휆 (0, 휇) 푒 푑휇 − 2휋 ∫ 퐼휆 (휏0휆, −휇) 푒 푑휇 − 푑휏휆 0 0

휏휆 푘휆 휏0휆 푘휆 푘휆 2 ∫ 푒푏휆(t) 퐸1(휏휆 − 푡)푑푡 − 2 ∫ 푒푏휆(t) 퐸1(푡 − 휏휆)푑푡 + 4 { 푒푏휆(휏휆)} (2.42) 0 훽휆 휏휆 훽휆 훽휆

Multiplying Equation (2.42) by (-1) further gives:

−휏 −(휏 −휏 ) 푑푞′′(휆) 1 휆 1 0휆 휆 푅 + 휇 − 휇 − = 2휋 ∫ 퐼휆 (0, 휇) 푒 푑휇 + 2휋 ∫ 퐼휆 (휏0휆, −휇) 푒 푑휇 + 푑휏휆 0 0

휏휆 푘휆 휏0휆 푘휆 푘휆 2 ∫ 푒푏휆(t) 퐸1(휏휆 − 푡)푑푡 + 2 ∫ 푒푏휆(t) 퐸1(푡 − 휏휆)푑푡 − 4 { 푒푏휆(휏휆)} (2.43) 0 훽휆 휏휆 훽휆 훽휆

Next, the gray medium approximation can be applied. Which says that the absorption and extinction coefficient are independent of the wavelength. This permits Equation (2.43) to be re-written in as: [26, 63]

−휏 −(휏 −휏) 푑푞′′ 1 1 0 휏 푘 푅 + 휇 − 휇 0 4 − = 2휋 ∫ 퐼휆 (0, 휇) 푒 푑휇 + 2휋 ∫ 퐼휆 (휏0, −휇) 푒 푑휇 + 2 ∫ σ푇 (푡) 퐸1|휏 − 푑휏휆 0 0 0 훽

푘 푡|푑푡 − 4 σ푇4(푡) (2.44) 훽

+ − From reference [62], the terms for diffuse surfaces, 퐼휆 (0, 휇) and 퐼휆 (휏0, −휇), can be expressed in terms of surface radiosities as:

−휏 1 휆 +( ) 휇 ( ) 2휋 ∫0 퐼휆 0, 휇 푒 푑휇 = 2퐵w휆퐸2 휏휆 (2.45)

−(휏 −휏 ) 1 0휆 휆 −( ) 휇 ( ) 2휋 ∫0 퐼휆 휏0휆, −휇 푒 푑휇 = 2퐵2휆퐸2 휏0휆 − 휏휆 (2.46)

Upon integrating Equation (2.44) for all wave lengths from 0 → ∞ Equation (2.44) can be written as:

′′ 푑푞푅 ∞ ∞ τ 푘 4 − = ∫ [2퐵푤휆퐸2(휏휆) + 2퐵2휆퐸2(휏0휆 − 휏휆)]푑휆 + 2 ∫ ∫ σ푇 (푡) 퐸1(τ − 푡)푑푡푑휆 + 푑휏휆 0 0 0 훽

∞ 휏 푘 ∞ 2 ∫ ∫ 0 σ푇4(푡) 퐸 (τ − 푡)푑푡푑휆 − 4 ∫ (1 − 휔)σ푇4(푡) dλ (2.47) 0 τ 훽 1 0

In the present study, the spectral albedo of scattering was neglected. The mixture was assumed under local thermodynamic equilibrium inside a bubble has a black boundary surface.

By applying these boundary conditions, Equation (2.47) can be simplified, after setting [57]

4 where, 퐵w휆 = 휖푤휎푇푤 where 휖푤 = 1 and 퐵2휆 = 0 at the centre of the bubble.

′′ τ 휏0 푑푞푅 4 4 4 4 = 2σ푇푤퐸2(휏) + 2 ∫ σ푇 (푡) 퐸1(τ − 푡)dt + 2 ∫ σ푇 (푡) 퐸1(휏휆 − 푡)dt − 4σ푇 (푡) 푑휏휆 0 τ

(2.48)

The black body emissive power can be factored out from the above integral [56].

Thus, at t = 푡′ =≫ 푇4(푡, 휏) = 푇4(휏)

′′ 휏 휏0 푑푞푅 4 4 −퐸2(휏−푡) 4 −퐸2(푡−휏) − = 2σ푇푤퐸2(휏) + 2σ푇휏 [ ] + 2σ푇휏 [ ] (2.49) 푑휏 −1 0 1 휏

푑푞′′ − 푅 = 2σ푇4퐸 (휏) + 2σ푇4퐸 (휏 − τ) + 2σ푇4 − 2σ푇4 퐸 (휏) − 2σ푇4퐸 (휏 − τ) + 2σ푇4 − 푑휏 푤 2 휏 2 0 휏 ∞ 2 휏 2 0 휏

4 4σ푇휏 (2.50)

푑푞′′ − 푅 = 2σ푇4 퐸 (τ) − 2σ푇4 퐸 (τ) (2.51) 푑휏 푤 2 휏 2

푑푞′′ − 푅 = 2퐸 (τ)휎(푇4 − 푇4) (2.52) 푑휏 2 푤 휏

The approximation that was used to simplify Equation (2.48) to Equation (2.52) was

4 4 used in many references and gave an acceptable result [56]. Further, since 푇휏 ≫ 푇푤 Equation

(2.52) can be written in a final form as:

푑푞′′ 푅 = 2 퐸 (τ) 휎 푇4(휏) 0 < 휏 ≤ 휏 (2.53) 푑휏 2 표

By substituting Equation (2.53) into Equation (2.39) the divergence of the radiative flux becomes:

′′ ∞ 1 휕 2 ′′ ∞ 퐵휆 푑푞푅 (휆) ∞ 2퐵휆 ′′ ∫ 2 (푟 푞푅 (휆))푑휆 = ∫ 푑휆 + ∫ 푞푅 (휆) 푑휆 (2.54) 0 푟 휕푟 0 푑휏휆 0 휏

By recalling Equation (2.35) the second term in Equation (2.54) can be simplified as

−휏 −(휏 −휏 ) 1 휆 1 0휆 휆 ′′( ) +( ) 휇 −( ) 휇 푞푅 휆 = 2휋 ∫0 퐼휆 0, 휇 푒 휇 푑휇 − 2휋 ∫0 퐼휆 휏0휆, −휇 푒 휇 푑휇 +

휏휆 푘휆 휏0휆 푘휆 2 ∫ 푒푏휆(t) 퐸2(휏휆 − 푡)푑푡 − 2 ∫ 푒푏휆(t) 퐸2(푡 − 휏휆)푑푡 (2.55) 0 훽휆 휏휆 훽휆

Next, the gray medium approximation can be applied. This permits Equation (2.55) to be re-written in as: [62] [26]

−휏 −(휏 −휏 ) 1 휆 1 0휆 휆 ′′( ) +( ) 휇 −( ) 휇 푞푅 휆 = 2휋 ∫0 퐼휆 0, 휇 푒 휇 푑휇 − 2휋 ∫0 퐼휆 휏0휆, −휇 푒 휇 푑휇 +

휏휆 푘 휏0휆 푘 2 ∫ 푒푏휆(t) 퐸2(휏휆 − 푡)푑푡 − 2 ∫ 푒푏휆(t) 퐸2(푡 − 휏휆)푑푡 (2.56) 0 훽 휏휆 훽

−휏 1 휆 +( ) 휇 ( ) 2휋 ∫0 퐼휆 0, 휇 푒 휇 푑휇 = 2퐵푤휆퐸3 휏휆 (2.57)

−(휏 −휏 ) 1 0휆 휆 −( ) 휇 ( ) 2휋 ∫0 퐼휆 휏0휆, −휇 푒 휇 푑휇 = 2퐵2휆퐸3 휏0휆 − 휏휆 (2.58)

Upon substituting Equations (2.57) and (2.58) into Equation (2.56), the terms for diffuse surfaces,

+ − 퐼휆 (0, 휇) and 퐼휆 (휏0, −휇), can be expressed in terms of surface radiosities as:[62]

∞ ∞ ∞ 휏 푘 ∫ 푞′′(휆) 푑휆 = ∫ [2퐵 퐸 (휏 ) − 2퐵 퐸 (휏 − 휏 )] 푑휆 + 2 ∫ ∫ 휆 σ푇4(푡) 퐸 (휏 − 0 푅 0 1휆 3 휆 w휆 3 0휆 휆 0 0 훽 2 휆

∞ 휏0휆 푘 4 푡)푑푡푑휆 − 2 ∫ ∫ σ푇 (푡) 퐸2(푡 − 휏휆)푑푡푑휆 (2.59) 0 휏휆 훽

′′ ( ) ( ) 휏( ) 4( ) ( ) 휏0( 푞푅 = 2퐵푤퐸3 휏 − 2퐵2퐸3 휏0 − τ + 2 ∫0 1 − 휔 σ푇 푡 퐸2 τ − 푡 푑푡 − 2 ∫휏 1 −

4 휔)σ푇 (푡)퐸2(푡 − τ)푑푡 (2.60)

By applying these boundary conditions, Equation (2.60) can be simplified, after setting [57]

4 퐵w휆 = 휖w휎푇푤 where 휖푤 = 1 and =≫ 퐵2 = 0 at the center of the bubble.

′′ 4 ( ) 휏 4( ) ( ) 휏0 4( ) ( ) 푞푅 = 2휎푇푤퐸3 휏 + 2 ∫0 σ푇 푡 퐸2 τ − 푡 푑푡 − 2 ∫휏 σ푇 푡 퐸2 푡 − τ 푑푡 (2.61)

휏 4( ) ( ) In the second term of Equation (2.61) 2 ∫0 σ푇 푡 퐸2 τ − 푡 푑푡

0 ≤ 휏 − 푡 ≪ 휏 but −휏 ≤ 푡 − 휏 ≪ 0 =≫ 0 ≪ |푡 − 휏| ≪ 휏

So, if 0 ≤ 휏 − 푡 ≪ 휏 then 퐸2(휏 − 푡) = 퐸2(|푡 − 휏|)[63]

휏0 4( ) ( ) Similarly, in the third term of Equation (2.62) −2 ∫휏 σ푇 푡 퐸2 푡 − τ 푑푡

0 < 푡 − 휏 < 휏표 − 휏 since 푡 ≫ 휏 then =≫ 0 < |푡 − 휏| < 휏표 − 휏

So, if 0 < 푡 − 휏 < 휏표 − 휏 then 퐸2(푡 − 휏) = 퐸2(|푡 − 휏|) and Equation (2.61) becomes

′′ 4 ( ) 휏표 ( ) 4( ) (| |) 푞푅 = 2휎푇푤퐸3 휏 + 2 ∫0 sgn 휏 − 푡 σ푇 푡 퐸2 τ − 푡 푑푡 (2.62)

Back substituting Equation (2.62) into Equation (2.54) gives us

1 휕 2 휏 (휏2푞′′) = 2 퐸 (휏)휎푇4 + {2휎푇4퐸 (휏) + 2 ∫ 표 sgn(휏 − 푡)σ푇4(푡)퐸 (|τ − 푡|)푑푡} (2.63) 휏2 휕휏 푅 2 휏 휏 푤 3 0 2

Where 0 < 휏 ≤ 휏표

2-1-1 The Conservation Equations

The mixture in the bubble contains three components: the non-condensable gas Xenon,

Uranium dioxide vapor, and fog. Initially there are only two components: Xenon and Uranium dioxide vapor at high temperature. But when the temperature of the mixture reaches the saturation temperature at time t = tf, (condensation of UO2 vapor to small droplets) fog starts to form and moves to the colder regions at the bubble’s wall. While the Uranium dioxide vapor moves to the cold regions and cools to form fog, the Xenon gas starts to move toward the hotter regions. This motion of non-condensable Xenon gas impedes the motion of UO2 vapor and subsequently reduces the condensation process of the Uranium dioxide vapor. This effect impedes the formation of a liquid but causes the UO2 vapor to condense as a fog. In other words, the ordinary diffusion process between the Xenon and UO2 vapor reduces heat and mass transfer related to the phase change. To describe the motion of the system inside the bubble, the conservation equations of the individual components are considered, and they can be solved together for the magnitude of each species at a given pressure and temperature [57, 15].

The conservation equations of the components (in molar units) can be represented as

∂C 1 ∂ 1 + (r2N ) = r (2.64) ∂t r2 ∂r 1 1

∂C 1 ∂ 2 + (r2N ) = 0 (2.65) ∂t r2 ∂r 2

∂C 1 ∂ 3 + (r2N ) = r (2.66) ∂t r2 ∂r 3 3

Where r1 the amount of UO2 vapor per unit volume, and r3 is the amount of fog per unit volume. When r1 equals r3, the condensed vapor is transferred from the hot region to the cold

region at the same amount that is transformed to a fog [56]. From Fick’s first law of diffusion, the molar flux of species can be written as [25, 27]:

∗ 푁1 = 푥1퐶푣 − 퐶퐷∇푥1 (2.67)

∗ Where 푁1 + 푁2 = 퐶푣 (2.68)

∗ Where 푣 is the molar – average velocity for the mixture and 푁1, 푁2 are the molar fluxes of UO2 vapor and Xe gas.

휌푖 푐𝑖 = (2.69) 푀푖

퐶 푥 = 푖 where I = 1,2 and 3 (2.70) 𝑖 퐶

퐶 = 퐶1 + 퐶2 (2.71)

S. H. Chan et al. [ 56] assumed that the volume of the fog is insignificant and that it moves at the molar average velocity. Because the fog doesn’t interfere with the primary diffusion components (UO2 vapor and Xenon gas) the fog diffusion can be neglected, thus giving a binary diffusion model. Here, the binary diffusion has been also neglected between the UO2 vapor and

Xenon gas because expansion of the mixture in the bubble is very fast due to the associated shock wave effects while the mixing process between the gases is a slow process [50]. According to this assumption the conservation Equations of the components (2.64), (2.65) and (2.66) becomes:

∂퐶 1 = r (2.72) ∂t 1

∂퐶 2 = 0 (2.73) ∂t

∂퐶 3 = r (2.74) ∂t 3

퐶 Upon substituting equation ( 푥 = 푖 ) into the equations (2.72), (2.73) and (2.74) and 𝑖 퐶 differentiating with respect to t we get the following equations. will include bubble collapse times,

휕푥 휕퐶 퐶 1 + 푥 = − 푅 (2.75) 휕푡 1 휕푡 1

휕푥 휕퐶 퐶 2 + 푥 = 0 (2.76) 휕푡 2 휕푡

휕푥 휕퐶 퐶 3 + 푥 = 푅 (2.77) 휕푡 3 휕푡 3

3 Where R3 = - R1 = R is the fog rate (mole/m ) and from Equation (2.76) we can write [56],

휕퐶 −퐶 휕푥 = 2 (2.78a) 휕푡 푥2 휕푡

휕푥 휕푥 since 푥 + 푥 = 1 =≫ 2 = − 1 (2.78b) 1 2 휕푡 휕푡

Upon substituting Equation (2.78b) into (2.78a) Equation (2.78a) becomes

휕퐶 퐶 휕푥 = 1 (2.79) 휕푡 푥2 휕푡

Back substituting into Equation (2.75) gives us

휕푥 푥 퐶 휕푥 퐶 1 + 1 1 = −푅 (2.80) 휕푡 푥2 휕푡

휕푥 푥 휕푥 퐶 1 + 1 1 = −푅 (2.81) 휕푡 (1−푥1) 휕푡

휕푥 푥 퐶 1 [1 + 1 ] = −푅 (2.82) 휕푡 (1−푥1)

휕푥 1 퐶 1 [ ] = −푅 (2.83) 휕푡 (1−푥1)

휕푥 퐶 1 = −푅(1 − 푥 ) (2.84) 휕푡 1

Now, by adding Equations (2.75) and (2.76) we find

휕푥 휕푥 휕퐶 퐶 1 + 퐶 2 + (푥 + 푥 ) = −푅 (2.85) 휕푡 휕푡 1 2 휕푡

휕푥 휕푥 휕퐶 퐶 1 − 퐶 1 + (1) = −푅 (2.86) 휕푡 휕푡 휕푡

휕퐶 =≫ = −푅 (2.87) 휕푡

Back substituting Equation (2.77)

휕푥 퐶 3 + 푥 (−푅) = 푅 (2.88) 휕푡 3

휕푥 퐶 3 = (1 + 푥 )푅 (2.89) 휕푡 3

Based on the above assumptions and definitions and, after the simplifications, the conservation equations of the three components can be re-written as:

휕푥 퐶 1 = −(1 − 푥 )푅 (2.90) 휕푡 1

휕푥 퐶 2 = 0 (2.91) 휕푡

휕푥 퐶 3 = (1 + 푥 )푅 (2.92) 휕푡 3

Now, the conservation equations of UO2 vapor and fog can be solved together by eliminating R and solving for x1 and x3 as function of temperature. Note that the diffusion term is neglected in Equation (2.91) because it was assumed that the fog is moving with molar average velocity based on the S. H. Chan et al. solution [56].

As mentioned above, within the fog free region, there are only two species- UO2 vapor and Xenon gas where no fog particles are existing in this region. Therefore,

퐶(1 + 푥3) = 퐶0 (2.93)

When 푍 ≤ 푍푓 =≫ 푥3 = 푅 = 0 , =≫ 퐶3 = 퐶푥3 and 퐶3 = 0 where Z is physical coordinate from the center of the sphere and Zf is the fog boundary as show on Figure (2).

∵ 퐶 = 퐶1 + 퐶2 + 퐶3 (2.94a)

=≫ 퐶 = 퐶1 + 퐶2 in fog free region 퐶 = 퐶1 + 퐶2 = constant (2.94b)

Where 퐶0, is the initial value of the molar concentration and can be calculated from the

푃푈푂2 ideal gas relationship as: 퐶표 = (2.95) 푥표푅푇∞

푃푈푂2 푥표 = (2.96) 푃푡표푡푎푙

Where 푃푈푂2 is the UO2 vapor pressure that can be calculated from Menzie’s Equation such as used by M. L. Corradini et al. [49].

1 76800 푃푠푎푡 = exp (69.979 − − 4.34푙푛푇푠푎푡) (2.97) 10 푇푠푎푡

Where P is the pressure in (Pa) and temperature T is in (K). The total pressure inside the cavity is the sum of the partial the pressures of the UO2 spices and Xe species [25].

Ptotal = PUO2 + PXe (2.98)

We assumed that there is no surface vaporization and sodium entrainment occurring at the surface. The partial pressure of non-condensable gas Xe can be computed from the ideal gas equation [27].

푚 푅 푇 푃 = 푋푒 푋푒 (2.99) 푋푒 푉

The gas constant for the Xenon is ( 푅푋푒 63.323 J/kg K) [27].

In fog region 풁 > 풁풇 the conservation Equations (2.84) and (2.86) may be solved together by eliminating R and applying the fog boundary condition.

When 푍 = 푍푓 =≫ 푥1 = 푥푠푎푡 푎푛푑 푥3 = 0

1 휕푥 1 휕푥 − 1 = 3 (2.100a) 1−푥1 휕푡 1+푥3 휕푡

푥1 = 푥푠푎푡 at t = 푡푓 = { (2.100b) 푥3 = 0

By integrating both sides of Equation (2.100b) with respect to 휕푥1 and 휕푥3, gives

−1 1 ∫ ∂x1 = ∫ ∂x3 1−x1 1+x3

ln(1 − 푥1) = ln(1 + 푥3) + ln(푐표푛푠푡푎푛푡)

(1 − 푥1) = (1 + 푥3) 퐶 (2.101)

The final solution, after applying the boundary condition (2.100b), is where 퐶표 = 1 −

푥푠푎푡, such that:

1−푥1 = 1 − 푥푠푎푡 (2.102) 1+푥3

Using Equations (2.102) and (2.95), a solution for 푥1 , 푥3 can be obtained as function of temperature T:

푃푈푂2 푃푣(푇) 퐶(1 + 푥3) = 퐶표 , 퐶표 = =≫ 퐶 = 푥표푅푇∞ 푥1푅 푇

푃푣(푇) (1 + 푥3) = 퐶0 (2.103) 푥1푅푇

푃푣(푇) 푃푣(푇) Let (1 + 푥3) = 푥1 and let 푔0(푇) = 퐶0푅푇 퐶표푅푇

푔0(푇)(1 + 푥3) = 푥1 (2.104)

By using Equation (2.104), (2.102) and solving for 푥1, gives:

푔0(푇) 푥1(푇) = [ ] (2.105) 1−푥푠푎푡+푔0(푇)

Also, by using Equations (2.104) and (2.102) and solving for 푥1, results in:

푥푠푎푡−푔0(푇) 푥3(푇) = [ ] (2.106) 1−푥푠푎푡+푔0(푇)

Using Equations (2.105) and (2.106) and solving for the local fog concentration 퐶3, gives:

푥3(푇) 푔0(푇) = 푥1(푇) [푥푠푎푡 − 푔0(푇)] (2.107)

푃푣(푇) 푃푣(푇) By substituting 푔0(푇) = and 푥1 = into Equation (2.107) results in: 퐶표푅푇 퐶 푅 푇

퐶3 = 퐶0 [푥푠푎푡 − 푔0(푇)] (2.108)

Figure 6. The fog thickness.

The total fog R produced can be computed by integrating Equation (2.108) from RF to RE as it indicated in Figure 6.

푅퐸 푅푡표푡푎푙 = ∫ 퐶3 푑푉 (2.109) 푅퐹

푅퐸 2 푅푡표푡푎푙 = ∫ 4휋 퐶0 [푥푠푎푡 − 푔0(푇)] 푟 푑푟 (2.110) 푅퐹

푅퐸 2 푚푓표푔 = 4휋 퐶0푀푈푂 ∫ [푥푠푎푡 − 푔0(푇)] 푟 푑푟 (2.111) 2 푅퐹

Equation (2.110) gives the amount of fog produced in moles, and Equation (2.111) gives us the amount of fog produced in grams.

휏 푑휏 Where 푀 is the molecular weight of UO2, and let 푟 = => 푑푟 = 푈푂2 푘 푘

Another form,

4휋 퐶0푀푈푂2 휏표 2 푚푓표푔 = ∫ [푥푠푎푡 − 푔0(푇)] 휏 푑휏 (2.112) 푘3 휏푓

Here, k is the absorption coefficient.

2-1-2 The Energy Equation

To simulate fuel bubble’s geometry reasonably well, according to experimental observation, the energy equation must be reformulated into a spherical coordinate system. This equation was used with the radiative flux heat transfer equation (Equation 2.54) to obtain the effect of the fuel bubble’s geometry on the transient radiative heat flux. It was also used to predict the transient temperature distribution in the participating medium during a hypothetical core disruptive accident (HCDA) for a liquid metal fast breeding reactor (LMFBR) [24, 63].

DT 1 ∂ dT ∂x ∂T 1 ∂ (퐶퐶 ) = (r2k ) + R h + 퐶D r2(C − C ) 1 . − (r2q" ) (2.113) 푝 Dt r2 ∂r dr fg p1 p2 ∂r ∂r 푟2 ∂r rad

The left-hand side in Equation (2.108) [24] represents the convection heat transfer among the three components inside the cavity. The total specific heat of the mixture can be calculated from:

3 퐶푝 = ∑𝑖=1 푥𝑖퐶푝𝑖 (2.114)

푔 (푇)퐶 0 푝1 (푥푠푎푡−푔0(푇)) 퐶푝 = + (1 − 푥1)퐶푝 + 퐶푝 (2.115) 1−푥푠푎푡+푔0(푇) 2 1−푥푠푎푡+푔0(푇) 3

(퐶 −퐶 )푔 (푇)+[푥 −푔 (푇)]퐶 푝1 푝2 0 푠푎푡 0 푝3 퐶푝 = 퐶푝 + (2.116) 2 1−푥푠푎푡+푔0(푇)

The first term on the right-hand side of Equation (2.113) represents the conduction heat transfer in a radial direction. The second term represents the latent heat of condensation to fog formation. The fourth term represents the binary diffusion between Xenon gas and Uranium dioxide vaper.

The last term is radiative heat flux. The body force and heating due to friction have been neglected in this derivation of the energy equation, Moreover, the work done on the control volume due to friction is negligible because the flow is normal to the bubble – coolant interface.

Equation (2.113) was simplified under these assumptions by assuming that the system is under thermodynamic equilibrium, and the radiation dominates. Under these conditions, it was also assumed that the effects of convection, conduction and diffusion can be neglected as compared with radiation heat transfer [56].

2-1-3 In Fog Free Region

The radiative heat transfer Equation (2.113) in this region can be reduced to

∂T 1 ∂ (퐶퐶 ) = − (푟2 q" ) (2-117) 푝 ∂t 푟2 ∂r rad

When 푍 < 푍푓 because there are no fog droplets in this zone as shown in Figure 7.

Figure 7. Temperature distribution for two components and a two-phase mixture.

Upon substituting Equation (2.63) into Equation (2.117) gives us

∂T 2 휏 −퐶퐶 = 푘 [2휎푇4퐸 (τ) + {2휎푇4 퐸 (τ) + 2 ∫ 표 sgn(휏 − 푡)σ푇4(푡)퐸 (|τ − 푡|)푑푡}] (2.118) 푝 ∂t 휏 2 휏 푤 3 0 2

푇 Let 휃 = =≫ 휃 푇∞ = 푇 (2.119a) 푇∞ and 푑휃 푇∞ = 푑푇 (2.119b)

By substituting Equations (2.119a) and (2.119b) into Equation (2.118) and multiplying both sides by 퐶퐶푝, gives the following equation

3 휕휃 2푘휎푇∞ 4 2 4 휏표 4 − = [휃휏 퐸2(τ) + { 휃푤퐸3(τ) + ∫ sgn(휏 − 푡)휃휏 퐸 (|τ − 푡|)푑푡}] (2.120) 휕푡 퐶퐶푝 휏 0 2

3 4 휏표 4 2 푘 휎 푇∞ Let C = 2 {휃푤퐸3(τ) + ∫ sgn(휏 − 푡)휃휏 퐸2(|τ − 푡|)푑푡} and Let β = and Back 0 퐶퐶푝 substituting into Equation (2.120) and integrating both sides from t = 0 to t = t gives us,

휕휃 −휏 = 훽 [ 휏 휃4퐸 (τ) + 퐶] (2.121) 휕푡 휏 2

The term C in equation (2.121) can be determined numerically by using an iterations

4 method and by apply the S. H. Chan et al. solution [56] as initial guess where 휃휏 is known.

The initial condition for Equation (2.121) is when t = 0 =≫ θ = 1. Back substitution into Equation

(2.121) gives,

1 푑휃 β t/τ = ∫ 4 0 < 휏 ≤ 휏표 (2.122) 휃 [휏 휃휏 퐸2(τ)+퐶]

Equation (2.122) represents the temperature distribution in the fog free region which starts from the saturated temperature to the super-heated vapor temperature 푇∞. Calculating the time 푡푓required to reach the saturation point 휃푠푎푡(휏푓, 푡푓) by inserting 휃푠푎푡 into Equation

(2.122) and solving for 푡푓 result in: [56]

휏푓 1 푑휃 푡푓 = ∫ 4 0 < 휏 ≤ 휏표 (2.123) 훽 휃푠푎푡 [휏푓 휃푠푎푡퐸2( 휏푓)+퐶푠푎푡]

2-1-4 The Wall Heat Flux in Fog Free Region

The total heat flux in fog free region can be calculated from Equations (2.63):

1 휕 (휏2푞′′) = 2 휎 푇4 [휏 휃4 퐸 (휏 ) + 퐶] 0 < 휏 ≤ 휏 (2.124) 휏 휕휏 푅 ∞ 휏 2 푓 표

4 ( ) 휏표 ( ) 4 (| |) Where C = 2 {휃푤퐸3 휏 + ∫0 sgn 휏 − 푡 휃휏 퐸2 τ − 푡 푑푡} (2.125)

By integrating Equation (2.124) with respect to τ, gives the total wall heat flux.

휏 2 ′′ 4 0 4 ( ) 휏 푞푅 = 2 휎 푇∞ ∫0 [휏 휃휏 퐸2 휏 + 퐶] 휏 푑휏 0 < 휏 ≤ 휏표 (2.126)

Where 휃(휏, 푡) is given by Equation (2.122).

2-1-5 Transient Temperature Profile in The Fog Region

The transient temperature distribution profile and the wall heat flux can be extracted in the fog region from the solution of the energy equation (2.113) using the continuity equations

(2.64) and (2.66) for the vapor and fog species, respectively. This solution follows that of S. H.

Chan et al. [56] to reach the final form of the transient temperature distribution. It uses the wall heat flux in spherical coordinate assuming local thermodynamic equilibrium with thermal radiation. The conduction, convection, and diffusion components of the energy equation have

been neglected compared with the thermal radiation. Based on these assumptions, the energy equation and the conservation equations can be written as:

휕푥 퐶 1 = −(1 − 푥 )푅 (2.127) 휕푡 1

휕푥 퐶 3 = (1 + 푥 )푅 (2.128) 휕푡 3

휕푇 1 ∂ (퐶퐶 ) = R h − (푟2 q" ) (2.129) 푝 휕푡 fg 푟2 ∂r rad

퐶 휕푥 From Equation (2.127) 푅 = − 1 (2.130) (1−푥1) 휕푡

Upon substituting Equation (2.130) into Equation (2.129) gives

휕푇 퐶 휕푥1 1 ∂ 2 " (퐶퐶푝) + hfg = − 2 (푟 qrad) (2.131) 휕푡 (1−푥1) 휕푡 푟 ∂r

휕푥 휕푥 휕푇 The chain rule, can be written as 1 = 1 × (2.132) 휕푡 휕푇 휕푡

휕푥 휕푇 1 = 푥′ (2.133) 휕푡 1 휕푡

Upon substituting Equation (2.133) into Equation (2.131), results in:

휕푇 퐶 ′ 휕푇 1 휕 2 ′′ (퐶퐶푝) + 푥1 hfg = − 2 (휏 푞 ) (2.134) 휕푡 (1−푥1) 휕푡 휏 휕휏 푅

휕푇 Noting there is a common factor, 퐶 , on the left-hand side of Equation (2.134) can give: 휕푡

′ 휕푇 hfg푥1 1 휕 2 ′′ (퐶퐶푝) [1 + ] = − 2 (휏 푞 ) (2.135) 휕푡 퐶푝(1−푥1) 휏 휕휏 푅

Substituting Equation (2.63) into Equation (2.135) produces:

′ 휕푇 hfg푥1 4 2푘 4 휏표 (퐶퐶푝) [1 + ] = −2k 퐸2(휏)휎푇휏 + {2휎푇푤 퐸3(휏) + 2 ∫ sgn(휏 − 휕푡 퐶푝(1−푥1) 휏 0

4 푡)σ푇 (푡)퐸2(|τ − 푡|)푑푡} 0 < 휏 ≤ 휏표 (2.136)

푇 Let 휃 = =≫ 푑휃 푇∞ = 푑푇 (2.137) 푇∞

Upon substituting Equation (2.137) into Equation (2.136) and multiplying both sides by dt, gives the following equation

3 4 −2 푘 휎 푇∞ [휏 휃휏 퐸2(휏)+퐶] 휏퐶퐶푝푑휃 = ′ (2.138) hfg푥1 [1+ ] 퐶푝(1−푥1)

Dividing Equation (2.138) by 퐶∞퐶푝∞ ( fog free region at 푇 = 푇∞)

3 4 휏퐶퐶푝 −2 푘 휎 푇∞[휏휃휏 퐸2(휏)+퐶] 푑휃 = ′ 푑푡 (2.139) 퐶∞퐶푝∞ hfg푥1 퐶∞퐶푝∞[1+ ] 퐶푝(1−푥1)

퐶퐶 Let 휂(푇) = 푝 (2.140) 퐶∞퐶푝∞

h 푥′ 훾(푇) = fg 1 (2.141) 퐶푝(1−푥1)

By substitute Equations (2.140) and (2.141), into Equation (2.138) gives,

−2 푘 휎 푇3 [휏휃4퐸 (휏)+퐶] 휏휂(푇) 푑휃 = ∞ 휏 2 푑푡 (2.142) 퐶∞퐶푝∞[1+훾(푇)]

2 푘 휎 푇3 Let 훽 = ∞ (2.143) 퐶∞퐶푝∞

β [휏휃4퐸 (휏)+퐶] 휏휂(푇) 푑휃 = − 휏 2 푑푡 (2.144) [1+훾(푇)]

By separating the variables and integrating from 휃푓 → 휃 and from 푡푓 → 푡 Equation

(2.144) becomes:

휃 휂(푇)(1+훾(푇))푑휃 푡 − ∫ 4 = ∫ 훽/휏 푑푡 (2.145) 휃푓 [휏휃휏 퐸2(휏)+퐶] 푡푓

휃푓 휂(푇)(1+훾(푇))푑휃 ∫ 4 = 훽(푡 − 푡푓 )/휏 (2.146) 휃 [휏휃휏 퐸2(휏)+퐶]

Upon substituting Equation (2.122b) into equation (2.146), results in

훽푡 1 푑휃 휃푓 휂(푇)(1+훾(푇))푑휃 = ∫ 4 + ∫ 4 (2.147) 휏 휃푠푎푡 [휏푓휃푠푎푡퐸2( 휏푓)+퐶푠푎푡] 휃 [휏휃휏 퐸2(휏)+퐶]

Equation (2.147) represents the transient temperature distribution in the fog region, and numerical solution should use to solve this equation. The trapezoid method will be used for

numerical integration. The temperature profile in the fog region is appears up implicitly on the left-hand side of Equation (2.147). This equation was calculated numerically by using MAT LAB.

2-1-6 The Wall Heat Flux

The wall heat flux at the bubble wall can be obtained by the following relationship:

1 휕 (휏2푞′′) = 2 휎 푇4 [휃4 휏 퐸 (휏) + 퐶] 0 < 휏 ≤ 휏 (2.148) 휏 휕휏 푅 ∞ 휏 2 표

4 ( ) 휏표 ( ) 4 (| |) Where 퐶 = { 휃푤퐸3 휏 + ∫0 sgn 휏 − 푡 휃휏 퐸2 τ − 푡 푑푡} (2.149)

푞푟푤(푡) 휏푓 4 ∞ 4 4 = ∫ (휏 휃휏 퐸2(휏) + 퐶) 푑휏 + ∫ [(휏 휃휏 퐸2(휏) + 퐶]푑휏 (2.150) 2휎푇∞ 0 휏푓

Equation (2.129) has two regions. The first term on the right-hand side represents the wall heat flux in the fog free region, and 휃4(푡, 휏) is the transient temperature distribution in the fog free region. The second term represents the total heat flux in the fog region. Numerical integration was used to solve Equation (2.149) by using successive substitutions method.

2-2 Scattering Model

In this section, a scattering model will be investigated using a spherical coordinate system.

The objective of this model is to study the thermal radiation heat transfer processes occurring with a participating medium. The system, as shown in Figure 2-3, is composed of two concentric spheres, and the space between the spheres contains 푈푂2 embryos forming from homogenous nucleation at high saturation temperature T. The inner sphere is designated as sphere 1 with properties 푇1, 휖1 and 휌1which represents the hot gases (Uranium dioxide and Xenon). While the outer sphere is designated as sphere 2 with surface properties 푇2 , 휖2, and 휌2 and represents the coolant region (Sodium or water). The medium between the two spheres is the participating medium which is absorbs, scatters, and emits thermal radiation. The thermal radiation emitted

from the inner sphere interacts with the particles inside the medium and, as result, the radiation intensity at the boundary surface is reduced [28, 30].

2-2-1 Radiative Analysis in Isothermal Spherical Media

The physical system in Figure 2-8 consists of two concentric spheres. The intensity of the radiation at the boundaries is independent to the direction, and it is composed of emitted and diffusely reflected components as shown in Figure 2-8.

Figure 8. Physical model of a bubble contains UO2 fuel and fission gas.

The region between the two spheres is a homogenous, absorbing, emitting and scattering medium at its saturated temperature as shown in Figure 2-8 [28]. The surfaces of the spheres are gray, diffusely emitting, and diffusely reflecting, and the region between the two spheres is a participating medium. This medium contains 푈푂2 particles with a particle - size distribution as it obtained by K. Chen et al. [50]. To model the problem the radiative heat transfer equation in spherical coordinates for the intensity I (r, μ) can be written as [26, 30]

∂I 1−μ2 ∂I 1 1 μ v + v + βI = kI (T) + σ ∫ P(μ, μ′)I (r, μ′) dμ′ (2.151) ∂r r ∂μ v vb 2 −1 v

β the extinction coefficient

Θ the angle between the radius vector and the unit vector in the direction of beam radiation

Iv Specific intensity of radiation

R Radius of the sphere

K Spectral absorption coefficient

σ Spectral scattering coefficient

Ivb (T) The black body radiation intensity at some temperature

ω Spectral albedo of scattering coefficient

µ the direction cosine of propagating beam radiation

P(μ, μ′) is the phase function that can be calculated from the following series [26]:

′ ∞ ′ P(μ, μ ) = ∑푛=0(2푛 + 1) 푎푛푃푛(휇)푃푛(휇 ) (2.152)

The coefficients, 푎푛, are determined by [26]:

1 1 푎 = ∫ 푃(휇) 푃 (휇)푑휇 (2.153) 푛 2휔 −1 푛

′ 푃푛(휇) 푎푛푑 푃푛(휇 ) are the Legendre polynomials of the scattering and incident radiation respectively, and the asymmetry parameter 푎𝑖 is provided in Appendix C.

푟 the Now letting 푅 = , we obtain 푟 = 푅 푟2 =≫ 푑푟 = 푑푅 푟2. Substituting these 푟2 dimensionless quantities into Equation (2.151) gives us

2 ∂Iv 1−μ ∂Iv 1 1 ′ ′ ′ μ + + βIv = kIvb(T) + σ ∫ P(μ, μ )Iv(r, μ ) dμ (2.154) 푟2휕푅 R 푟2 ∂μ 2 −1

For gray medium I-intensity is independent of frequency and therefore Equation (4) may be written as [26, 63]

∂I(R,μ) 1−μ2 ∂I(R,μ) 휔휏 1 μ + + 휏 I(R, μ) = (1 − 휔)휏 I (T) + 2 ∫ P(μ, μ′)I(R, μ′) dμ′ (2.155) 휕푅 R ∂μ 2 2 b 2 −1

Where R is dimensionless in range 푅1 < 푅 < 1 and the spherical harmonics method will be used to solve the radiative heat transfer equation in spherical coordinates [26].

2-2-2 Spherical Harmonics Method

The spherical harmonics method was formulated by Jeans (1917). This method is used to solve the radiative heat transfer equation in spherical coordinates by converting it to a non- homogenous Bessel equation [26]

휕I (푅,μ) 1−μ2 휕I(푅,μ) 휔휏 1 μ + + 휏 퐼 (R, μ) = (1 − ω)휏 퐼 (T) + 2 ∫ 푝(휇, μ′)퐼(r, μ′) dμ′ (2.156) 휕R R 휕μ 2 2 푏 2 −1

′ In this method I (푅, μ) and 푝(휇, μ ) are represented by a series of Legendre polynomials

2푚+1 퐼(푅, 휇) = ∑∞ 푃 (휇)훹 (푅) (2.157) 푚=0 4휋 푚 푚

Where 푃푚(휇) is the Legendre polynomial, and 훹푚(푅) is an unknown function in R direction, such that, if this function is characterized by specific intensity, then the radiation can be determined [26].

Equation (2.157) can be written in other form as [26]:

2푚+1 퐼(휏, 휇) = ∑∞ 푃 (휇)훹 (휏) (2.158) 푚=0 4휋 푚 푚

Upon substituting Equations (2.152) and (2.157) into Equation (2.156) gives the following

2푚+1 휕훹 (푅) 2푚+1 휕 1−휇2 ∑∞ 푃 (휇) [휇 푚 + 훹 (푅)휏 ] + {∑∞ [푃 (휇)훹 (휇)]} = (1 − 푚=0 4휋 푚 휕푅 푚 2 푚=0 4휋 휕휇 푚 푚 푅

휔휏 2푚+1 −1 휔)휏 퐼 (푇) + 2 ∑∞ ∑∞ (2푛 + 1) 푎 푃 (휇) 훹 (푅) ∫ 푃 (휇′)푃 (휇′)푑휇′ (2.159) 2 푏 2 푚=0 푛=0 푛 푛 4휋 푚 1 푛 푚

Now by differentiating the second term and applying the orthogonality of Legendre polynomials in the last term of Equation (2.159) [26, 76].

−1 0 , , 푚 ≠ 푛 ∫ 푃 (휇)푃 (휇)푑휇 = { 2 (2.160) 1 푛 푚 , 푚 = 푛 (2푚+1)

2푚+1 휕훹 (푅) 2푚+1 휕훹 (푅) ∑∞ 푃 (휇) [휇 푚 + 훹 (푅)휏 ] + {∑∞ [푃 (휇) 푚 + 푚=0 4휋 푚 휕푅 푚 2 푚=0 4휋 푚 휕푅

휕푃 (휇) 1−휇2 휔휏 훹 (푅) 푚 } = (1 − 휔)휏 퐼 (푇) + 2 ∑∞ (2푚 + 푚 휕휇 푅 2 푏 2 푚=0

2푚+1 2 1) 푎 푃 (휇) 훹 (푅) (2.161) 푚 푚 4휋 푚 (2푚+1)

Equation (2.161) may be simplified further to yield:

2푚+1 휕훹 (푅) 2푚+1 휕푃 (휇) 1−휇2 ∑∞ 푃 (휇) [휇 푚 + 훹 (푅)휏 ] + {∑∞ 훹 (푅) 푚 } = (1 − 푚=0 4휋 푚 휕푅 푚 2 푚=0 4휋 푚 휕휇 푅

(2푚+1) 휔)휏 퐼 (푇) + 휔휏 ∑∞ 푎 푃 (휇)훹 (푅) (2.162) 2 푏 2 푚=0 4휋 푚 푚 푚

(m+1)P (μ)+mP (μ) μP (μ) = m+1 m−1 (2.163) m (2m+1)

dP (1 − μ2) m(μ) = mP (μ) − mμP (μ) (2.164) dμ m−1 m

Upon substituting the recurrence formulas for Legendre polynomials of Equations (2.163) and (2.164) [76] in Equation (2.162) and multiplying both sides of Equation (2.162) by 4π, Equation

(2.165) is obtained:

∞ ′ ∞ ∞ ∑푚=0(2푚 + 1) 휇 푃푚(휇) 훹푚(푅) + ∑푚=0(2푚 + 1) 푃푚(휇) 훹푚(푅)휏2 + ∑푚=0(2푚 +

휕P [1−휇2] 1) 훹 (푅) m(μ) = 4휋(1 − 휔)휏 퐼 (푇) + 휔휏 ∑∞ (2푚 + 1) 푎 푃 (휇)훹 (푅) 푚 ∂μ 푅 2 푏 2 푚=0 푚 푚 푚

(2.165)

Substituting Equation (2.138) into the first term of Equation (2.165) yields

∞ ′ ∞ ∑푚=0 훹푚(푅) [(m + 1)Pm+1(μ) + mPm−1(μ)] + ∑푚=0(2푚 +

휕P [1−휇2] 1)푃 (휇) 훹 (푅)휏 + ∑∞ (2푚 + 1) 훹 (푅) m(μ) = 4휋(1 − 휔)휏 퐼 (푇) + 푚 푚 2 푚=0 푚 ∂μ 푅 2 푏

∞ 휔휏2 ∑푚=0(2푚 + 1) 푎푚푃푚(휇)훹푚(푅) (2.166)

The above series can be rearranged as a function of 푃푚(휇). Starting with the first term,

∞ ′ I = ∑푚=0 훹푚(푅) [mPm−1(μ) + (m + 1)Pm+1(μ)] (2.167)

∞ ′ ′ I = ∑푚=0 mPm−1(μ) 훹푚(푅) + (m + 1)Pm+1(μ)훹푚(푅) ] (2.168)

Equation (2.168) can be written as the sum of two series

∞ ′ ∞ ′ I = ∑푚=0(m + 1)Pm+1(μ) 훹푚(푅) + ∑푚=0 mPm−1(μ) 훹푚(푅) (2.169)

Let k = m + 1 => m = k -1 and letting k = m-1 => m = k + 1.

Upon substitution of m by (k-1) and (k+1) in the first and second terms of Equation (2.169) respectively, Equation (2.170) is obtained

푑훹 (푅) 푑훹 (푅) I = ∑∞ 푘 (k−1) P (μ) + ∑∞ (푘 + 1) (k+1) P (μ) (2.170) 푘=0 푑푅 k 푘=0 푑푅 k

Now let k = m

푑훹 (푅) 푑훹 (푅) I = ∑∞ 푚 (m−1) P (μ) + ∑∞ (푚 + 1) (m+1) P (μ) (2.171) 푚=0 푑푅 m 푚=0 푑푅 m

Pm(μ) is a common factor between the two terms of the above, therefore factoring yields:

푑훹 (푅) 푑훹 (푅) I = ∑∞ [푚 (m−1) + ∑∞ (푚 + 1) (m+1) ]P (μ) (2.172) 푚=0 푑푅 푚=0 푑푅 m

The third term III is

휕P [1−휇2] IIII = ∑∞ (2푚 + 1) 훹 (푅) m(μ) (2.173) 푚=0 푚 ∂μ 푅

Substitute the recurrence formula Equation (2.173) into Equation (2.171) gives

훹 (푅) III = ∑∞ (2푚 + 1) [ 푚 (mP (μ) − mμP (μ))] (2.174) 푚=0 푅 m−1 m

Substituting the recurrence formula Equation (2.163) into Equation (2.174) gives

훹 (푅) (m+1)P (μ)+mP (μ) III = ∑∞ (2푚 + 1) [ 푚 (mP (μ) − m ( m+1 m−1 ))] (2.175) 푚=0 푅 m−1 (2m+1)

훹 (푅) (푚2+m)P (μ)+푚2P (μ) III = ∑∞ (2푚 + 1) [ 푚 (mP (μ) − ( m+1 m−1 ))] (2.176) 푚=0 푅 m−1 (2m+1)

훹 (푅) P (μ)(2푚2+m) (푚2+m)P (μ)+푚2P (μ) III = ∑∞ (2푚 + 1) [ 푚 ( m−1 − ( m+1 m−1 ))](2.177) 푚=0 푅 (2m+1) (2m+1)

훹 (푅) III = ∑∞ [ 푚 ((푚2 + m)P (μ) − (푚2 + m)P (μ))] (2.178) 푚=0 푅 m−1 m+1

Factoring the common term (푚2 + m) from Equation (2.178), gives

훹 (푅)(푚2+m) III = ∑∞ [ 푚 (P (μ) − P (μ))] (2.179a) 푚=0 푅 m−1 m+1

Equation (2.179a) can be written as the sum of two series:

1 III = (∑∞ (푚2 + m)훹 (푅)P (μ) − ∑∞ (푚2 + m) 훹 (푅)P (μ))(2.179b) 푅 푚=0 푚 m−1 푚=0 푚 m+1

Again, let k = m + 1 => m = k -1, and letting k = m-1 => m = k + 1. Substituting m by (k+1) and

(k-1) in the above series (2.179a) for the first and second terms respectively gives

1 III = (∑∞ ((k + 1)2 + k + 1)훹 (푅)P (μ) − ∑∞ ((k − 1)2 + k − 푅 푘=0 k+1 k+1−1 푘=0

1) 훹k−1(푅)Pk−1+1(μ)) (2.180)

1 I = (∑∞ (푘 + 1)(푘 + 2)훹 (푅) − ∑∞ 푘(푘 − 1) 훹 (푅))P (μ) (2.181) 푅 푘=0 k+1 푘=0 k−1 k

Now let m = k to get

1 III = (∑∞ (푚 + 1)(푚 + 2)훹 (푅) − ∑∞ 푚(푚 − 1) 훹 (푅))P (μ) (2.182) 푅 푚=0 m+1 푚=0 m−1 k

1 III = (∑∞ (푚 + 1)(푚 + 2)훹 (푅) − ∑∞ 푚(푚 − 1) 훹 (푅))P (μ) (2.183) 푅 푚=0 m+1 푚=0 m−1 k

Now going back to Equation (2.165), move the fourth term from the right-hand side to the left-hand side, and factor Pk(μ) as a common factor to yield:

훹 (푅) 훹 (푅) III = ∑∞ [(푚 + 1)(푚 + 2) m+1 − 푚(푚 − 1) m−1 ] P (μ) (2.184) 푚=0 푅 푅 k

∞ ∞ The second and the fifth terms II&V = ∑푚=0(2푚 + 1)푃푚(휇) 훹푚(푅)휏2 − 휔 ∑푚=0(2푚 +

1) 푎푚푃푚(휇)훹푚(푅) 휏2 (2.185)

The above two term series can be combined into a single series by factoring the common term (2푚 + 1)푃푚(휇) 훹푚(푅)휏2:

∞ II&V = ∑푚=0(2푚 + 1)푃푚(휇) 훹푚(푅)휏2(1 − 휔푎푚) (2.186)

The fourth term IV in Equation (2.165), 4휋(1 − 휔)휏2퐼푏(푇), can be put on the left-hand side with a minus sign:

1 for m = n IV = −4휋(1 − 휔)휏 퐼 (푇) 훿 푃 (휇) , and 훿 { (2.187) 2 푏 0푚 푚 0푚 0 otherwise

Substituting Equations (2.172), (2.183), (2.186), and (2.187) into Equation (2.165) yields:

푑훹 (푅) 푑훹 (푅) ∑∞ [푚 (m−1) + ∑∞ (푚 + 1) (m+1) ]P (μ) + ∑∞ [(푚 + 1)(푚 + 푚=0 푑푅 푚=0 푑푅 m 푚=0

훹 (푅) 훹 (푅) 2) m+1 − 푚(푚 − 1) m−1 ] P (μ) + ∑∞ (2푚 + 1)푃 (휇) 훹 (푅)휏 (1 − 휔푎 ) − 푅 푅 k 푚=0 푚 푚 2 푚

4휋(1 − 휔)휏2퐼푏(푇) 훿0푚푃푚(휇) = 0 (2.188)

푑훹 (푅) 푑훹 (푅) 훹 (푅) ∑∞ [(푚 + 1) (m+1) + 푚 (m−1) + (푚 + 1)(푚 + 2) m+1 − 푚(푚 − 푚=0 푑푅 푑푅 푅

훹 (푅) 1) m−1 + 휏 (2푚 + 1) 훹 (푅)(1 − 휔푎 ) − 4휋(1 − 휔)휏 퐼 (푇) 훿 ] 푃 (휇) = 0 푅 2 푚 푚 2 푏 0푚 푚

(2.189)

The coefficients of the Legendre polynomials of Equations (2.189), 푃푚(휇), must be disappear identically to satisfy the equation for any values of μ [28, 76].

푑훹 (푅) 푑훹 (푅) 훹 (푅) 훹 (푅) (푚 + 1) (m+1) + 푚 (m−1) + (푚 + 1)(푚 + 2) m+1 − 푚(푚 − 1) m−1 + 푑푅 푑푅 푅 푅

휏2(2푚 + 1)(1 − 휔푎푚) 훹푚(푅) − 4휋(1 − 휔)휏2퐼푏(푇) 훿0푚 = 0 , 푚 = 0,1, … (2.190)

Equation (2.190) gives N - ordinary differential equations. However, in this study, the P1 approximation method is used to solve Equation (2.190).

푛2휎푇4(휏) By setting ,m = 0,1 and using 퐼 (푇) = [30, 63] Equation (2.190) gives two ordinary 푏 휋 differential equations Ψ0, Ψ1, and Ψ2 as shown in the following equations[28].

푑훹 훹 1 + 2 1 + 휏 (1 − 휔푎 )훹 − 4휏 (1 − 휔)휎푛2푇4(휏) = 0 (2.191) 푑푅 푅 2 0 0 2

푑훹 푑훹 훹 2 2 + 0 + 6 2 + 3휏 (1 − 휔푎 )훹 = 0 (2.192) 푑푅 푑푅 푅 2 2 1

Using the spherical harmonic method, the incident radiation can be calculated from

Equation (2.158), and the orthogonality of the Legendre polynomials can be calculated from

Equation (2.160) [26, 76].

′ Upon multiplying Equation (2.158) from both sides by 푃푚(휇 ) and utilizing the orthogonality of the Legendre polynomials Equation (2.158) we obtain

2푚+1 푃 (휇′) 퐼(휏, 휇′) = ∑∞ 푃 (휇′) 푃 (휇) 훹 (휏) (2.193) 푚 푚=0 4휋 푚 푚 푚

Upon multiplying Equation (2.163) by 푑휇′ and integrating from 휇′ = −1 to = 1 , results in

[26].

1 2푚+1 1 ∫ 푃 (휇′) 퐼(휏, 휇′) 푑휇′ = ∑∞ ∫ 푃 (휇′) 푃 (휇) 푑휇′ 훹 (휏) (2.194) −1 푚 푚=0 4휋 −1 푚 푚 푚

1 2푚+1 2 ∫ 푃 (휇′) 퐼(휏, 휇′) 푑휇′ = ∑∞ × 훹 (휏) (2.195) −1 푚 푚=0 4휋 (2푚+1) 푚

Equation (2.195) can be simplified and solving for 훹푚(휏) ,it may be rewritten as

( ) 1 ( ′) ( ′) ′ 훹푚 휏 = 2휋 ∫−1 푃푚 휇 퐼 휏, 휇 푑휇 (2.196)

( ) 1 ( ′) ( ′) ′ ( ′) When m = 0 =≫ 훹0 휏 = 2휋 ∫−1 푃0 휇 퐼 휏, 휇 푑휇 and 푃0 휇 = 1 then

( ) 1 ( ′) ′ 훹0 휏 = 2휋 ∫−1 퐼 휏, 휇 푑휇 (2.197)

Therefore 훹0(휏) = 퐺(휏) where 퐺(휏) is the incident radiation [26].

1 ( ′) ′ 퐺(휏) = 2휋 ∫−1 퐼 휏, 휇 푑휇 (2.198)

( ) 1 ( ′) ( ′) ′ ( ′) When m = 1 =≫ 훹1 휏 = 2휋 ∫−1 푃1 휇 퐼 휏, 휇 푑휇 and 푃0 휇 = 휇 then

( ) 1 ( ′) ′ 훹1 휏 = 2휋 ∫−1 휇 퐼 휏, 휇 푑휇 (2.199)

Therefore 훹1(휏) = 푞(휏), where 푞(휏) is the net radiative heat flux [26].

1 ( ′) ′ 푞(휏) = 2휋 ∫−1 휇 퐼 휏, 휇 푑휇 (2.200)

By definition, 훹0(휏) and 훹1(휏), are the incident radiation and the radiant heat flux respectfully [26, 30].

Now, after preforming the substitutions 푞(휏) , G(휏), 휏 ( 휏 = 푅 휏2)and neglecting 훹2,

Equations (2.191) and (2.192) simplify to

푑푞푟(휏) 푞푟(휏) + 2 + (1 − 휔푎 )퐺(휏) − 4(1 − 휔)휎푛2푇4(휏) = 0 (2.201) 푑휏 휏 0

푑퐺(휏) + 3(1 − 휔푎 )푞푟(휏) = 0 (2.202) 푑휏 1

Upon differentiating Equation (2.202) with respect to τ we get

푑푞푟(휏) 1 푑2퐺 = − 2 (2.203) 푑휏 3(1−휔푎1) 푑휏

Further, by substituting Equations (2.202) and (2.203) into Equation (2.201) we find

2 1 푑 퐺 2 푑퐺(휏) 2 4 − 2 − + (1 − 휔푎0)퐺(휏) = 4(1 − 휔)휎푛 푇 (휏) (2.204) 3(1−휔푎1) 푑휏 3휏(1−휔푎1) 푑휏

Multiplying Equation (2.204) by −3(1 − 휔푎1) gives:

푑2퐺 2 푑퐺(휏) + − 3(1 − 휔푎 )(1 − 휔푎 )퐺(휏) = −12 (1 − 휔푎 )(1 − 휔)휎푛2푇4(휏) (2.205) 푑휏2 휏 푑휏 1 0 1

2 Let 푏 = 3 (1 − 휔푎1)(1 − 휔푎0),this reduces to:

푑2퐺 2 푑퐺(휏) + − 푏2퐺(휏) = −4 푏2휎푛2푇4 (2.206) 푑휏2 휏 푑휏

푑2퐺 푑퐺(휏) 휏2 + 2휏 − 푏2 휏2퐺(휏) = 0 (2.207) 푑휏2 푑휏

Equation (2.207) is a homogeneous Bessel equation with the following solution [29]:

퐴 퐶 퐶 퐺(휏) = 휏 exp(퐵휏) [퐶1퐽푛(퐷휏 ) + 퐶2푌푛(퐷휏 )] (2.208)

푥2푢′′ + [(1 − 2퐴)푥 − 2퐵푥2]푢′ + [퐶2퐷2푥2퐶 + 퐵2푥2 − 퐵(1 − 2퐴)푥 + 퐴2 −

퐶2푛2]푢 = 0 (2.209)

By comparing the coefficients in Equation (2.209) and Equation (2.187) the values of constants A, B, C, D and n can be determined [29].

1 (1 − 2퐴) = 2 =≫ 퐴 = − 2

2퐵 = 0 =≫ 퐵 = 0

2 퐶 = 2 =≫ 퐶 = 1

퐶2퐷2 = −푏2 =≫ 퐷2 = −푏2 =≫ 퐷 = 푖푏

1 퐴2 − 퐶2푛2 = 0 =≫ 푛 = 2

Substituting the coefficients A, B, C, D and n into Equation (2.183) gives us the complementary solution of Equation (2.210) [29].

ퟏ − 퐺(휏) = 휏 ퟐ exp (0 × 휏) [퐶3퐽ퟏ(푖푏휏) + 퐶4푌ퟏ(푖푏휏)] (2.210) ퟐ ퟐ

1 퐺(휏) = [퐶3퐽ퟏ(푖푏휏) + 퐶4푌ퟏ(푖푏휏)] (2.211) √휏 ퟐ ퟐ

Upon utilizing the recurrence relations, the 퐽푘(푖푏휏) and 푌푘(푖푏휏) can be converted to

퐽푘(푏휏) and to 푌푘(푏휏)[29].

1 푘 1 퐽푘(푖푥) = 푖 퐼푘(푥) , 푘 = =≫ 퐽1(푖푥) = 푖2 퐼1(푥) =≫ 퐽1(푖푥) = 퐶3 퐼1(푥) (2.212) 2 2 2 2 2

2 1 푌 (푖푥) = 푖푘 [푖퐼 (푥) − (−1)푘푘 (푥)] , 푘 = 푘 푘 휋 푘 2

1 2 1 푌1(푖푥) = 푖2 [푖 퐼1(푥) − (−1)2 푘1(푥)] , 푥 = (푏휏) 2 2 휋 2

푌1(푖푥) = 퐶5퐼1(푏휏) + 퐶6 푘1(푏휏) (2.213) 2 2 2

Back substituting Equations (2.212) and (2.216) into Equation (2.211) gives us

1 퐺(휏) = [퐶1 퐼1(푏휏) + 퐶2퐾1(푏휏)] (2.214) √휏 2 2

Equation (2.214) is the solution of homogenous Bessel Equation (2.187), and the general solution of Equation (2.206) is given by [28]:

1 2 4 퐺(휏) = [퐶1 퐼1(푏휏) + 퐶2퐾1(푏휏)] + 4 휎푛 푇 (2.215) √휏 2 2

Where T is the saturation temperature of the medium between the isothermal surfaces.

By recalling Equation (2.203), the radiative heat flux, can be written in this form.

−1 푑퐺(휏) 푞푟(휏) = (2.116) 3(1−휔푎1) 푑휏

By differentiating Equation (2.115) with respect to τ and utilizing the following derivatives of Bessel functions the radiative heat flux can be obtained [29].

푘 퐼′ (푥) = 퐼 (푥) − 퐼 (푥) (2.217) 푘 푘−1 푥 푘

푘 퐾′ (푥) = −퐾 (푥) − 퐾 (푥) (2.218) 푘 푘−1 푥 푘

′ 1 퐼ퟏ(푏휏) = 푏퐼 1(푏휏) − 퐼1(푏휏) (2.219) − 2휏 ퟐ 2 2

′ 1 퐾ퟏ(푏휏) = −푏퐾 1(푏휏) − 퐾1(푏휏) (2.220) − 2휏 ퟐ 2 2

1 1 ′ 1 1 ′ 푑퐺(휏) − ′ − − ′ = 퐶1휏 2 퐼1(푏휏) + 퐶1퐼1(푏휏) (휏 2) + 퐶2휏 2퐾1(푏휏) + 퐶2퐾1(푏휏) (휏2) (2.221) 푑휏 2 2 2 2

1 1 1 3 푑퐺(휏) − 2 − 1 − 2 1( ) 1( ) 2 1( ) 2 = 퐶1휏 퐼− 푏휏 푏 − 퐼 푏휏 퐶1휏 푏 + 퐶1퐼 푏휏 (− ) 휏 + 푑휏 2 푏휏 2 2 2

1 3 − 푏 1 − 퐶 휏 2 (−푏퐾 1(푏휏) − 퐾1(푏휏)) + 퐶 퐾1(푏휏)(− )휏 2 (2.222) 2 − 2 2 2 2휏푏 2 2

퐶 푏퐼 1(푏휏) 퐶 푏퐼1(푏휏) 퐶 푏퐾 1(푏휏) 퐶 퐾1(푏휏) 푑퐺(휏) 1 − 1 2 − 2 푑퐸 (휏) = 2 − 2 − 2 − 2 − 푏푣 (2.223) 푑휏 1 3 1 3 푑휏 휏2 휏2 휏2 휏2

Upon substituting Equation (2.213) into Equation (2.216) the radiative heat flux can be written as [28].

퐼1(푏휏) 푏퐼 1(푏휏) 푘1(푏휏) 푏푘 1(푏휏) 1 − − 푞(휏) = {[ 2 − 2 ] 푐 + [ 2 + 2 ] 푐 } (2.224) 3[1−휔푎 ] 3 1 1 3 1 2 1 휏2 휏2 휏2 휏2

Where 퐶1 and 퐶2 are the integration constants of the non-homogeneous Bessel equation

2 of Equation (2.224). The units of 퐶1 and 퐶2 are in W/m and U1 and U2 are Bessel functions of

−1 1 , orders [29]. 2 2

2 푏 퐼 3(푏휏) 2퐼1(푏휏) 푏 퐼 1(푏휏) − − 2 2 2 푈1 = [ 1 + 5 − 3 ] (2.225) 휏2 휏2 휏2

2 푏 푘 3(푏휏) 푘1(푏휏) 푏 푘−1(푏휏) − 2 2 2 푈2 = [ 1 + 5 + 3 ] (2.226) 휏2 휏2 휏2

푎1 is the symmetry coefficient of Legendre Polynomials of phase function, and its value depending on the scattering type and its polarization components [37, 38].

ω is the elbedo of scattering coefficient. n is the index of refraction of UO2, which is equals to 1 in vapor phase [56].

τ is the optical thickness.

2-2-3 Boundary Conditions

The surfaces of the spheres are assumed to be gray, diffusely emitting and diffusely reflecting boundaries, and an isothermal boundary conditions were assumed at these surfaces

[28]. Inside the medium a steady state temperature profile was assumed. Whereas this physical model consists of three regions, the first region contains hot gases at 4500 K, and the second region is the sodium coolant at 811 K. The third region is the participating medium confined between the two concentric spheres at 4000 K shown in Figure (2.4). To formulate the problem, the integration constants in the radiative heat flux Eq. (2.224), must be determined by applying the boundary conditions at the internal and external surfaces as shown in Figure (2.9) [26, 28].

휖 휎푇4 1 푖(휏 , 휇) = 1 1 + 휌푠푖(휏 , −휇) + 2휌푑 ∫ 푖(휏 , −휇′ )휇′푑휇′, 휇 > 0 (2.227) 1 휋 1 1 1 0 1

휖 휎푇4 1 푖(휏 , −휇) = 2 2 + 휌푠푖(휏 , 휇) + 2휌푑 ∫ 푖(휏 , 휇′ )휇′푑휇′, 휇 > 0 (2.228) 2 휋 2 2 2 0 2

Figure 9. Physical model at T1 ≠ 0, T2 ≠ 0, and T ≠ 0

Upon substituting Equation (2.227) and Equation (2.228) in Marsak’s boundary condition

[26], it can be rewritten as:

1 1 휖 휎푇4 1 ∫ 푖(휏 , 휇) = ∫ [ 1 1 + 휌푠푖(휏 , −휇) + 2휌푑 ∫ 푖(휏 , −휇′ )휇′푑휇′] 휇푑휇, 휇 > 0 (2.229) 0 1 0 휋 1 1 1 0 1

1 1 휖 휎푇4 1 ∫ 푖(휏 , −휇) = ∫ [ 2 2 + 휌푠푖(휏 , 휇) + 2휌푑 ∫ 푖(휏 , 휇′ )휇′푑휇′ ] 휇푑휇, 휇 > 0 (2.230) 0 2 0 휋 2 2 2 0 2

Where i(τ, μ) is defined as,

2푚+1 푖(휏, 휇) = ∑∞ 푃 (휇) 훹 (휏) (2.231) 푚=0 4휋 푚 푚

For the P1 approximation method, m = 1 and Equation (2.231) becomes:

1 푖(휏, 휇) = [ 푃 (휇) 훹 (휏) + 3 푃 (휇) 훹 (휏)] (2.232) 4휋 0 0 1 1

Where 푃0(휇) = 1, 푃1(휇) = 휇 , 훹0(휏) = 퐺(휏) and 훹1(휏) = 푞(휏)

Equation (2.232) can be written as,

1 푖(휏, 휇) = [퐺(휏) + 3휇 푞(휏)] (2.233) 4휋

Upon substituting Equation (2.233) into Equation (2.229) gives

1 1 휖 휎푇4 1 1 1 ∫ [퐺(휏 ) + 3휇 푞(휏 )] 휇푑휇 = 1 1 ∫ 휇푑휇 + 휌푠 ∫ [퐺(휏 ) − 3휇 푞(휏 )]휇푑휇 + 4휋 0 1 1 휋 0 4휋 1 0 1 1

2 1 1 휌푑 ∫ 휇푑휇 ∫ [퐺(휏 ) − 3휇 푞(휏 )]휇′푑휇′ (2.234) 4휋 1 휇=0 휇′=0 1 1

After preforming the integration, Equation (2.234) simplifies to,

푞(휏 ) 퐺(휏 ) (1 + 휌푠 + 휌푑) 1 − [휌푠 + 휌푑 − 1] 1 = ∈ 휎푛2푇4 (2.235) 1 1 2 1 1 4 1 1

Upon substituting the radiative heat flux 푞(휏)휏=휏1 (Equation (2.224)) and the irradiation

퐺(휏)휏 = 휏1 (Equation (2.215)) into Equation (2.235) the following solution is obtained as shown in reference [28].

I1(b휏1) bI 1(b휏1) k1(b휏1) bk 1(b휏1) d s − − (1+ρ1+ρ1) 1 2 2 2 2 {[ 3 − 1 ] c1 + [ 3 + 1 ] c2} = 2 3[1−ωa1] 휏12 휏12 휏12 휏12

d s (ρ1+ρ1−1) 1 2 4 2 4 { [퐶1 퐼1(푏휏1) + 퐶2퐾1(푏휏1)] + 4 휎푛 푇 } + ∈1 휎푛 푇1 (2.236) 4 √휏1 2 2

d s d s (1+ρ2+ρ2) 1 (ρ2+ρ2−1) Let 퐸2 = and let 퐹2 = upon substituting E2 and F2 into 2 3[1−ωa1] 4

Equation (2.236) gives :

퐸2 퐹2 퐸2푏 퐸2 퐹2 퐸2푏 − [ 3 + ] 푐1퐼1(푏휏2) + [ 1 ] 푐1퐼 1(푏휏2) − [ 3 − ] 푐2푘1(푏휏2) − [ 1 ] 푐2푘 1(푏휏2) = √휏2 − √휏2 − 휏22 2 휏22 2 휏22 2 휏22 2

2 4 2 4 4퐹2휎푛 푇 +∈2 휎푛 푇2 (2.237)

퐸2 퐹2 퐸2푏 퐸2 퐹2 퐸2푏 푐1 {− [ 3 + ] 퐼1(푏휏2) + [ 1 ] 퐼 1(푏휏2)} − 푐2 {[ 3 + ] 푘1(푏휏2) − [ 1 ] 푘 1(푏휏2)} = √휏2 − √휏2 − 휏22 2 휏22 2 휏22 2 휏22 2

2 4 2 4 4퐹2휎푛 푇 +∈2 휎푛 푇2 (2.238)

Likewise, Equations (2.233) and (2.230) yields

1 1 휖 휎푇4 1 1 1 ∫ [퐺(휏 ) − 3휇 푞(휏 )] 휇푑휇 = 2 2 ∫ 휇푑휇 + 휌푠 ∫ [퐺(휏 ) + 3휇 푞(휏 )]휇푑휇 + 4휋 0 2 2 휋 0 4휋 2 0 2 2

2 1 1 휌푑 ∫ 휇푑휇 ∫ [퐺(휏 ) + 3휇 푞(휏 )]휇′푑휇′ (2.239) 4휋 2 휇=0 휇′=0 2 2

푞(휏 ) 퐺(휏 ) −(1 + 휌푠 + 휌푑) 2 − [휌푠 + 휌푑 − 1] 2 = ∈ 휎푛2푇4 (2.240) 2 2 2 2 2 4 2 2

Upon substituting 푞(휏)휏=휏2 and 퐺(휏)휏 = 휏2 into Equation (2.240), the same solution as in reference is obtained [28].

I1(b휏2) bI 1(b휏2) k1(b휏2) bk 1(b휏2) d s − − (1+ρ2+ρ2) 1 2 2 2 2 − {[ 3 − 1 ] c1 + [ 3 + 1 ] c2} = 2 3[1−ωa1] 휏22 휏22 휏22 휏22

d s (ρ2+ρ2−1) 1 2 4 2 4 { [퐶1 퐼1(푏휏2) + 퐶2퐾1(푏휏2)] + 4 휎푛 푇 } + ∈2 휎푛 푇2 (2.241) 4 √휏2 2 2

d s d s (1+ρ2+ρ2) 1 (ρ2+ρ2−1) Let 퐸2 = and let 퐹2 = upon substituting E2 and F2 into Equation 2 3[1−ωa1] 4

(2.228) gives :

2 4 2 4 4퐹2휎푛 푇 +∈2 휎푛 푇2 (2.242)

2 4 2 4 4퐹2휎푛 푇 +∈2 휎푛 푇2 (2.243)

Equations (2.238) and (2.243) can be used with Cramer’s rule, to find 푐1and 푐2 [77]:

Where Equations (2.238) and (2.243) are written as:

퐴1푐1 + 퐴2푐2 = 퐴3 (2.244)

퐴4푐1 + 퐴5푐2 = 퐴6 (2.245)

The integration constants 푐1and 푐2 can be written using Cramer’s rule [77]:

푐1 = [퐴3 퐴5 − 퐴2 퐴6]/[퐴1 퐴5 − 퐴2 퐴4 ] (2.246)

푐2 = [퐴1 퐴6 − 퐴3 퐴4]/[퐴1 퐴5 − 퐴2 퐴4 ] (2.247)

Where

퐸1 퐹1 퐸1푏 퐴1 = [ 3 − ] 퐼1(푏휏1) − [ 1 ] 퐼 1(푏휏1) (2.248) √휏1 − 휏12 2 휏12 2

퐸1 퐹1 퐸1푏 퐴2 = [ 3 − ] 푘1(푏휏1) + [ 1 ] 푘 1(푏휏1) (2.249) √휏1 − 휏12 2 휏12 2

2 4 2 4 퐴3 = 4퐹1휎푛 푇 +∈1 휎푛 푇1 (2.250)

퐸2 퐹2 퐸2푏 퐴4 = − [ 3 + ] 퐼1(푏휏2) + [ 1 ] 퐼 1(푏휏2) (2.251) √휏2 − 휏22 2 휏22 2

퐸2 퐹2 퐸2푏 퐴5 = − [ 3 + ] 푘1(푏휏2) − [ 1 ] 푘 1(푏휏2) (2.252) √휏2 − 휏22 2 휏22 2

2 4 2 4 퐴6 = 4퐹2휎푛 푇 +∈2 휎푛 푇2 (2.253)

After the integration constants 푐1and 푐2 have computed, the net radiative heat flux q(τ), and the incident radiation G(τ), can be calculated from Equations (2.224) and (2.215).This solution will be used to analyze and interpret the data from CVD experiments data that were conducted in the FAST facility at ORNL [4] and for the FFTF reactor in Argonne National Laboratory ANL.

2-3 Boundary Work

While analyzing the FAST experiments, it is worth noting that the work transfer, which is the work performed by the expanding bubble which compresses the covergas. If the work is assumed to be sufficiently large, it would stimulate the contents of the bubble to condense.

Condensation could plausibly shorten the bubble lifetime, or perhaps even reduce the probability that a large enough bubble would reach the covergas boundary. However, it should be noted that the work transfer by itself cannot completely eliminate the super heat vapor temperature, as also the bubble contains non-condensable species which would hinder the condensation process.

However, it is reasonable to assume that work transfer could decrease the amount of UO2 vapor within in the bubble as it reached the covergas. Since work transfer comprises one parameter in the overall energy balance, the work transfer to the covergas could be calculated using covergas pressure measurements in conjunction with the boundary work law [64].

푓(푡) 푊 = 푃푑푉 1→2(푡) ∫𝑖 (2.254)

The initial state of the covergas is represented by i, this is the state before capacitor banks are fired, and f(t) at the upper integration limits represents an arbitrary state while the covergas is being compressed due to expanding bubble while f representing the time elapse in reaching the above-mentioned arbitrary state (maximum volume)[64].

We have

∆푉퐵 = −∆푉푐푔 (2.255)

Based on this change in bubble volume, the current bubble volume becomes,

푉퐵 = 푉퐵0 + ∆푉퐵 (2.256)

The covergas volume change can be related to covergas pressure readings through the isentropic property relation, [64]

(푃푉훾) = (푃푉훾) (2.257) 푐푔0 푐푔

From Equation (2.257) the covergas volume becomes,

푃 1 푐푔0 ⁄훾 푉푐푔 = 푉푐푔 ( ) (2.258) 0 푃푐푔

Then, the bubble volume results,

∆푉푐푔 = 푉푐푔 − 푉푐푔0 (2.259)

1 ⁄훾 푃푐푔0 ∆푉푐푔 = 푉푐푔 (( ) − 1) (2.260) 0 푃푐푔

Substituting Equation (2.260) into Equations (2.255) and (2.256) yields:

1 ⁄훾 푃푐푔0 푉퐵 = 푉퐵 − 푉푐푔 (( ) − 1) (2.261) 0 0 푃푐푔

Finally, the volume formula for a sphere provides the bubble radius, [64]:

1 3 3 푃 훾 푅 = √ ( 푉 + 푉 (1 − ( 푐푔0) )) (2.262) 4휋 퐵0 푐푔0 푃 푐푔 푐푔

Using Equation (3-262), 푃푚푎푥 can be written as:

3 −훾 4휋푅푚푎푥 푃푚푎푥 = 푃𝑖 [1 − ] (2.263) 3푉푖

푉퐵푂 3 ≈ 0 because 푉퐵푂 is an initial bubble volume of 15,000 mm [4], and it is small compared 푉푖 with the initial volume. Thus, it can be neglected.

The previous isentropic property relation, 푉푚푎푥 can be written as: [64]

3 훾 4휋푅푚푎푥 푉푚푎푥 = 푉𝑖 (1 − ) (2.264) 3푉푖

푃 푉 −푃 푉 Equation (2.254) can be rewritten as, 푊 = 푚푎푥 푚푎푥 푖 푖 (2.265) 퐵표푢푛푑푎푟푦 1−푘

Substituting Equation (2.263) and Equation (2.264) into Equation (2.265) gives the work transfer at the boundary:

3 1−훾 푃1푉1 4휋푅푚푎푥 푊퐵표푢푛푑푎푟푦 = [(1 − ) − 1] (2.266) (1−훾) 3푉1

3 Where, P1 and V1 are the initial pressure (Pa) and volume (m ) for the initial state, as it is tabulated in Table (9) Appendix C, and γ is the specific heat ratio for Argon gas (1.67).

CHAPTER III

RESULTS

In this chapter, modeling results are presented and discussed. The first section describes results from the Rayleigh non-scattering model. This is followed by a section that treats the Mie scattering model. Lastly, the work transfer evidenced in out-of-reactor experiments conducted during 1980’s for oxide UO2 fueled reactors in Fuel Aerosol Simulant Test (FAST) facility at Oak

Ridge National Laboratory (ORNL) is considered. These analyses are applied to the bubble collapse time to study the safety profile of LMR reactors during HCDA. Lastly, the results of the models are compared to R. L. Webb et al. [53].

3-1 Rayleigh Non-Scattering Model Results

In the Rayleigh non-scattering model, the mixture was assumed to be in local thermodynamic equilibrium inside the bubble which has a black body surface boundary. The mixture in the bubble contains three components: the xenon, uranium dioxide vapor, and fog. To simulate fuel bubble geometry as realistically as possible, the energy equation in a spherical coordinates system was solved with the radiative flux heat transfer equation (RTE) to explore the effect of bubble geometry on the transient radiative heat flux and to predict the transient temperature distribution in the participating medium during a hypothetical core disruptive accident (HCDA) for liquid metal fast breeding reactor (LMFBR).

3-1-1 Compare of Current Method by Comparison With Past Work

It is important to compare simulations with accepted past work for plate geometry condensable mixture. For this purpose, the transient radiative heat flux at the outer surface and bubble temperature distributions are compared with those from S. H. Chan et al. [56].

Fog Free Region

FogwWith Fog

FogwWithout Fog

Figure 10. Transient temperature profile for semi-infinite mixture [56].

Figure 10 shows the temperature distributions in the fog and fog free regions at t = 0.05 s, t = 0.35 s, t = 1.15 s and t = 2.07 s from S. H. Chan et al. [56], and the fog boundary (that was defined by Equation 2.100b) at these times are 1.5, 3, 4, and 4.5. As the bubble cools down, the wall temperature drops, and the optical thickness increases as indicated in Figure 10. Figure 10 shows that, the presence of fog resulting from adiabatic expansion of UO2 and Xenon gas enhances the temperature distributions in the fog region as it obtained in Figure 10 at time t =

2.07 s. For example, at τ = 2.5, the bubble temperature is 3995 K in the fog region, but without fog, the bubble wall temperature is far below this temperature at 3607 K which is a large difference.

We cannot compare the calculated results of the two different geometries directly.

However, the simulated results should have similar behavior. In following paragraph, the results

from the present simulations are described. Figures 12, 13, and 14 show the temperature distributions in the fog region and fog free region at R = 0.1 m and t = 1.85 ms, R = 0.1 m and t =

2.5 ms, R = 0.1 and t = 3.5 ms. These figures show the fog penetration depth values at t = 5 ms equals to τf = 0.32 and t = 2.5 ms equals to τf = 0.42. It’s clear from these results that the fog penetration depth increases as the time increases and its values smaller than one (i. e. τ < 1). In other words, the medium is optically thin at these inputs which agrees with the experimental photo that had conducted on ORNL under water experiments as shown in Figure 11 [3]. As shown in Figure 11 the fog has formed, because of the pressure of non-condensable gas inside the bubble, the condensate fuel around the coolant appears as fog [49]. As the time increases fogs gets thicker until all the UO2 that evaporated is condensed. The ORNL pictures of water experiments were taken in sequences to show when and how the fog started forming (time between sequence 0.4 ms). For example, in the first image, Figure 11 shows small amounts of fog, but the fourth image shows large amounts of fog.

Fog Free

Fog

Figure 11. Fog formation under water tests at ORNL [3, 63].

In these figures, the temperature distribution profile in the fog free region is greater than that in the fog region, and its values at the boundary are lower than the saturation temperature.

This reduction resulted from spherical boundary effects and non-condensable gas (Xenon).

Comparing Figures 12 and 14, the wall temperature decreased from 3945 K to 3888 K at R = 0.1 m. The reduction of wall temperature means no sodium vaporization will occur at the bubble surface. This reduction in wall temperature was due to absorption effects inside the bubble which is higher in small bubbles. Moreover, no collapse will occur in the stainless-steel vessel of the reactor because the bubble cools down rapidly due to high levels of thermal radiation emissions to the cooler boundary [56]. In the fog region, the latent heat of condensation sustains the temperature distribution profiles and keeps it at high levels and drops suddenly at the bubble’s surface due to heat loss by radiation through the sodium coolant which has high thermal conductivity and low emissivity [53].

Figure 12. Transient temperature profile in spherical coordinates at t=1.85 ms, k=10 m-1 and R=100 mm.

However, the presence of Xenon gas lowers the saturation temperature of the UO2 vapor

(Dew Point Suppression) in the mixture, and the non-condensable gas inhibits condensation in either heterogeneous or homogenous (fog formation) by dew point suppression. From Figure 12 the thickness of the fog is 3.2 cm. This value was calculated by using Equation (2.3) where the fog

-1 boundary ( 휏푓 )and the absorption coefficient (k) were 0.32 and 10 m respectively. In addition, the amount of fog condensed can be computed by using the temperature distribution in fog region and Equation (2.112) as indicated in section (3-4-1). The intersection point of the fog free region profile and fog profile is shown in Figures 11, 12, and 13 which is important to compute the fog thickness and mass.

Figure 13. Transient temperature profile in spherical coordinates at t=2.5 ms, k=10 m-1 and R=100 mm.

Figure 14. Transient temperature profile in spherical coordinates at t=3.5 ms, k=10 m-1 and R=100 mm.

Lastly, the behavior of transient temperature distribution profile in spherical coordinates of the present study, is similar to the semi-infinite plate study of S. H. Chan et al. [56]. For example, the wall temperature drops as the bubble cools down, and the fog penetration dept increases.

The effect of non-condensable gas is the same in both geometries, and the wall temperature for slab and sphere geometries were 3889 K and 3894 K respectively. However, the bubble surface temperature of spherical geometry is higher than in semi-infinite plate geometry at same dimension and time because more energy was absorbed and converted into heat within the system as internal heat generation as indicated in Figure 15.

Figure 15. Transient temperature profile for slab at 3.5 ms, τ = 1 and k = 10 m-1.

Figure 15 shows the transient temperature distribution for slab at τ = 1 which means that the unrealistic (slab) bubble geometry cools faster than the realistic (sphere) bubble geometry.

However, the exact temperature distribution for the bubble was that provided in Figure 14 not that provided in Figure 15 where the bubble needs some time to reach this cooling stage. This shows us how is important to simulate the problem with spherical bubble shape rather than unrealistic shape such as slab because it did not give us the exact bubble temperature distribution.

3-1-2 The Wall Heat Flux

Figure 16 shows the comparison between the black body emission and the transient heat flux with fog formation at wall that was investigated by S. H. Chan et al. [56].

Figure 16. Transient heat flux for semi-infinite condensable mixture [56].

The important conclusion from their study is, that the fog formation is a significant parameter that must be considered in computing the heat transfer from a very high-temperature condensable mixture. Figure 17 shows the transient heat flux for spherical coordinate system vs time near to the bubble wall at R (90, 100, 110, and 130 mm) and k = 10 m-1. The curves represent the values of transient heat flux with fog formation. Figure 17 shows that the presence of fog resulting from the adiabatic expansion of uranium dioxide and xenon gas enhances the radiative heat flux at the boundary because the latent heat of condensation sustains the temperature of the fog and keeps it near the saturation temperature [56].

Figure 17. Transient wall heat flux for bubble at different sizes.

Figure 17 shows that as the volume of bubble increases, the wall heat flux increases because of the thermal absorption effect inside the bubble. This is higher in small bubbles compared with large bubbles that contain the same mass of UO2 vapor. In small bubbles more energy is absorbed by the medium because the medium is optically dense compared with large bubbles. In what follows, the temporal effects of a rising, thermally-active bubble are compared with the previous assessments of radiative heat transfer for purposes of determining whether sufficient time is available for a travelling bubble to break the coolant-covergas free surface or whether bubble lifetime is sufficiently short so that the bubble remains wholly submerged during the energy transfers so that little if any aerosol material could be expected to vent into the covergas [63].

3-2 The Bubble Rise Time

This assessment is based on a comparison of the bubble rise time, computed using the

Taylor formula, against the elapsed time required for energy transfer by radiation out of the bubble [63].

H t = (3.1) √g R

Figure 18. Bubble position relative to pool [63].

Here H is the vertical distance above the bubble. R is the bubble radius, and g is the acceleration of gravity (9.81 m/s2) as shown in Figure 18. Using Equation 3.1 and using the second and the third columns in Table 1, the time rise of bubble in sodium coolant has been calculated

[63].

Table 1 describes experiments conducted at ORNL (sodium experiments). The first column shows the name of the experiments and the second column represents the coolant level above the bubble measured from the center of the bubble to the covergas as shown in Figure 18. The third column lists the maximum bubble radius. The last column is the rise time of bubble that was calculated using Equation 3.1[63].

Table 1. The bubble rise time.

Experiment # H (m) R (m) Rise Time (s)

FAST104 1.06 0.1 1.0702

FAST106 0.24 0.11 0.2310

FAST107 0.14 0.1 0.1413

FAST108 0.08 0.08 0.0903

FAST109 0.1 0.09 0.1064

FAST113 0.03 0.13 0.0266

3-3 The Bubble Collapse Time

The theoretical and numerical results, which include bubble collapse times, will be compared with capacitor discharge vaporization (CDV) experiments conducted in the FAST Facility at Oak Ridge Laboratory [63]. First, let us describe the MATLAB code of the non-scattering model.

This code uses three parameters. The first parameter is the temperature distribution in the fog region, which varies from Tw to Tsat = 4000 K as shown in Figure 19.

The second parameter is the mass of UO2 vaporized from CDV experiments (1 gram for CDV tests), and the maximum bubble’s radius that was measured. The third parameter is the absorption coefficient which is k = 10 m-1 as provided in Table 14-C of Appendix C. The magnitude of the absorption coefficient depends on the particle’s surface area, absorption efficiency, and the number of particles in the medium.

Figure 19. Transient temperature profile and fog penetration depth at bubble collapse time.

In addition, as the absorption coefficient increases, the loss of radiation due to absorption increases, and the attenuation of the thermal radiation inside the participating medium occurs toward the bubble wall. In other words, more energy is absorbed by the medium. Figure 3-19 shows that the radiant energy absorbed by the participating medium inside the large bubble is smaller than the small bubbles, and large bubbles temperature distribution is higher than the small bubbles at constant mass (1 g for CDV tests) as provided in Appendix C Table 12 [50].

3-3-1 Sample Calculations of The Bubble Collapse Time

Equation 2-112 was used to compute the condensation time of UO2 vapor (bubble collapse time) based on the temperature distribution indicated in Figure 19. Referring to Figure

19, the temperature in fog region varies from Twall to Tsat in sodium experiments, and from

Appendix C the maximum radius is listed in Table 12 under sodium data. Using the MATLAB trapz

function, the integral of the Fog Production Equation 2.112 is used to compute the amount of condensed UO2 vapor (fog). Table 2 shows the bubble collapse times in Rayleigh model for sodium tests.

Table 2. The bubble collapse time at k = 10 m-1

Experiment. # Calculated Aerosol External Radius Collapse Time

Mass mfog (g) REdge (m) (mS)

FAST104 1 0.1 1.85 ms

FAST106 1 0.11 1.5 ms

FAST107 1 0.1 1.85 ms

FAST108 1 0.08 6.8 ms

FAST109 1 0.09 2.65 ms

FAST113 1 0.13 1.1 ms

Based on these calculations the bubble’s collapse time for (CDV) experiments conducted in FAST Facility at Oak Ridge Laboratory, is smaller than the bubble’s rising time as indicated in

Tables 1 and 2. This means that there is no aerosol release from the FAST vessel to the cover gas because all UO2 particles condense before the bubble reaches the cover-gas or break the surface and release bubble contents [63].

The effect of the absorption coefficient on bubble collapse time resulted from the e- folding length (푟∗) effect which can be defined as, the distance at which the intensity (I) drops to

퐼 ∗ 38 % of its initial value. For pure absorption = 푒−푘푟 , so when 푘 = 10 푚−1 the e-folding 퐼0 length 푟∗ = 10 푐푚 which means a large absorption coefficient provides to high time collapse values.

Table 3. The bubble collapse time at k = 2.2 m-1

Experiment # Calculated Aerosol Mass fog External Radius Time Collapse

(g) REdge (m) (mS)

FAST106 6 0.11 44

FAST113 6 0.13 21

R. L. Webb et al. [53] shows that at FAST 106 test (R = 0.11 m), the bubble collapse time was 22.95 ms. Good agreement was achieved when the results in Table 3 were compared with those of R. L. Webb et al. [53]. The results were generated at k = 2.2 m-1 according to the values of the absorption coefficient provided in Table 14 of Appendix C.

Figure 20. Comparison of temperature distribution at K = 10 m-1 and k = 30 m-1.

Figure 20 shows that the radiant energy transferred at k = 10 m-1 is faster than when k =

30 m-1 for FAST 104 at time 5 ms, because the e-folding length for k = 30 m-1 is 3.3 cm and e- folding length of k = 10 m-1 is 10 cm. Finally, based on the experimental data conducted in the

1980’s at the Fuel Aerosol Simulant Test (FAST) facility at Oak Ridge National Laboratory and the results have been tabulated in Tables 1 and 2 regarding the bubble rising time and the bubble collapse time. The conclusion that can be drawn from the data 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. As we know that, the particle size distribution data of Chan et al. (1982), the measurement of the particle-size distributions for the CDV tests appear to be approximately lognormal (dmean = 0.014 ± 0.002 μm). It was assumed that the expansion of UO2 vapor, compressed the adjacent argon as in a spherical shock tube. The embryos were produced by homogenous nucleation and condensation process was resulting from the rarefaction wave. So, based on the Chan et al. [56] study, the radiative heat transfer could involve

Mie scattering. To study and analyze this model, a particle size of 0.07 μm was chosen in the scattering model results described below [50].

3-4 Scattering Model

Mie scattering in radiation heat transfer is used to study and interpret experiments conducted in the 1980’s for UO2 fueled reactors in the Fuel Aerosol Simulant Test (FAST) facility at Oak Ridge National Laboratory [63].The analysis can be applied to estimate the bubble collapse of Liquid Metal reactors (LMR’s) during a hypothetical core disruptive accident (HCDA). The

-1 scattering coefficient of the UO2 particles for the model (σ = 1.24 m ), was calculated by using

Mie theory as provided in Appendix C, at the same number of stable nuclei N ( 2.9 E15 nuclei/m3) that resulted from the absorption coefficient k = 0.082 m-1 [56]. The spherical harmonics method

(P1 approximation method) was used to solve the radiative transfer equation (RTE) in spherical

coordinates for a participating medium confined between the two concentric spheres. The surfaces of the spheres are assumed to be gray, diffusely emitting and diffusely reflecting boundaries, and isothermal. Marsak’s boundary condition was used to compute, the net radiative heat flux q(τ) and the incident radiation G(τ) and to analyze and interpret the CVD experimental data from the FAST facility at ORNL [4] and Fast Flux Test Facility reactor at ANL [52].

3-4-1 Reproduced Work for W. Li and T. W. Tong [28]

To make sure the accuracy of the model is reasonable, we reproduce the work done of

W. Li et al. [28]. In Case1 from their paper (in all cases) T1 is the initial temperature, T2 is final temperature, and T is the constant medium temperature. In addition, the figures of W. Li et al.’s paper were normalized to following the custom in the heat transfer literature.

Figure 21. Physical model of a bubble containing UO2 fuel and fission gas.

휏−휏 For the x-axis, the 1 ratio is used, which translates to the distance from the inner 휏2−휏1 surface to the outer surface. The dimensionless incident radiation or the dimensionless heat flux is used for the y- axis which defined by a suitable reference temperature. In case1 from their paper, T1 and T2 are zero, but T ≠ 0 as indicated in Figure 21. The medium is under the effect of

internal heat generation. Heat propagates from the center of the medium to the inner and outer surfaces of the system because the incident radiation has higher values in this region. Figure 22 shows that the irradiance has high energy photons in the interior region of the system, and the difference in the incident radiation at the center of the medium (the source point) and at the boundaries was due to the absorption and scattering losses that occurred inside the medium. As the ratios of 휏1/휏2 increases, the irradiance decreases because the beam encounters a high radiative thermal resistance inside the medium.

Figure 22. The variation of dimensionless incident radiation for case1 [28].

휏−휏 Figure 23 shows the net radiative heat flux profile is symmetrical at 1 = 0.5, and its 휏2−휏1

휏−휏 휏−휏 maximum values occurs at the system surface boundaries 1 = 1 , and 1 = 0. The 휏2−휏1 휏2−휏1

휏−휏 radiative heat flux releases the thermal energy from the center of the medium at 1 = 0.5 휏2−휏1 to both sides, and its maximum value occurs at the boundaries. Figure 3-23 shows that the

휏−휏 dimensionless radiative flux has negative values at the inner wall 1 = 0 because the 휏2−휏1 radiative heat flux propagates in -µ direction.

Figure 23. The variation of dimensionless radiative flux for case1 [28].

Now we examine case 2 from W. Li et al.’s paper[28], in case 2 assume that, T = T2 = 0, but

T1 ≠ 0 .This means that the net radiative heat transfer propagates from the inner surface to the outer surface through the medium at zero temperature as indicated in Figure 24 and Figure 25.

Figure 24 shows the effect of thermal radiation loss on incident radiation, and Figure 25 shows the effect of thermal radiation loss on radiative heat flux. When the ratio of 휏1/휏2 decreases the medium becomes optically dense (more energy absorbed by the medium) and both the irradiance

휏−휏 and heat flux decreases as 1 increases. Moreover, the forward scattering coefficient of the 휏2−휏1

Legendre polynomials, a1, makes the transport of thermal radiative energy gradient more easily transmitted from the inner surface to the outer surface (forward direction).

Figure 24. Dimensionless incident radiation variation for case2 [28].

Figure 25. Dimensionless radiative flux variation for cas2 [28].

The last case in W. Li et al.’s paper is case3 which was defined as, T = T1 = 0, but T2 ≠ 0. This means the medium is under the effect of the outer surface emission. The propagation of radiant energy from the outer surface to the inner surface through the medium at zero temperature as per the assumption T = T1 = 0, but T2 ≠ 0, is shown in Figure 26 and Figure 27. Figure 26 shows the effect of radiation losses on incident radiation due to the absorption and scattering of radiation from the high level at the outer surface to low level at the inner surface through participating medium at zero temperature.

Figure 27 shows that as the ratio of 휏1/휏2 decreases the medium becomes denser and photons travel over longer distances while also making many collisions with the particles inside the medium, and, losing energy to the medium. This means that at a lower 휏1/휏2 ratio, the system has a higher thermal resistance. In this case, the transport of the radiant energy occurs at low levels due to radiative interchange and radiative interactivity effects inside the medium. For high

휏1/휏2 , each particle in the medium exchanging its radiative energy directly with the boundary, and therefore minimal radiative exchange inside the medium will occur (optically thin medium).

Figure 26. The effect of dimensionless incident radiation for case3 [28].

Figure 27. The effect of dimensionless radiative flux for case3 [28].

3-4-2 Mie Scattering Model on Sodium Experiments

The experiments conducted in the FAST facility used a variety of cover gas pressures and sodium pool levels, Although the pool level was one of the most critical experimental parameters affecting release as adjustments account for the physical extremes of low release, corresponding to high pool level. to total release, corresponding to subvaporizer levels, subcooling of the pool was also recognized as an important contributor to suppressing bubble transport. Pool level settings spanned a wide range of values (-300 mm to +1060 mm) the -300 mm pool level experiment (FAST-111) was designed to gauge the aerosol yield potential of CDV. Variations in argon pressure were used to determine the effect of cover gas compressibility on bubble size, pool dynamics and aerosol release. Covergas pressure varied over an order of magnitude during the series of experiments. Xenon pressure levels were varied to account for the effect of non- condensable on bubble transport. Values of experimental parameters, bubble size estimates and pool levels are listed in Table 12-C for selected experiments provided in Appendix C [63]. In addition, the experiments conducted at Argonne National Laboratory was by Fauske et al. [52].

Fauske et al. (1973) found that the bubble consists of 116 lb. of sodium vapor, 174 lb. of liquid sodium, 680 lb. of UO2 vapor, and 35 lb. of xenon gas. The bubble had a large diameter of 11.3 ft and rising from the bottom of the Sodium pool at 8.5 - 8.8 ft/s, and it reached the upper surface of the Sodium pool in approximately 2 s [2, 53].

In this model T1, T2, and T are not zero, but have values of 4500 k, 811 k, and 4000 K, respectively, following Oak Ridge National laboratory FAST experiments [4] and T1 = 4662 K, T =

4605 K and T2 = 838 K for FFTF reactor experiments at Argonne National Laboratory [2, 53].

Besides the Mie scattering model, the internal radius of the FAST and FFTF tests that were used in the MATLAB code are 0.012 m and 0.82 m respectively as show in the physical model of Figure

3-21. These values were selected according to the amount of xenon that was used in the

experiments [63, 53]. For the scattering model, the sodium experiments were investigated using

-1 -1 the following parameters β = 1.322 m , k = 0.082 m , and ω = 0.9379 ( n = 1 for UO2 vapor, m

= 2.42+ 0.009i, and a1 = 0.0975 for nonlinear- anisotropic forward scattering as indicated in

Appendix C). In this research, the mie_teta (m, x, 180) [40] function was used to compute ai- factors as provided in Table 7.

Figure 28. Dimensionless irradiance behavior under sodium experiments

Figures 28 and 31 show that the irradiance has high energy photons in the interior region of the system, and the difference between the incident radiation at the center of the medium (the source point) and at the boundaries was due to the absorption and scattering losses that occurred inside the medium. Figures 29 and 32 show the net radiative heat flux distribution from the inner surface of the bubble that contains hot gasses to the cooled outer bubble surface. It is clear from

Figure 3-29 that the net radiative heat flux is higher at the inner surface 7.140 × 106 푊/푚2

compared with the bubble outer surface 2.506 × 105 푊/푚2, and this reduction in the net radiative heat flux was due to the absorption and scattering losses.

Figure 29. Radiative Heat Flux behavior under sodium experiments

It is clear from Figures 28, 29, 31, and 32 when the radiant energy propagates in the medium, the thermal losses are high at small bubble volumes. The thermal losses were observed to be low when the bubble volume was large because of a low concentration of particles within the medium. This means that the large bubble size has low losses under these conditions of nonlinear - anisotropic forward scattering, and the small bubble sizes have high losses in energy. In the same time, we cannot neglect the bubble volume effects on volumetric heat generation component and the radiative heat transfer at the outer surface as shown in Figures 30 and 33.

Figure 30. Effect of 휏1/휏2 ratio on radiative heat transfer under sodium tests.

Table 4. Shows the magnitudes of dimensionless heat fluxes for sodium tests at the outer bubble surface

Table 4. Dimensionless heat fluxes at the outer surface.

Experimental NO. Maximum Radius Dimensionless Heat Fluxes m FAST 104 0.1 0.0107 FAST 105 0.08 0.0117 FAST 106 0.11 0.0106 FAST 107 0.1 0.0106 FAST 108 0.08 0.0118 FAST 109 0.09 0.011 FAST 110 0.06 0.0153 FAST 113 0.13 0.0108

Figure 31. Dimensionless irradiance behavior under sodium experiments.

Figure 32. Dimensionless radiative heat flux behavior under sodium experiments.

Figure 33. Effect of 휏1/휏2 ratio on radiative heat transfer under sodium tests.

3-4-3 Application of Current Model to Sodium Experiments

Since the model is composite from three basic cases (superposition solution) as indicated from W. Li et al. (1989), it’s important to study each case alone and determine which case has a significant effect on the scattering model. For case A1 in the current study, the situation is T1 = T2

= 0, and T = Tsat = 4000 K. The system is under the effect of volumetric heat generation which

휏−휏 equals to the total emission minus the total absorption. As the ratio 1 increases, the 휏2−휏1

푞′′ 퐺(휏) dimensionless radiative heat flux ( 푟 ) increases and dimensionless irradiance decreases as 휎푇4 4휎푇4 shown in Figures 34 and 35. The reduction in the irradiance causes a reduction on the total absorption component which leads to the rise in the magnitude of dimensionless radiative heat flux from the inner surface to the outer surface as indicated in Figure 34.

Figure 34. Radiative heat flux variation under sodium tests for case1.

Figure 34 shows that as the volume of the bubble increases, the stimulated emission increases and enhances the net radiative heat flux at the boundaries. The bubbles with high UO2 particle-concentrations produce low values of radiative flux compared with bubbles with low concentrations that give high rates.

In this case, the symmetric point occurs at (0.2, 0) where the net radiative heat flux is zero. The heat dissipates along the medium toward the outer surface, which is outside of the system, as well as inside the system to the inner surface which both are at a temperature of zero.

Lastly, case A1 has a significant effect on the bubble collapse time because it is controlled by the volumetric heat generation, but in the same time we must know the other cases such as case A2.

Figure 35. The variation of dimensionless irradiance under sodium tests for case1.

Case A2

The situation in this case assumed T = T2 = 0, and T1 = 4500 K. In this case, only the inner surface is emitting thermal radiation toward the outer surface inside the system. The forward Mie scattering forced the thermal radiative heat energy to transfer from high level radiative thermal energy to low level thermal energy. The reduction in thermal radiation between the inner surface and outer surface in the medium was because of absorption and scattering losses of the radiation during propagation through the participating media. More losses were observed when the medium is optically thick (i.e., 휏1⁄휏2 has a minimum value), and less losses occurred at high 휏1⁄휏2 ratio as shown in Figures 3-36 and 3-37.

Figure 36. Radiative heat flux variation under sodium tests for case2.

Figure 37. The variation of dimensionless irradiance under sodium tests for case2.

Case A3

The situation in this case is assumed that T = T1 = 0, and T2 = 811 K. This case is same as case A2 in physical meaning except that the emission is by the outer surface as indicated in Figures

38 and 39. Because the radiant energy flows in (- µ) direction, the magnitude of the dimensionless

푞′′ heat flux 푟 is negative as show in Figure 38. Furthermore, the magnitude of the dimensionless 휎푇4

퐺(휏) irradiance at the outer surface is greater than the inner surface as shown in Figure 39, and 4휎푇4 the attenuation on the irradiance resulted from the absorption and scattering losses.

Figure 38. Dimensionless heat flux variation under sodium tests for case3.

Figure 39. The variation of dimensionless irradiance for sodium tests for case3.

3-4-4 Bubble Collapse Time

For unsteady state problem, the energy balance of the system shown in Figure (3-40) can be written as, the volumetric heat generation rate = the total emission – the total absorption.

Here T1 = 4500 K, T = 4000 K, and T2 = 811 K for FAST experiments, and for the FFTF reactor, T1 =

4662 K, T = 4605 K, and T2 = 838 K.

Figure 40. Physical model drawing and coordinates.

Then, the volumetric heat generation rate can be calculated from the following formula based on the conservation of energy [28].

푛2휎푇4 푞̇(휏) = 푘 ∫ 푑휔 − 푘 퐺(휏) (3.2) 4휋 휋

푛2휎푇4 푞̇(휏) = 푘 ∫ 푑휔 − 푘 퐺(휏) (3.3) 휋 4휋

푞̇(휏) = 4 푘 푛2휎 푇4 − 푘 퐺(휏) (3.4)

Equation (3-2) represents the net heat generation inside the system, and the incidence radiation can be calculated from Equation (2.215).

1 2 4 퐺(휏) = [퐶1 퐼1(푏휏) + 퐶2퐾1(푏휏)] + 4 휎푛 푇 (3.5) √휏 2 2

To calculate the bubble collapse times for the FAST experiments and the (FFTF) reactor, the total net heat generation inside the system can be found by integrating Equation (3-4) from

휏1 → 휏2 because G(τ) is not constant. The irradiance G(τ) varies with τ as shown in Figures 31 and

Figure 41 where it has a maximum value in the inner surface where 푄̇푔푒푛 is minimum and decreases from 휏1 → 휏2, where 푄̇푔푒푛 is maximum as shown in Figure 41.

Figure 41. The variation of the volumetric heat generation inside the system

푟2 2 4 2 퐸푔푒푛̇ (휏) = ∫ (4 푘 푛 휎 푇 − 푘 퐺(휏))4휋푟 푑푟 (3.6) 푟1

Let 휏 = 훽 푟 ( 3.7)

Then, 푑휏 = 훽 푑푟 (3.8)

Back substituting into Equations 3-5, 3-7 and 3-8 into Equation 3-6 gives,

4휋푘 휏2 2 4 1 2 4 2 퐸̇ (휏) = ∫ (4 푛 휎 푇 − [퐶 퐼1(푏휏) + 퐶 퐾1(푏휏)] − 4 푛 휎푇 ) 휏 푑휏 (3.9) 푔푒푛 훽3 휏 휏 1 2 1 √ 2 2

4휋푘 휏2 1 2 퐸̇ (휏) = − ∫ ( [퐶 퐼1(푏휏) + 퐶 퐾1(푏휏)]) 휏 푑휏 (3.10) 푔푒푛 훽3 휏 1 2 1 √휏 2 2

3 4휋푘 휏2 ̇ ( ) 1( ) 1( ) 2 퐸푔푒푛 휏 = − 3 {∫휏 [퐶1퐼 푏휏 + 퐶2퐾 푏휏 ]휏 푑휏} (3.11) 훽 1 2 2

For a time, interval ∆t, the global energy balance around the physical model indicated in

Figure 3-40 can be written as:

(퐸 −퐸 ) 퐸̇ + 퐸̇ − 퐸̇ = ∆퐸̇ = 푖푛푖푡푖푎푙 푓푖푛푎푙 = ∆푈 + ∆퐾퐸 + ∆푃퐸 (3.12) 𝑖푛 푔 표푢푡 푠푡 ∆푡

The effects of kinetic and potential can be neglected, Equation (3-13) becomes

(퐸̇𝑖푛 + 퐸푔̇ − 퐸̇표푢푡)푡 = ∆푈푡 + ∆푈푙푎푡 (3.13)

Since the medium is in an isothermal state, then the sensible heat is zero

(퐸̇𝑖푛 + 퐸푔̇ − 퐸̇표푢푡)푡 = ∆푈푙푎푡 = 푚ℎ푓푔 (3.14)

퐸̇𝑖푛 = 퐴1푞(휏1)𝑖푛 (3.15)

퐸̇표푢푡 = 퐴2푞(휏2)표푢푡 (3.16)

Where 푞(휏1)𝑖푛 and 푞(휏2)표푢푡 can be computed from Equation 2.224 at τ = τ1 and τ = τ2 respectively.

퐼1(푏휏) 푏퐼 1(푏휏) 푘1(푏휏) 푏푘 1(푏휏) 1 − − 푞(휏) = {[ 2 − 2 ] 푐 + [ 2 + 2 ] 푐 } (3.17) 3[1−휔푎 ] 3 1 1 3 1 2 1 휏2 휏2 휏2 휏2

The bubble collapse time can be computed from this the following formula that resulted from Equations 3.11, 3.15, 3.16 and 3.17:

푚ℎ푓푔/푀 푡 = 3 (3.18) 4휋 2 푘 휏2 2 { 휏 푞(휏 ) − ∫ [퐶 퐼1(푏휏) + 퐶 퐾1(푏휏)] 휏2푑휏 − 휏 푞(휏 ) } 훽2 1 1 푖푛 훽 휏 1 2 2 2 표푢푡 1 2 2

Where 퐼1(푏휏) and 퐾1(푏휏) are the modified Bessel functions of the first and second kind 2 2

1 ( )( ) of order 2, and 푏 = √3 1 − 휔푎1 1 − 휔푎0 . a1 and a0 are the asymmetry coefficients of

Legendre Polynomials of the phase function, and its value depends on the scattering type and its polarization components. Also, m, is the mass of condensed vapor (grams), and M is the molecular weight of UO2 (g/moles). Where r1 is the inner radius (m), and τ1 is the optical depth

(dimensionless). Likewise, r2 is the outer radius (m) and τ2 is the optical depth (dimensionless). t, is the bubble collapse time (s), and n is the refractive index of UO2 in the vapor phase. Moreover,

A1 is the surface at the inner surface m2, and A2 is the surface at the outer surface m2.

A MATLAB code was used to compute the bubble collapse time (t) according to Equation

4.18 by using the trapezoid method to approximate the definite integral with 100 subintervals.

This code appears in Appendix B. It was run under T1 = 4500 K, T = 4000 K, and T2 = 811 K for FAST experiments [3][4], and at T1 = 4662 K, T = 4605 K and T2 = 838 K for an FFTF reactor [2, 52]. In the calculations, the absorption coefficient was set to k = 0.082 m-1, scattering coefficient was σ = 1.24

-1 m , and a1 = 0.0975 which is a slightly nonlinear - anisotropic forward scattering coefficient

(presented in Appendix C Table 7). The magnitude of the absorption coefficient used in S. H. Chan

[56] is used in the present study, and the results are tabulated in Table 3-5.

The conclusion that can be drawn from the data is that the reactor has a larger margin of safety since the bubble rising time (indicated in Table 3.5) is greater than the bubble collapse time.

Table 5. The bubble collapse time at k = 0.082 m-1

Expert. # Aerosol Mass External Radius Collapse Time Rise Time

(g) (m) (ms) (s)

FAST104 1 0.1 24.49 ms 1.0702 s

FAST105 1 0.08 34.68 ms 1.17 s

FAST106 1 0.11 20.47 ms 0.231 s

FAST107 1 0.1 24.49 ms 0.1413 s

FAST108 1 0.08 34.68 ms 0.0903 s

FAST109 1 0.09 29.23 ms 0.1064 s

FAST110 1 0.06 47.3 ms 0.1824 s

FAST113 1 0.13 14.37 ms 0.0266 s

FAST 104 6 0.1 0.146 ms 1.07 s

FAST 106 6 0.11 0.122 ms 0.231 s

FFTF 308440 1.72 2.12 sec 2.3 s

As we know the absorption coefficient that was used by S. H. Chan et al. [56] is the absorption coefficient of CO2 not for UO2, and we know that the absorption coefficient of UO2 is greater than 0.082 m-1[63]. Since we do not have the exact values of the absorption and the scattering coefficients, but we know the number of particles per unit volume in FAST tests and

FFTF reactor.

In the current study, the number of particles per unit volume that was selected (N = 4

× 1016 particles/m3 ) which is in between (N = 2.9 × 1015 particles/m3) for S. H. Chan et al.

[56] and (N = 9 × 1017 particles/m3) for R. L. Webb et al. [53] at particle size d = 0.7 μm[50].

Özisik found that when radiation propagates through the medium containing spherical particles with the same composition and uniform size, the absorption and scattering coefficients can be computed from the following equations [19]:

2 푘 = 퐶푎푁 = 휋푟 푄푎푁 (3.19)

2 휎 = 퐶푠푁 = 휋푟 푄푠푁 (3.20)

The scattering and extinction efficiencies can be calculated from the following functions provided in Appendix C [26, 35]. By using mie (m, x) [37] function the scattering and the absorption efficiencies are 0.1102 and 0.0073 respectively, and by using mie_gi (m, x,1,12) [40] a1 was 0.097 as indicated in Appendix C.

Upon using Equations 3.19 and 3.20 the absorption and scattering coefficients are 1.1 m-1 and 15 m-1. A comparison of the calculated results with measured data of W. L. Webb et al. [53] are shown in Table 6.

Table 6. The bubble collapse time at k = 1.1 m-1

Test # Aerosol Mass External Radius Collapse Time Collapse Time

(g) (m) (ms) (ms) [53]

FAST109 6 0.09 36 ms 32.9 ms

FAST104 6 0.1 27.9 ms 17 ms

FAST106 6 0.11 22.1 ms 22.95 ms

Good agreement was achieved when the collapse times of the experiments of FAST 104,

FAST 106 and FAST 109 at 6 grams of UO2 mass are compared with measured data of R. L. Webb and A. B. Reynolds [53].

In the participating medium, the radiative transfer equation (RTE) controls the radiative intensity (shown in Figure 42) as the Navier Stokes Equations controls the flow in fluid mechanics

[30]. In the scattering model, the total emission term in RTE is the most effective parameter in this model. This parameter depends on five variables, the participating medium temperature (T =

4000 K), the refraction index of UO2 (n = 1), the emissivity of UO2 (0.28), and the absorption

-1 coefficient of UO2 (k = 0.082 m ) [56]. The other parameter is the scattering coefficient σ, for example, in FAST 106 [63], if the albedo of scattering (ω) increases from 0.5 to 0.96, the bubble collapse time (t) decreases from 25.6 ms to 20.3 ms at constant k (0.082 m-1) because the radiative heat transfer rate is enhanced by the scattered radiation everywhere inside the bubble to the boundary.

Figure 42. The attenuation of incident radiation in participating medium [30].

In energy balance principle, the volumetric energy generation, the net radiative heat transfer rate at the boundaries and the irradiance should also be considered as indicated in

Equation (3.18). However, there is no significant effect for the nonlinear- anisotropic forward scattering coefficient (a1, in-scattering coefficient) on the bubble collapse times, but the more prevalent factor in the scattering processes is the out-scattering term. Hence, the bubble collapse time in the scattering model was controlled by the net emission and out-scattering radiation parameters [30]. While analyzing the scattering model (Mie scattering) for FAST experiments and for the (FFTF) reactor, the bubble rise time is greater than the bubble collapse time, thus, increasing the safety profile of the reactor. From the analysis of Mie scattering model, we can conclude that there is a greater margin of safety when the bubble rising time is greater than the bubble collapse time since the bubble collapses (UO2 condenses) before it can reach the top of the vessel. Therefore, there is less chance of aerosol release.

3-4-5 Boundary Work and Fuel-Gas Bubble

In the previous section we discussed the bubble collapse time for the FAST experiments and FFTF reactor. Now, we are going to discuss and study the boundary work resulting from fuel

-gas bubble expansion.

For an extremely low probability, ‘’severe accident scenarios which have been postulated for liquid metal cooled fast reactors, large bubble cavities containing fuel vapor and fission products transit a layer of coolant and release this material to the cover gas thereby presenting a contribution to an accident-specific source term. So that a more mechanistic assessment of these types of events can be developed, analyses has recently been performed to account for the work transfer observed in out-of-reactor source term experiments conducted during the 1980’s for oxide fueled reactors in the Fuel Aerosol Simulant Test (FAST) facility at Oak Ridge National

Laboratory” [63]. A. B. Reynolds et al. [75] reported that “the question of how much mechanical energy, the primary system must be able to contain in an energetic core disruptive accident has not been answered in the U.S. The cutoff in licensing activity for CRBR left the question suspended, with the NRC suggesting 1200 MJ and the CRBR project proposing not more than 600 MJ” [75].

M. L. Corradini et al. [49] indicated that the work done by the bubble expansion equals the amount of mechanical energy of a HCDA accident [49].

In this section, the magnitude of mechanical energy induced from a bubble expansion containing fuel vapor and xenon gas in the test has been computed as shown in Figure 43. Figure

43 shows that the maximum value of the mechanical energy for the FAST (111) test was 3.7 kJ, because the vaporizer was positioned within the cover gas. In addition, the conclusion drawn from the study of the boundary work as the sodium coolant level decreases, the magnitude of the boundary work increases. In FAST 113, the height of the coolant was 0.03 m, and the boundary work was 1.117 kJ. However, in FAST 112, the height of the coolant was 0.25 m, and the

mechanical energy resulted from the boundary work which was 0.267 kJ as shown in Figure 43 below. The x-axis represents the CDV power supply system used to energize the sample, and the y-axis represents the work transfer.

Figure 43. Work transfer trends (Na Tests).

At the same time, it should be concluded that the work transfer by itself couldn’t completely eliminate the superheated vapor temperature, as also the bubble contains non-condensable species which would hinder the condensation process. However, it is reasonable to assume that work transfer could decrease the amount of UO2 vapor in the bubble as it reached the covergas

[63]. Petrykowski and Mohamed [63] concluded that “to make the modelling more complete with respect to radiation, identified as the most probable, dominant mode in this study, an extension of the physical configuration to include the presence of a fog or particulate layer may be needed.

The presence of a fog layer may reduce radiation heat transfer thereby increasing bubble lifetimes and subsequent transport of aerosol to the covergas, significantly altering the claim that radiation strongly impedes bubble transport towards the covergas” [63].

CHAPTER IV

SUMMARY AND CONCLUSIONS

The main objective of the present study is to place the assessment of previous experimental data on a more mechanistic footing. Previous experiments were designed to examine primarily on a phenomenological basis, the transport to a covergas space of a bubble containing oxide-fuel vapor and fission product mixtures. Bubble formation occurred during capacitor discharge vaporization (CDV) of simulant material positioned in sodium pools. In what follows, the transport characteristics of the bubble were consistent with the energy transfer that occurred between the pool, the bubble, and the covergas [63]. The results extracted from the

Rayleigh model (non-scattering model) of radiation heat transfer, Mie model (scattering model) of radiation heat transfer, and boundary work and fuel-gas bubble model were used to interpret the phenomenology and existing database associated with severe accident assessments. (e.g.,

Oak Ridge Experiments and Reactor-Scale Events).

4-1 Rayleigh, Mie, and Expansion Work Models

The Rayleigh model in radiation heat transfer has been investigated to analyze and interpret the experiments that were conducted for UO2 fueled reactors in the Fuel Aerosol

Simulant Test (FAST) facility at Oak Ridge National Laboratory (ORNL).These analyses are applied to estimate the bubble collapse of Liquid Metal reactors (LMR’s) during a hypothetical core disruptive accident (HCDA). In this model, the radiative heat equation was simplified and solved using spherical coordinates. Scattering effects were neglected in the one-dimensional thermal radiation transfer (r-direction) model. Then by substituting equations for radiant heat transfer into the energy equation, they were solved together with the continuity equations of the mixture species under local thermodynamic equilibrium.

A MATLAB code was used to solve the radiative transfer equation (RTE) in spherical coordinates. The mixture of three components: the non-condensable gas Xenon, Uranium dioxide vapor, and fog was assumed to be in local thermodynamic equilibrium inside the bubble which was assumed to have a black body surface boundary. The effect of the fuel bubble’s geometry on the transient radiative heat flux and the transient temperature distribution in the participating medium were studied. It was found that as the volume of bubble increases, the wall heat flux increases because of the thermal absorption effect inside the bubble. This is higher in small bubbles compared with large bubbles that contain the same mass of UO2 vapor. In small bubbles more energy is absorbed by the medium because the medium is optically dense compared with large bubbles. Furthermore, a model of the fuel-gas bubble with transient temperature distributions is presented, and the differences between the results of realistic sphere bubble geometry and unrealistic slab bubble geometry were investigated and compared the results with

S.H Chan et al. [56]. The transient temperature distribution in the fog region was used to predict the amount of condensable UO2 vapor. In the fog region, the latent heat of condensation sustains the temperature distribution profiles and keeps it at high levels and drops suddenly at the bubble’s surface due to heat loss by radiation through the sodium coolant which has a high thermal conductivity and a low emissivity [53]. However, the presence of Xenon gas lowers the saturation temperature of the UO2 vapor (Dew Point Suppression) in the mixture, and the non- condensable gas inhibits condensation in either heterogeneous or homogenous (fog formation) by dew point suppression. There was good agreement with the results of R. L. Webb et al. [53].

These analyses were used to estimate the bubble collapse time and to study the safety profile of

LMR reactors during hypothetical core disruptive accident (HCDA). From the results of the non- scattering model, it was shown that there is a greater margin of safety when the bubble rise time

is greater than the bubble collapse time since the bubble collapses (UO2 condenses) before it reaches the top of the vessel. Thus, the release of an aerosol is unlikely.

With the Mie scattering model, a spherical harmonics method (P1 approximation) was used to solve the radiative transfer equation (RTE) in spherical coordinates for a participating medium confined between two concentric spheres. The surfaces of the spheres were assumed to be gray, diffusely emitting and diffusely reflecting boundaries. In addition, 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 the CVD experiments data that were conducted in the FAST facility at ORNL [4] and Fast Flux Test Facility reactor (FFTF) at ANL.

A MATLAB function (nonlinear -anisotropic scattering model) was used to compute, the scattering, extinction, and absorption efficiencies by using the mie (m, x) function [39]. The asymmetry parameter ai represents the strength of the radiation scattering, and its magnitude dependent on the type of scattering being studied; for example, Mie scattering type has a higher value of ai compared with Rayleigh scattering type. Moreover, the influence of the scattering coefficient on the rate of radiative heat transfer and bubble lifetime in spherical coordinates containing UO2 particles during (HCDA) for (LMR) was investigated.

In a participating medium, the radiative transfer equation (RTE) governs the radiative intensity just as the Navier Stokes Equations govern the flow in fluid mechanics [30]. In the Mie scattering model, the total emission term in the RTE is dominant. This parameter depends on four variables, the participating medium temperature, the refraction index of UO2, the emissivity of

UO2, and the absorption coefficient of UO2. However, there is no significant effect for the nonlinear- anisotropic forward scattering coefficient (a1, in-scattering coefficient) on the bubble collapse times, but the more prevalent factor in the scattering processes is the out-scattering

term. Hence, the bubble collapse time in the scattering model was controlled by the net emission and out-scattering radiation parameters [30]. Good agreement was achieved when the results of the sodium experiments of FAST 104, FAST 106 and FAST 109 at 6 grams of UO2 mass were compared with measured data (with the same bubble size) of R. L. Webb and A. B. Reynolds [53].

The scattering model (Mie scattering) showed that the bubble rise time was greater than the bubble collapse time, thus, increasing the safety profile of the reactor.

Lastly, in the FAST experiments, work transfer, (work done by the expanding bubble in compressing the covergas) could, if sufficiently large, cause components of the bubble to condense, perhaps shortening bubble lifetime or reducing the probability that a large bubble would reach the covergas. Although it is not reasonable to assume that work transfer alone could bring about complete elimination of high temperature vapor, if for no other reason than the bubble also contains non-condensable components, work transfer could reduce the amount of

UO2 vapor contained by the bubble as it reached the covergas. Since work transfer comprises one factor in the overall energy balance, the work transfer to the covergas was computed by using covergas pressure readings and the experimental data provided in Appendix C [63]. At the same time, it should be concluded that the work transfer by itself couldn’t completely eliminate the superheated vapor temperature, as also the bubble contains non-condensable species which would hinder the condensation process. However, it is reasonable to assume that work transfer could decrease the amount of UO2 vapor in the bubble as it reached the covergas [63]. J. C.

Petrykowski and H. Mohamed (2015) concluded that “to make the modelling more complete with respect to radiation, identified as the most probable, dominant mode in this study, an extension of the physical configuration to include the presence of a fog or particulate layer may be needed.

The presence of a fog layer may reduce radiation heat transfer thereby increasing bubble lifetimes

and subsequent transport of aerosol to the covergas, significantly altering the claim that radiation strongly impedes bubble transport towards the covergas” [63].

4-2 Recommendations

It is important to know the effect of the spectral albedo of scattering in fog formation.

This can be formulated by including the scattering coefficient parameter to the gray medium analysis shown in the current research. In this model, the radiative transfer equation (RTE) and the energy equation in spherical or semi-infinite condensable mixture, should be solved together along with continuity equations of mixture components under local thermodynamic equilibrium, surrounded by gray surface.

This model is similar to Mie scattering model in the present study, except it would be solved with different method such as (PN approximation method). The spherical harmonics method (PN approximation method) could be used to solve the radiative heat transfer equation

(RTE) in spherical coordinates for a participating medium confined between the two concentric spheres. The surfaces of the spheres should be assumed to be gray, diffusely emitting and diffusely reflecting boundaries, and isothermal. Marsak’s boundary condition would be used to computed, the net radiative heat flux q(τ), and the incident radiation G(τ), to analyze and interpret the CVD experimental data.

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APPENDIX A

Matzler’s Functions

A-1 Computes a Matrix of Mie Coefficients

% Computes a matrix of Mie Coefficients, an, bn,

% of orders n=1 to nmax, for given complex refractive-index

% ratio m=m'+im" and size parameter x=k0*a where k0= wave number in ambient

% medium for spheres of radius a;

% Eq. (4.88) of Bohren and Huffman (1983), BEWI:TDD122

% using the recurrence relation (4.89) for Dn on p. 127 and

% starting conditions as described in Appendix A.

% C. Mätzler, July 2002

z=m.*x; nmax=round(2+x+4*x.^(1/3)); nmx=round(max(nmax,abs(z))+16); n=(1:nmax); nu = (n+0.5);

sx=sqrt(0.5*pi*x); px=sx.*besselj(nu,x); p1x=[sin(x), px(1:nmax-1)]; chx=-sx.*bessely(nu,x); ch1x=[cos(x), chx(1:nmax-1)]; gsx=px-i*chx; gs1x=p1x-i*ch1x; dnx(nmx)=0+0i; for j=nmx:-1:2 % Computation of Dn(z) according to (4.89) of B+H (1983)

104

dnx(j-1)=j./z-1/(dnx(j)+j./z); end; dn=dnx(n); % Dn(z), n=1 to nmax da=dn./m+n./x; db=m.*dn+n./x;

an=(da.*px-p1x)./(da.*gsx-gs1x); bn=(db.*px-p1x)./(db.*gsx-gs1x);

result=[an; bn];

A-2 Computation of Mie Efficiencies

% Computation of Mie Efficiencies for given

% complex refractive-index ratio m=m'+im"

% and size parameter x=k0*a, where k0= wave number in ambient

% medium, a=sphere radius, using complex Mie Coefficients

% an and bn for n=1 to nmax,

% s. Bohren and Huffman (1983) BEWI:TDD122, p. 103,119-122,477.

% Result: m', m", x, efficiencies for extinction (qext),

% scattering (qsca), absorption (qabs), backscattering (qb),

% asymmetry parameter (asy=) and (qratio=qb/qsca).

% Uses the function "mie_ab" for an and bn, for n=1 to nmax.

% C. Matzler, May 2002, revised July 2002.

if x==0 % To avoid a singularity at x=0

result=[0 0 0 0 0 1.5]; elseif x>0 % This is the normal situation

nmax=round(2+x+4*x.^(1/3));

n1=nmax-1;

105

n=(1:nmax);cn=2*n+1; c1n=n.*(n+2)./(n+1); c2n=cn./n./(n+1);

x2=x.*x;

f=mie_ab(m,x);

anp=(real(f(1,:))); anpp=(imag(f(1,:)));

bnp=(real(f(2,:))); bnpp=(imag(f(2,:)));

g1(1:4,nmax)=[0; 0; 0; 0]; % displaced numbers used for

g1(1,1:n1)=anp(2:nmax); % asymmetry parameter, p. 120

g1(2,1:n1)=anpp(2:nmax);

g1(3,1:n1)=bnp(2:nmax);

g1(4,1:n1)=bnpp(2:nmax);

dn=cn.*(anp+bnp);

q=sum(dn);

qext=2*q/x2;

en=cn.*(anp.*anp+anpp.*anpp+bnp.*bnp+bnpp.*bnpp);

q=sum(en);

qsca=2*q/x2;

qabs=qext-qsca;

fn=(f(1,:)-f(2,:)).*cn;

gn=(-1).^n;

f(3,:)=fn.*gn;

q=sum(f(3,:));

qb=q*q'/x2;

asy1=c1n.*(anp.*g1(1,:)+anpp.*g1(2,:)+bnp.*g1(3,:)+bnpp.*g1(4,:));

asy2=c2n.*(anp.*bnp+anpp.*bnpp);

asy=4/x2*sum(asy1+asy2)/qsca;

qratio=qb/qsca;

result=[qext qsca qabs qb asy qratio]; end;

106

A-3 Computation and Plot of Mie Power Scattering

% Computation and plot of Mie Power Scattering and diffraction functions

% for complex refractive-index ratio m=m'+im", size parameters x=k0*a,

% according to Bohren and Huffman (1983) BEWI:TDD122

% plot of scattered intensity S, and degree of polarization

% computes gi-factors (coeffs of Legendre Polynomials of Phase Function).

% Input: m: complex refractive indes,x: size parameter,

% nj: number of Legendre coefficients

% nsteps: number of scattering angles (recommendation: nsteps=22*x)

% C. Matzler, June 2005.

m1=real(m); m2=imag(m); nx=(1:nsteps); dteta=pi/nsteps;

Q=mie(m,x); Qext=Q(1); Qsca=Q(2); Qabs=Q(3); Qb=Q(4); asy=Q(5); nmax=round(2+x+4*x^(1/3)); ab=mie_ab(m,x); an=ab(1,:); bn=ab(2,:); teta=(nx-0.5).*dteta; tetad=teta*180/pi; u=cos(teta); s=sin(teta); px=pi*x^2; st=pi*s*dteta/Qsca; for j = 1:nsteps,

pt=mie_pt(u(j),nmax);

pin =pt(1,:);

tin =pt(2,:);

n=(1:nmax);

n2=(2*n+1)./(n.*(n+1));

pin=n2.*pin; tin=n2.*tin;

S1=(an*pin'+bn*tin');

S2=(an*tin'+bn*pin');

107

SR(j)= real(S1'*S1)/px;

SL(j)= real(S2'*S2)/px; end; z=st.*(SL+SR); zR=st.*SR; zL=st.*SL;

nj=nj+1; % Phase fct decomposition in Legendre Polynomials for jj=1:nj,

xa=legendre(jj-1,u);

x0=xa(1,:);

gi(jj) =x0*z'; % beam efficiency, asymm. factor and gi's

giR(jj)=x0*zR';

giL(jj)=x0*zL'; end;

%gi=gi(2:nj);giR=giR(2:nj); giL=giL(2:nj); z=cumsum(z); % Beam Efficiency vs. tetalim

S=(SL+SR); % Intensity dS=(SR-SL)./S; % Degree of polarization result.s=[tetad',S',dS',z']; result.Q=[Qext,Qsca,Qabs,Qb,asy]; result.gi=gi; result.giR=giR; result.giL=giL;

A-4 Computation and Plot of Mie Power Scattering and Diffraction Functions

% Computation and plot of Mie Power Scattering and diffraction functions

% for complex refractive-index ratio m=m'+im", size parameters x=k0*a,

% according to Bohren and Huffman (1983) BEWI:TDD122

108

% 1) polar diagram, linear or in dB scale with respect to minimum, with

% SL in upper semicircle, SR in lower semicircle and 3 cartesian diagrams

% 2) same for SL0 and SR0 without diffraction pattern,

% 3) scattered intensity S (lin or log scale), and degree of polarization

% 4) scattered intensity without diffraction peak S0 (lin or log scale),

% 5) beam efficiencies of S and S0, diffraction efficiency Qd

% 6) gi-factors (coefficients of Legendre Polynomials of Phase Function).

% nsteps: number of scattering angles (for accurate comp. use nsteps=22*x)

% nsmooth: number of values to be averaged in polarization, S and S0

% (for 'log' type only)

% type:= 'log' or 'lin' for logarithmic or linear plots

% C. Matzler, April 2004.

nstart=round(min(0.5*nsteps,nsteps*pi/x+nsmooth)); m1=real(m); m2=imag(m); nx=(1:nsteps); dteta=pi/nsteps;

pt=mie_pt(u(j),nmax);

pin =pt(1,:);

tin =pt(2,:);

n=(1:nmax);

n2=(2*n+1)./(n.*(n+1));

109

pin=n2.*pin; tin=n2.*tin;

S1=(an*pin'+bn*tin');

S2=(an*tin'+bn*pin');

xs=x.*s(j);

if abs(xs)<0.00002 % Diffraction pattern according to BH, p. 110

S3=x.*x*0.25.*(1+u(j)); % avoiding division by zero

else

S3=x.*x*0.5.*(1+u(j)).*besselj(1,xs)./xs;

end;

S4=S1-S3;

S5=S2-S3;

SR(j)= real(S1'*S1)/px;

SL(j)= real(S2'*S2)/px;

SD(j)= real(S3'*S3)/px;

SR0(j)=real(S4'*S4)/px;

SL0(j)=real(S5'*S5)/px; end; z=st.*(SL+SR); z0=st.*(SL0+SR0);

nj=11; % Phase fct decomposition in Legendre Polynomials for jj=1:nj,

xa=legendre(jj-1,u);

x0=xa(1,:);

gi(jj)=x0*z'; % beam efficiency, asymm. factor and higher gi's

g0i(jj)=x0*z0'; % same as gi's, but diffraction signal removed end; etab=gi(1); gi=gi/etab; gi=gi(2:nj); etab0=g0i(1); g0i=g0i/etab0; g0i=g0i(2:nj);

110

Qd=Qsca*(1-etab0); % Qd = diffraction efficiency z=cumsum(z); % Beam Efficiency vs. tetalim z0=cumsum(z0);

S=(SL+SR); S0=(SL0+SR0); % Intensity

Ss=smooth(S,nsmooth); S0s=smooth(S0,nsmooth); dS=(SR-SL)./S; % Degree of polarization dSs=smooth(dS,nsmooth);

figure; if type=='lin' % linear plots

y=[teta teta+pi;SR SL(nsteps:-1:1)]';

polar(y(:,1),y(:,2)),

title(sprintf('Mie Scattering Diagram: m=%g+%gi, x=%g',m1,m2,x)),

xlabel('Scattering Angle'), figure; subplot(2,1,1);

plot(tetad,S,'k-')

title(sprintf('Mie Angular Scattering: m=%g+%gi, x=%g',m1,m2,x)),

xlabel('Scattering Angle'),

ylabel('S'); subplot(2,1,2);

plot(tetad,dS,'k-')

xlabel('Scattering Angle'),

ylabel('Polarization Degree '); elseif type=='log', % logar. plots

y=[teta teta+pi;10*log10(SR) 10*log10(SL(nsteps:-1:1))]';

ymin=min(y(:,2)); % Minimum for normalization of log-polar plot

y(:,2)=y(:,2)-ymin;

111

polar(y(:,1),y(:,2)),

title(sprintf('Mie Scattering Diagram: m=%g+%gi, x=%g, min(dB)= %g',m1,m2,x,ymin)),

xlabel('Scattering Angle (deg)');

y=[teta teta+pi;10*log10(SR0) 10*log10(SL0(nsteps:-1:1))]';

ymin=min(y(:,2)); % Minimum for normalization of log-polar plot

y(:,2)=y(:,2)-ymin; figure;

polar(y(:,1),y(:,2)),

title(sprintf('No-Peak Scattering Diagram: m=%g+%gi, x=%g, min(dB)= %g',m1,m2,x,ymin)),

xlabel('Scattering Angle (deg)'); figure; subplot(2,1,1);

semilogy(tetad,Ss,'k-'),

title(sprintf('Mie Angular Scattering: m=%g+%gi, x=%g',m1,m2,x)),

xlabel('Scattering Angle'),

ylabel('S'); subplot(2,1,2);

plot(tetad,dSs,'k-'),

xlabel('Scattering Angle'),

ylabel('Polarization Degree '); figure;

semilogy(tetad(nstart:nsteps),Ss(nstart:nsteps),'r:',tetad,S0s,'k-'),

title(sprintf('No-Peak Angular Scattering: m=%g+%gi, x=%g',m1,m2,x)),

xlabel('Scattering Angle'),

ylabel('S0'); figure;

xmin=min(tetad/180);

semilogx(tetad/180,z,'r-',tetad/180,z0,'k--',tetad/180,z-z0,'b:'),

112

xlabel('Maximum Scattering Angle/180°'),

axis([xmin, 1, 0, 1.1]); end; result.s=[tetad',SR',SL',SR0',SL0',SD',dS',z',z0']; result.Q=[Qext,Qsca,Qabs,Qb,Qd,asy]; result.gi=gi; result.g0i=g0i;

113

APPENDIX B

MATLAB Codes

B-1 Mie Scattering for FAST Tests clear all ;close all;clc, n = 1 ; % Index of refraction of UO2 in vapor phase (dimensionless) % sigma = 5.67e-8; ; % Boltz man constant W/ m2.K4 %

Ts1 = 4500; % Vapor Temperature (K) %

T = 4000; % Saturated Temperature (K) %

Ts2 = 811; % Coolant Temperature (K) %

Ka = 1.1; sca = 15; % Absorption and scattering coefficients (1/m) %

B = Ka+sca; % Extinction coefficient (1/m)% w = sca/B; % ALBEDO ,coeffs of Legendre Polynomials of Phase Function (dimensionless)% a1 = 0.0975; a0 = 1 ; %computes gi-factors (coeffs of Legendre Polynomials of Phase Function) (dimensionless) % b = sqrt(3*(1-w*a0)*(1-w*a1)); % dimension less function (dimensionless)%

R1 = 0.01212882; %% The radius of the inner sphere (m)%%

R2 = 0.11; %% The radius of out sphere Table 3.(m) %% mass = 6; %% mass of UO2 vapor grams (g) %% hfg = 516382 - 22.946*Ts1 ; %% The latent of heat of condensation (J/mole) %%

M = 270.03; %% The molecular weight of UO2 (g/mole) %% rho1s = 0; rho1d = 0.72; %% diffuse reflectivity of UO2 (dimensionless)%% rho2s = 0 ; rho2d = 0.80; %% diffuse reflectivity of Na (dimensionless) %%

Ebcilon1 = 1- rho1s-rho1d; %% emissivity at UO2 surface (dimensionless)%%

Ebcilon2 = 1- rho2s - rho2d ; %% emissivity at Na surface (dimensionless)%%

F1 = ( rho1s + rho1d -1)/4; % dimensionless %

F2 = ( rho2s + rho2d -1)/4; % dimensionless %

E1 = ( 1 + rho1s + rho1d )/[6*(1-w*a1)];% dimensionless %

E2 = (1 + rho2s + rho2d )/[6*(1-w*a1)]; % dimensionless %

114

tau1 = R1* B; %%The optical thickness at the inner sphere (dimensionless)%% tau2 = R2 * B; %% The optical thickness at the outer sphere(dimensionless)%% eta1 = b * tau1; %dimension less function (dimensionless) % eta2 = b * tau2; %dimension less function (dimensionless) %

A6 = sigma*n^2*(Ebcilon2*Ts2^4+4*F2*T^4); % W/m2 %

A3 = sigma*n^2*(Ebcilon1*Ts1^4+4*F1*T^4); % W/m2 %

A1 = [E1/tau1^1.5-F1/tau1^0.5]*besseli(0.5,eta1)-[E1*b/tau1^0.5]*besseli(-0.5,eta1); %dimensionless%

A2 = [E1/tau1^1.5-F1/tau1^0.5]*besselk(0.5,eta1)+[E1*b/tau1^0.5]*besselk(-0.5,eta1); %dimensionless%

A4 = -[ E2/tau2^1.5+ F2/tau2^0.5]*besseli(0.5,eta2)+[E2*b/tau2^0.5]*besseli(- 0.5,eta2);%dimensionless%

A5 = -[ E2/tau2^1.5 + F2/tau2^0.5]*besselk(0.5,eta2)-[E2*b/tau2^2]*b*besselk(- 0.5,eta2);%dimensionless%

C1 = (A3*A5-A2*A6)/(A1*A5-A2*A4); %% Cramer's rule (W/m2)%%

C2 = (A1*A6-A3*A4)/(A1*A5-A2*A4); %% Cramer's rule (W/m2)%% delta = (tau2-tau1)/100; %% number of shells (dimensionless)%% tau5 = tau1:delta:tau2; % dimensionless % yy = b* tau5; % dimensionless %

U30 = (- b*besseli(-0.5,yy))./tau5.^(0.5)+ besseli(0.5,yy)./tau5.^(1.5);%dimensionless%

U40 = ( b*besselk(-0.5,yy))./tau5.^(0.5)+ besselk(0.5,yy)./tau5.^(1.5) ;%dimensionless% denominator =3*(1-w*a1); %dimensionless% qrw = ( C1*U30 + C2*U40)/denominator; %% Equation 42 Radiative Heat Flux (W/m2)%%

QR = ((4*pi)*(tau5).*(tau5).*qrw)/(B^2); %% HEAT RATE (W)%%

G = (C1*besseli(0.5,yy))./tau5.^(0.5)+ (C2*besselk(0.5,yy))./tau5.^(0.5)+ 4*n^2*sigma*T^4;%% Irradiance (W/m2) %%

XXX= (tau5-tau1)/(tau2-tau1);

for kk = 1:length(tau5) %% Numerical Integration of The Heat Generation %%

115

EE(kk) = ((C1*besseli(0.5,b*tau5(kk)))*tau5(kk)^(3/2)) + ((C2*besselk(0.5,b*tau5(kk)))*tau5(kk)^(3/2));

end

% Qgeneration = (tau2-tau1)*trapz(EE) % TRAPZ FUNCTION %

Qgeneration = trapz(tau5,EE) ; % TRAPZ FUNCTION %

Qgen = ((- 4* pi *Ka *Qgeneration) /B^3) %% watts%%

Est = QR(1)+ Qgen +QR(end); % ENERGY BLANCE (W)%

Hfg = ( mass*hfg)/(M); %% The latent of heat of condensation (Jouls)%%

Bubbletimecollapse = 1000*(Hfg)/(Est) %% Bubble Time Collapse in milli Seconds %%

B-2 Bubble Collapse Time for Fast Tests

clear all;close all;clc,

Twall = 811; %% Sodium Temperature K %%%

R = 8.314;%% The gas constant (J/K.mole)%%

Xa = 0.885;%% The mole fraction of UO2 vapor in the bubble mixture (dimensionless)%%

MUO2 = 270.03;%% The molecular mass of UO2 (g/mol)%%

Tsat = 4000;%% The saturation temperature of UO2 vapor (K) %%

K = 8.314; %%The gas constant (J/K.mole)%%

P2 = 0.2655E6;%% The initial pressure of UO2 vapor (Pa)%%

T2 = 4050;%% The initial temperature of UO2 vapor (K)%%

C0 = P2/(Xa*R*T2);%% The molecular mass of UO2 (g/mol)%%

Xsat = P2/(C0*R*Tsat);%%The saturation molar fraction of UO2 (dimensionless) %%

Cp1 = 20.02; %% The specific heat at constant pressure of UO2 vapor (J/mol. K)%%

Cp2 = 12.55; %% The specific heat at constant pressure of Xenon gas (J/mol. K)%%

Cp3 = 135.56; %% The specific heat at constant pressure of FOG (J/mol. K)%%

116

RO = 0.1; %% The sphere radius (m) %%CC

Ka = 10 ; %% The absorption coefficient of UO2 (m^-1)%% tauz = RO*Ka ; %% The optical thickness (dimensionless) %% sigma = 5.67e-8; %%Boltzman constant (W/m^2. K^4) %%

Tinf = T2;%%The initial temperature of UO2 vapor (K)%% theta_f = Tsat/Tinf;% Dimensionless temperature distribution %% theta_sat = Tsat/Tinf;%% Dimensionless temperature distribution %% theta_wall = Twall/Tinf; %% Dimensionless temperature distribution %%

Enthalpy = 516382 - 22.946*T2 ;%% Enthalpy of vaporization (J/mol)%%

Gamma = Enthalpy;%% Enthalpy of vaporization (J/mol)%% ttt= 3.5e-3; %% Time in Seconds %% centervalue = 0.01 ;

T1 = linspace(3300,4050,2); for kk=1:length(T1)

P1(kk) = P2*exp( Enthalpy/K * (1/T2-1/T1(kk)));

g0(kk) = P1(kk)/(C0*R*T1(kk));

X1(kk) = g0(kk)/(1-Xsat+g0(kk));

X3(kk) = (Xsat-g0(kk))/(1-Xsat+g0(kk));

Cp(kk) = Cp2 + ( g0(kk)*(Cp1-Cp2)+(Xsat-g0(kk))*Cp3 )/(1-Xsat+g0(kk));

C(kk) = C0*(1-Xsat+g0(kk));

P1p(kk) = P2*Enthalpy/(K*T1(kk)^2)*exp(Enthalpy/K*(1/T2-1/T1(kk)));

g0p(kk) = (T1(kk)*P1p(kk)-P1(kk))/(C0*R*T1(kk)^2);

X1p(kk) = ((1-Xsat)*g0p(kk))/(1-Xsat+g0(kk))^2;

theta(kk) = T1(kk)/Tinf;

gamma(kk) = Gamma*X1p(kk)/(Cp(kk)*(1-X1(kk)));

eta = C.*Cp./(C(end)*Cp(end));

yu = C(end)*Cp(end);

Beta = (2*sigma*Tinf^3*Ka)/yu;

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end;

tau1= centervalue:0.01:tauz; %% from the center of the sphere to the walll %%

tau1r = tauz-tau1;

E1=expint(tau1r); %% The Exponential Integrals of tau1 %%

E2=exp(-tau1r)-tau1r.*E1; %% The Exponential Integrals of tau1 %%

E2(end)= 1;

y2 = (1/3)*(theta_sat.^(-3)-1);

y3 = Beta.*ttt*E2; %% solution of CCC in fog region %%

y4 = trapz((eta.*(1+gamma))./(theta.^4)); %% solution of CCC in fog region %%

yratio = (y3-y2)./y4; % solution of CCC in fog region, Nt == NN-1 %%

QQ = (0.987654321-yratio); %% solution of CCC in fog region %%

TFOG =4050*QQ; %% solution of CCC in fog region %%

TempF = 4050*QQ ; %% Temperature distribution of fog region in CCC's solution%%

tau2 =centervalue:0.01:tauz;%% from the center of the sphere to the walll %%

tau2r = tauz-tau2;

E1ff=expint(tau2r); %% The Exponential Integrals of tau1 %%

E2ff=exp(-tau2r)-tau2r.*E1ff; %% The Exponential Integrals of tau1 %%

E2ff(end)=1;

T_1 = 4050./(1+(3*Beta.*E2ff*ttt)).^(1/3); %% Temperature distribution of fog free region in CCC's solution%%

Temp =T_1 ;

Tempe = TempF;

HH = Temp;

GG = Tempe;

QQQ = 1./(1+(3*Beta.*E2ff*ttt)).^(1/3);

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for jj =1:100 %% This loop is to generate the temperature distribution of fog free region in Spherical Extension of CCC's solution%%

Temp = HH;

for kkK=1:length(Temp)

P1(kkK) = P2*exp( Enthalpy/K * (1/T2-1/Temp(kkK)));

g0(kkK) = P1(kkK)/(C0*R*Temp(kkK));

X1(kkK) = g0(kkK)/(1-Xsat+g0(kkK));

X3(kkK) = (Xsat-g0(kkK))/(1-Xsat+g0(kkK));

Cpp(kkK) = Cp2 + ( g0(kkK)*(Cp1-Cp2)+(Xsat-g0(kkK))*Cp3 )/(1-Xsat+g0(kkK));

CC(kkK) = C0*(1-Xsat+g0(kkK));

theta1(kkK) = Temp(kkK)/Tinf;

AAA(kkK)= CC(kkK)*Cpp(kkK);

Beta1(kkK) = (2*sigma*Tinf^3*Ka)./AAA(kkK);

end;

THETA1(:,jj)=theta1'; %% The dimension less temperature distributions in fog free region RESULTS from of CCC's solution %%

% the first column of THETA1 is QQQ then new theta1 untill converge %

DELTA = 0.01; %% DELTA is used in trapezoid method %%

tau4 = linspace(centervalue,tauz,length(tau2)); %% from the center of the sphere to the wall%

tau4r =tauz-tau4;

tau4r(end)= 0.01;

eee1=expint(tau4); %% The Exponential Integrals E1 %%

eee2=exp(-tau4)-tau4.*eee1; %% The Exponential Integrals E2 %%

eee2(end)= 1;

EE1=expint(tau4r); %% The Exponential Integrals E1 %%

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EE2=exp(-tau4r)-tau4r.*EE1; %% The Exponential Integrals E2 %%

EE2(end)= 1;

EE3=(exp(-tau4r)-tau4r.*EE2)/2; %% The Exponential Integrals E2%%

Y =abs(ttt-tau4r);

for u = 1: length(Y)

if Y(u)==0

EY2(u)=1;

EY1(u)=13.5;

disp('Y==0')

else

EY1(u)=expint(Y(u)); %% The Exponential Integrals of abs(Y) %%

EY2(u)=exp(-Y(u))-Y(u).*EY1(u); %% The Exponential Integrals of abs(Y) %%

end

term1 = theta_wall^4.*EE3;

term0 = 0; %% SYMMETRIC BOUNDARY CONDITION %

if jj==1

case1 = trapz(tau2r,(tau2r-ttt)/abs(tau2r-ttt).*QQQ.^(4).*EY2) %% Three methods, method 1 %% all the same

case2 = trapz(DELTA,(tau2r-ttt)/abs(tau2r-ttt).*QQQ.^(4).*EY2) %% Three methods, method 2 %%

term2= DELTA*trapz((tau2r-ttt)/abs(tau2r-ttt).*QQQ.^(4).*EY2) %% Three methods, method 3 %%

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else

thetaOLD = THETA1(:,jj-1)';

term2= DELTA*trapz((tau4r-ttt)/abs(tau4r-ttt).*thetaOLD.^(4).*EY2);%% the old theta1 %% Question 1

end

thetnew = THETA1(:,jj)';

CONSTANT = 2*(term1+term2); %% Numerical Integration of B (C1-constant) solution in fog region %%

Csat = 2*(term1+DELTA*trapz((tau2r-ttt)/abs(tau2r-ttt).*theta_sat.^(4).*EY2));

BM = trapz(1./(tau4r.*thetnew.^(4).*EE2 + CONSTANT ));%)% Three methods, method 3 %%

FA = (ttt*Beta1)./tau4r;

TEFF = (1- FA./BM); % theta

HH = 4050*(1- (FA./BM));

HHH(:,jj) = HH';

UUU(:,jj) = TEFF';

CCC(:,jj) = CONSTANT';

if jj~=1

Error = (HHH(:,jj)-HHH(:,jj-1))/HHH(:,jj-1);

if abs(Error) < 0.0005

disp('Converge')

break

end

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end

BBB =1000000*abs(HH-4000);

Mini = min(BBB);

x=find(BBB==Mini); % tauf %

TAU4 = tau4(1:x); % Fog Free Region from tauf to tauo %

TAU4r = tau4r(1:x);

hh = HH(1:x); % Temperature Distribution in Fog Free Region K %

teff = TEFF(1:x); % Dimension Less Temperature Distribution in Fog Free Region %

constant = CONSTANT(1:x); % C = 2{ ?_w^4 E_3 (?)+?_0^(?_o)??sgn(?-t) ??_?^4 E?_2 (|?-t|)dt?} %

ee2 = EE2(1:x);

ey2 = EY2(1:x);

BM1 = trapz(1./(TAU4r(end).*theta_sat.^(4).*ee2(end) +Csat ));

beta1 = Beta1(x:end);

eeeff = eee2(1:x);

tf = (TAU4r(end)./beta1(end)).*(DELTA).*BM1*1000;

QFF = HHH(:,end); %% The Exact Temperature Distribution in Fog free region %%

SSS = UUU(:,end); % The Exact Dimensionless Temperature Distribution in Fog Free Region (theta)%

C2 = CCC(:,end); % The constant %

ylim([3400 4200])

xlim ([0 1])

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for i =1:100

Tempe = GG;

for kkK=1:length(Tempe) %% The Spherical Extension of CCC.%%

P1(kkK) = P2*exp( Enthalpy/K * (1/T2-1/Tempe(kkK))); %% Clasius Clapeyron Equation %%

g0(kkK) = P1(kkK)/(C0*R*Tempe(kkK));

X1(kkK) = g0(kkK)/(1-Xsat+g0(kkK));

X3(kkK) = (Xsat-g0(kkK))/(1-Xsat+g0(kkK));

C3(kkK) = C0*(Xsat - g0(kkK));

Cpp(kkK) = Cp2 + ( g0(kkK)*(Cp1-Cp2)+(Xsat-g0(kkK))*Cp3 )/(1-Xsat+g0(kkK));

CC(kkK) = C0*(1-Xsat+g0(kkK));

P1p(kkK) = P2*Enthalpy/(K*Tempe(kkK)^2)*exp(Enthalpy/K*(1/T2- 1/Tempe(kkK)));

g0p(kkK) = (Tempe(kkK)*P1p(kkK)-P1(kkK))/(C0*R*Tempe(kkK)^2);

X1p(kkK) = ((1-Xsat)*g0p(kkK))/(1-Xsat+g0(kkK))^2;

X3p(kkK) = (- g0p(kkK))/(1-Xsat+g0(kkK))^2;

theta2(kkK) = Tempe(kkK)/Tinf;

gamma2(kkK) = Enthalpy*X1p(kkK)/(Cpp(kkK)*(1-X1(kkK)));

eta2 = CC.*Cpp./(CC(end)*Cpp(end));

DE(kkK)= CC(end)*Cpp(end);

Beta2(kkK) = (2*sigma*Tinf^3*Ka)./DE(kkK);

end;

THETA2(:,i)=theta2'; %% %% The dimensionless temperature distributions in fog region RESULTS from of CCC's solution %%

GAMMA2(:,i)=gamma2'; % The first column of THETA is QQ %

ETA2(:,i)=eta2';

DELTA1 = 0.01; %% used in trapz method %% tau1 = linspace(0,10,101) %%

tau3 = linspace(centervalue,tauz,length(tau1));

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tau3r = tauz-tau3;

tau3r(end) = 0.01;

EES11=expint(tau3r); %% The Exponential Integrals E1 %%

EES22=exp(-tau3r)-tau3r.*EES11; %% The Exponential Integrals E2 %%

EES22(end)= 1;

EES33=(exp(-tau3r)-tau3r.*EES22)/2;

EEE1=expint(tau3r); %% The Exponential Integrals %%

EEE2=exp(-tau3r)-tau3r.*EEE1; %% The Exponential Integrals %%

EEE2(1) = 1;

EEE3=(exp(-tau3r)-tau3r.*EEE2)/2; %% The Exponential Integrals %%

W =abs(ttt-tau3r);

for uu = 1: length(W)

if W(uu)==0

EEY2(uu)=1;

EEY1(uu)=13;

disp('W==0')

else

EEY1(uu)=expint(W(uu)); %% The Exponential Integrals of abs(Y) %%

EEY2(uu)=exp(-W(uu))-W(uu).*EEY1(uu); %% The Exponential Integrals of abs(Y) %%

end

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if i==1

term222 = trapz(tau1r,(tau1r-ttt)/abs(tau1r-ttt).*QQ.^(4).*EY2); %% Three methods, method 1 %% all the same

term22= trapz(DELTA1,(tau1r-ttt)/abs(tau1r-ttt).*QQ.^(4).*EY2);%% Three methods, method 2 %%

termf2= DELTA1*trapz((tau1r-ttt)/abs(tau1r-ttt).*QQ.^(4).*EEY2);%% Three methods, method 3 %%

else

thetaOLD2 = THETA2(:,i-1)';

termf2= DELTA1*trapz((tau3r-ttt)/abs(tau3r- ttt).*thetaOLD2.^(4).*EEY2);%% Three methods, method 3 %%

end

thetnew2 = THETA2(:,i)';

etanew = ETA2(:,i)';

gammanew = GAMMA2(:,i)';

CONSTANT1 = 2*(term1+termf2); %% Numerical Integration of B (C1- constant) solution in fog region %%

Z1 = (Beta2*ttt)./tau3r; %% solution of CCC in fog region %%

Z2 = (DELTA).*BM1; % ONE VALUE %

Z3 = trapz((etanew.*(1+gammanew))./((tau3r.*thetnew2.^(4).*EEE2 + CONSTANT1 ))); %% solution of CCC in fog region %%

ZZZ =(Z1-Z2)./Z3;

GG = (0.987654321-ZZZ)*4050;

GG(GG > 4000)=4000;

seta = (0.987654321-ZZZ);

GGG(:,i) = GG';

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if i~=1

Error = (GGG(:,i)-GGG(:,i-1))/GGG(:,i-1);

if abs(Error) < 0.0005

disp('Converge')

break

end

end end

%%%

BBBB = 1000000*abs(tau3-TAU4(end));

Mini1 = min(BBBB); x1=find(BBBB==Mini1);

TAU3 = tau3(x1:end); % MASS %

TAU3r = tau3r(x:end); gg = GG(x1:end); % this will u used for mass %

SETA = seta(x1:end); constant1 = CONSTANT1(x1:end); e2 = EEE2(x1:end); eeef =eee2(x1:end);

FS =10;LW=2.2;

axis_FS = FS;

p1 = plot(TAU3,gg,'blue','LineWidth',LW); hold on

p2 = plot(TAU4,hh,'red','LineWidth',LW);

V = [p1(1);p2];

legend(V,{'Fog Region','Fog Free Region'},'Fontsize',FS,'FontWeight','bold');

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grid

set(gca,'fontsize',axis_FS,'FontWeight','bold')

xlabel('\tau','Fontsize',FS,'FontWeight','bold')

ylabel('Temp. K','Fontsize',FS,'FontWeight','bold')

title(['Transient Temperature Profile at t = ',num2str(ttt*1000) ' ms and k = 10 m^{- 1}.'])

Tempp = 4050*SETA ; %% Temperature distribution range from Twall to Tsat %%

for kkK=1:length(Tempp)

P1(kkK) = P2*exp( Enthalpy/K * (1/T2-1/Tempp(kkK))); %% Clasius Clapeyron Equation %%

g0(kkK) = P1(kkK)/(C0*R*Tempp(kkK));

X1(kkK) = g0(kkK)/(1-Xsat+g0(kkK));

X3(kkK) = (Xsat-g0(kkK))/(1-Xsat+g0(kkK));

C33(kkK) = C0*(Xsat - g0(kkK));

end;

MASSUO1 = ((4*pi*MUO2)/Ka^3)*DELTA1*trapz(C33.*TAU3.^2)

MASSUO2 = ((4*pi*MUO2)/Ka^3)*trapz(DELTA1, C33.*TAU3.^2)

MASSUO3 = ((4*pi*MUO2)/Ka^3)*trapz(TAU3, C33.*TAU3.^2)

B-3 The Absorption Coefficient for Small Particles

clear all;close all;clc,

m = 2.51 - 0.076i; %% The complex refractive index of UO2 at lamda = 515 nm%%

% m = 2.42 - 0.009i; %% The complex refractive index of UO2 at lamda = 600 nm %%

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z = m^2-1; y = m^2+2 ;

% x = 0.1178; x = 0.07853981634; %% Size parameter based on dg = 0.015 micrometer %%

% mass = 0.001; %% The amount of UO2 was 1 kg in CDV- argon tests [50]%% mass = 78.0433526 %% The amount of UO2 was in reference [52] FAUSKE%% d = 8125.7; %% The liquid density of UO2 is 8125.7 Kg/m3 %%

R = 1.72212; % The bubble Radius m %

Lamda = 0.4e-6; % The thermal wavelength of UO2 was 0.4 µm %

BUBVOL = (4/3)*pi*R^3 ; % The bubble volume of the bubble %

W = z/y;

Magnitude = abs(W);

BB = Magnitude^2;

Qsca = (8/3)*BB * x^4 % The scattering efficiency %

Qabs = -4* imag(W)* x % The absorption efficiency %

VO = mass/d ; % The total volume of the UO2 particles % fv = VO/BUBVOL % The volume fraction ratio %

K = - imag(W)*(6*pi*fv)/(Lamda) %The absorption coefficient for small particles %

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APPENDIX C

Mie Theory

C-1 Introduction

The Mie theory will be employed to ascertain the amount of energy that is lost due to the presence of particles within the medium. The magnitude of the energy reduction can be determined by computing the scattering and absorbing coefficients of the participating medium. Thermal waves, consisting of photons, initially are travelling through the medium at the speed of light. However, the photons interact with particles or impurities within the medium resulting in a slowing of its speed. The interaction between photons and particles redirects the radiation into a direction close to or in the same direction of the incident angle on the particle surface. The loss of thermal energy could happen by absorption of the energy, the subsequent conversion to another form such as heat, as well as by scattering out from the direction of interest. In addition, the loss of thermal energy could result from the reflection and diffraction. Also, there exists a small space void of particles where photons may pass through without any collisions or resistance [63][30].

C-2 Rayleigh, Mie, and Optical Scattering

There are three types of scattering: Rayleigh, Mie, and optical scattering.

Rayleigh scattering occurs when the particle size (r) is less than 1/10 λ of the photon wavelength. Mie scattering region is in the range from 1/10 λ of particle size to less than

λ, and the optical scattering region is located when the particle size is larger than the photon wavelength. The parameters affecting the Mie theory are: first, the size of the radius dimension of the particles, where the dimension size x=k r, r is the radius of the sphere, and k is the free–space propagation constant such that k=2π/λ. Here λ is the wave

129

length of the incident radiation. The second parameter is the scattering angle between the incident radiation and the forward scattering radiation. Lastly, Mie theory and

Classical theory are provided in Appendix C in more details [30] [26].

In spherical coordinates system at a point P (r, θ, ∅) on homogenous sphere with refractive index (m) in vacuum medium (m = 1), the scalar wave equation is given by

∆훹 + k2푚훹 = 0 (C.1)

푥 = 푟푐표푠∅푠푖푛θ (C.2)

푦 = 푟푠푖푛∅푠푖푛θ (C.3)

푧 = 푟푐표푠θ (C.4)

Where θ and ∅ are the polar and azimuth angles of scattered radiation.

The formal solution of the scalar wave equation is:

cos 푙∅ 푙 훹𝑖푛 = ⟨sin 푙∅⟩푃푛(푐표푠θ)푍푛(푚kr) (C.5)

푛 ≥ 푙 ≥ 0 and 푛 and 푙 are integers

The first term of Equation (5-C) is sine or cosine; the second term is a Legendre

Polynomial and the third term is Spherical Bessel Function [35].

The outside (scattered) wave is given by the following equation:

2푛+1 푢 = 푒𝑖휔푡 cos 휑 ∑∞ −푎 (−푖)푛 푃1(cos 휃)ℎ2(푘푟) (C.6) 푛=1 푛 푛(푛+1) 푛 푛

2푛+1 푣 = 푒𝑖휔푡 sin 휑 ∑∞ 푚푑 (−푖)푛 푃1(cos 휃)푗 (푚푘푟) (C.7) 푛=1 푛 푛(푛+1) 푛 푛

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The inside wave is given by the following equation:

2푛+1 푢 = 푒𝑖휔푡 cos 휑 ∑∞ 푚푐 (−푖)푛 푃1(cos 휃)푗 (푚푘푟) (C.8) 푛=1 푛 푛(푛+1) 푛 푛

2푛+1 푣 = 푒𝑖휔푡 sin 휑 ∑∞ 푚푑 (−푖)푛 푃1(cos 휃)푗 (푚푘푟) (C.9) 푛=1 푛 푛(푛+1) 푛 푛

Where 휔 is the angular frequency in 푠−1and t is the time in 푠. The coefficients

푎푛, 푏푛, 푐푛, 푎푛푑 푑푛 are the Mie’s coefficients.

Since the electric field vector and magnetic field vector satisfy the vector wave equation, and the wave inside the sphere is equal to the wave outside the sphere, the magnitude of Mie’s coefficients was computed [35].

The parameters an and bn are Mie’s coefficients, and it can be calculated from the following equations:

′ 훹푛(푦) ′ 훹푛(푥)[ ]−푚훹푛(푥) 훹푛(푦) 푎푛 = ′ (C.10) 훹푛(푦) ′ 휉푛(푥)[ ]−푚휉푛(푦) 훹푛(푦)

′ 훹푛(푦) ′ 푚훹푛(푥)[ ]−훹푛(푥) 훹푛(푦) 푏푛 = ′ (C.11) 훹푛(푦) ′ 푚휉푛(푥)[ ]−휉푛(푦) 훹푛(푦)

Where 훹푛(푥), 휉푛(푥) are the Riccati- Bessel functions which are functions of first

1 kind Bessel function order (푛 + 2) and 푦 = 푚푘 푟.

1 휋푧 훹 (푧) = ( )2 퐽(푧) (C.12) 푛 2 푛+1/2

1 휋푧 푛 ( ) 2 1 휉푛 푧 = ( ) 퐽(푧)푛+ + (−1) 푖 퐽(푧)−푛−1/2 , 푖 = √−1 (C.13) 2 2

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MATLAB function has been used to compute: the matrix of Mie coefficients by using mie_ab (m, x) function [37]. Finally, by using this solution the scattering field distributions inside and outside the sphere can be determined, and the phase function can be computed [35].

C-3 Mie Efficiencies and Cross Sections

C Scattering cross section s is the ratio of the rate of energy scattered by the sphere to the incident energy flow rate per unit area. Absorption cross section C is the a ratio of the rate of energy absorbed by the sphere to the incident energy flow rate per

C unit area. Extinction cross section e, which is the sum of the scattering cross section and

C = C + C the absorption cross section. e s a where the cross section has a cross section units.

퐶 Efficiency factor for absorption 푄 = 푎 (C.14) 푎 휋푟2

퐶 Efficiency factor for scattering 푄 = 푠 (C.15) 푠 휋푟2

The efficiency factor for extinction is obtained by the following equation

퐶 푄 = 푒 (C.16) 푒 휋푟2

푄푒 = 푄푎 + 푄푠 . [26] (C.17)

The scattering and extinction efficiencies can be calculated from the following equations [26, 35]. can be determined by computing the scattering and absorbing coefficients of the participating medium.

2 푄 = ∑∞ (2푛 + 1) (|푎 |2 + |푏 |2) (C.18) 푠 푥2 푛=1 푛 푛

132

2 푄 = ∑∞ (2푛 + 1) 푅푒(푎 + 푏 ) (C.19) 푒 푥2 푛=1 푛 푛

Bohren and Huffman (1983) [53] indicated that the series can be truncated after

1/3 nmax, where 푛푚푎푥 = 푥 + 4푥 + 2 .

MATLAB function has been used to compute scattering, extinction and the absorption efficiencies by using mie (m, x) function [41].

C-4 Angular Scattering

In the spherical coordinate system, the scattering far field (퐸푠휃, 퐸푠휑) for a unit - amplitude incident field at a scattering angle is given by: [33] [43]

푒푖푘푟 퐸 = sin 휑 푆 (cos 휃) (C.20) 푆휑 𝑖푘푟 1

푒푖푘푟 퐸 = cos 휑 푆 (cos 휃) (C.21) 푆휃 −𝑖푘푟 2

Where 푆1(cos 휃) and 푆2(cos 휃) are the scattering functions or the scattering amplitudes, and given by

2푛+1 푆 (cos 휃) = ∑∞ (푎 휋 + 푏 휏 ) (C.22) 1 푛=1 푛(푛+1) 푛 푛 푛 푛

2푛+1 푆 (cos 휃) = ∑∞ (푎 휏 + 푏 휋 ) (C.23) 2 푛=1 푛(푛+1) 푛 푛 푛 푛

2푛−1 푛 Where: 휋 = cos 휃 휋 − 휋 푛 푛−1 푛−1 푛−1 푛−2

휏푛 = 푛 cos 휃 휋푛 − (푛 + 1) 휋푛−1[59]

Since the scattering functions are known from Equations (22-C) and (23-C), the scattered power can be computed from: [43]

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|푆 |2 푆 = 1 (C.24) 푅 휋푥2

|푆 |2 푆 = 2 (C.25) 퐿 휋푥2

푆푅 and 푆퐿 are the polarization perpendicular component and the polarization parallel component. The scattering efficiency can be calculated from the summation of Equation

(23) and (24). 푆 = 푆푅 + 푆퐿 (C.26)

Where 푆푅 and 푆퐿 are functions of 푆1(cos 휃) and 푆2(cos 휃). MATLAB function has been used to compute: the computation and plot of Mie power scattering and diffraction functions by using (mie_tetascanall (m, x, 120, 8,'lin’)) [39].

C-5 Phase Function

The phase function is used to derive the probability of the scattered

direction where the direction of the incident light is defined as (휃′, ∅′), and the

scattered direction is defined by (θ, ∅) within solid angle Ω [30].

The scattering angle θ can be calculated from.

휇 = 휇휇′ + √(1 − 휇)2(1 − 휇′)2 cos ∅ (C.27)

μ^' and μ are the cosine angles before and after the scattering where ∅ is the azimuth angle, and the phase function can be computed from the following equations.

2휋 푝(휇) = 푆(휇) (C.28) 푄푠푐푎

2휋 푝(휇) = 푆(휇) (C.29) 푄푒푥푡

The asymmetry parameter 푎𝑖 is given by

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1 1 푎 = ∫ 푃 (휇) 푝(휇)푑휇 (C.30) 𝑖 2 −1 𝑖

푃𝑖(휇) is a Legendre Polynomial.

푝(휇) is the phase function.

The asymmetry parameter 푎𝑖 represents the strength of the scattered in radiation, and its magnitude dependent on the type of scattering being studied; for example, Mie scattering type has a higher value of 푎𝑖 compared with Rayleigh scattering type.

MATLAB function has been used to compute: the coefficients of Legendre polynomials of phase function by using mie_gi (m, x,1,12) [40]. These MATLAB functions were modeled by C. Mätzler, and he computed the phase function accordioning to the scattered power 푆푅푎푛푑 푆퐿 as it explained previously.

C-6 Absorption and Scattering Coefficients

ÖZISIK had obtained that when the radiation propagates in the medium contains spherical particles with the same composition and uniform size, the absorption and scattering coefficients can be computed from the following equations:

2 푘 = 퐶푎푁 = 휋푟 푄푎푁 (C.31)

2 휎 = 퐶푠푁 = 휋푟 푄푠푁 (C.32)

Where:

푘 is the absorption coefficient.

휎 is the scattering coefficient.

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푁 is the number of particles per unit volume (number of particles/m3).

C-7 Nucleation Rate

From the reference [50] K. Chen et al. had reported in his paper that the homogeneous nucleation started at time t = 1.94 μs and ends at 2.8 μs, and the nucleation rate can be computed from the Becker and Döring formula based on the classic theory.

푡 푁 = ∫ 2 퐽 푑푡 (C.32) 푡1

1 P (T ) 2 2 σ m ⁄2 S2 4π σ r∗2 J = [ sat v ] ( ) exp (− ) (C.33) k Tv π ρl 3 k Tv

2 σ r∗ = (C.34) ρl Rv Tv lnS

J Nucleation rate (stable nuclei created/ m3. s).

σ Surface tension (N/m). m Mass per molecule (kg). k Boltzmann constant (1.38 × 10 -23 J/K).

ρl Liquid density (kg/m3).

r∗ Critical radius (m).

Tv Vapor temperature (K).

Psat(Tv) Saturation pressure at Tv (Pa).

An Excel program was created to compute the nucleation rate per unit volume by using The Becker and Döring formula (Equation C.33). The initial temperature starts at

4500 K and decrease until 3000 K is reached. A total of 1500 steps was calculated with

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one Kelvin decrement. At each step the nucleation rate (J) was calculated according to

Equation (33-C). Once the nucleation rate is calculated, the number of particles per unit

volume in each step (N) can be computed by multiplying the nucleation rate in each step

(J) by ∆푡 time step in seconds. The time step was 57.333 ns which equals to the nucleation

time as per reference [50] over the total number of steps. At the same time, the

absorption and scattering coefficients were calculated at each time step by using

Equation (C.31) and Equation (C.32).

Using reference [56] the absorption coefficients was 0.082 m-1 and the same

number is used in this calculation. The Excel program run until it reached the absorption

coefficient of 0.082 m-1 as in [56]. At this time step, the number of particles per unit

volume (N) can be computed from Equation (C.31) and as a result the scattering

coefficient (σ) can be evaluated from (C.32) where N is known, and the results are

tabulated in Table (15) and Table (17).

C-8 Angular Scattering Application

To compute the net radiative heat flux and the irradiance, the ai- factors of

Legendre polynomials of phase function should be determined to classify the type of

scattering, and to identify scattering strength and scattering direction. In this research

mie_teta (m, x, 180) [40] function was used to compute ai- factors. In Table (1) asymmetry

parameter a1 and higher ai coefficients for i ≤10, x = 2.42 +0.009i, for variable m of

Uranium dioxide has been calculated. [Christian Mätzler] [37] [38].

Table 7. Asymmetry factors of Legendre polynomials for phase function.

푎1 푎2 푎3 푎4 푎5 푎6 푎7 푎8 푎9 푎10

0.097 0.1053 0.0053 0.004 0.00084 0.0041 0.0009 0.0045 0.001 0.005

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Figure 47 shows the variation of the scattered power (thermal radiation) with the scattering angle. The upper semicircle represents the polarization perpendicular

|푆 |2 component 푆 = 1 , and the lower semicircle represents the polarization parallel 푅 휋푥2

|푆 |2 component 푆 = 2 [6]. In this Figure the scattering forward (0.01604) hemisphere is 퐿 휋푥2 larger than the scattering backward (0.01045) hemisphere.

Figure 48 shows the scattering function 푆 = 푆푅 + 푆퐿 over all scattering directions vs the scattering angle θ, and both figures was generated by this function mie_tetascanall (m, x, 120, 8,'lin’) [39].

Figure 44. Angular Mie scattering diagram [39].

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Figure 45. The scattering function [39].

In the same time the phase function can be computed from a series of Legendre polynomials as that was represented by Li et al’s model [28].

′ ∞ ′ 푝(휇, μ ) = ∑푛=0(2푛 + 1)푎푛푃푛(휇)푃푛(μ ) (C.35)

푎푛 factor depending on the scattering model.

′ 푃푛(μ )푃푛(휇) Legendre polynomials of the cosine angles before and after scattering.

Figure 46. The phase function at 휃′ = 0°.

Figure 44. shows the comparison between the phase function of Li et al.’s model and

C. Mätzler’s model. Li et al.’s model called linear -anisotropic scattering which used two

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terms to from the series of Legendre polynomials to quantify the phase function, but

Mätzler’s model used mie_teta (m, x, 180) [39] function to compute ai- factors for computing the phase function. In Figure 49. ten terms were used to compute the phase function of C. Mätzler’s model.

In Figure 44. both curves were expressed at incident angle 휃′= 0° with deviation of 101.29 %, but Figure 45. shows the phase function at incident angle 휃′= 180° with deviation of 103.95 %. From Figure 44. and Figure 45. one can conclude that, the phase function represented by Mätzler’s model is greater than that represented by Li et al.’s model for all scattering angle values especially at 휃′= 180°. Figure 46. shows the angular scattering diagram of phase function for non-linear anisotropic model and linear anisotropic model at incident angle 휃′= 180°.

Figure 47. The phase function at 휃′ = 180°.

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Figure 48. Angular scattering diagram.

Table 8. Mie model inputs.

D µm x Qabs Qsca Qext 0.09 0.707 0.0125 0.3298 0.3423 0.08 0.628 0.0095 0.1967 0.2062 0.07 0.549 0.0073 0.1102 0.1175

Table 9. Mie model parameters.

D µm N K σ β ω a1 Stable nuclei/m3 m-1 m-1 m-1 0.09 6.6 ×1014 0.069 2.21 2.279 0.9697 0.1676 0.08 1.79 ×1015 0.085 1.77 1.855 0.9541 0.1293 0.07 2.92 ×1015 0.082 1.24 1.322 0.9379 0.0975

C-9 Rayleigh Scattering Model

Table 10. Rayleigh model inputs.

D µm x Qabs Qsca Qext 0.04 0.314 0.0031 0.0105 0.0136 0.03 0.236 0.0022 0.0032 0.0054 0.02 0.157 0.0014 0.00062942 0.002 0.01 0.079 0.00067153 0.000038914 0.00071044

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Table 11. Rayleigh model parameters.

D µm N K σ β ω a1 Stable nuclei/m3 m-1 m-1 m-1 0.04 1.94 ×1016 0.076 0.256 0.332 0.7710 0.0314 0.03 4.83 ×1016 0.075 0.109 0.184 0.5923 0.0176 0.02 1.82 ×1017 0.08 0.036 0.116 0.3741 0.0078 0.01 1.5 ×1018 0.0789 0.00457 0.08247 0.5475 0.002

C-10 Sodium Data

Table 12. Sodium data.

Experimental Covergas Water Maximum Aerosol Mass Sodium NO. Volume Height Radius Detected Temperature m3 m m grams K FAST 104 0.09 1.06 0.1 0.0003 811 FAST 105 0.095 1.04 0.08 0.001 811 FAST 106 0.3 0.24 0.11 0.0002 811 FAST 107 0.32 0.14 0.1 0.00075 811 FAST 108 0.34 0.08 0.08 0.0007 811 FAST 109 0.33 0.1 0.09 0.00045 811 FAST 110 0.32 0.14 0.06 0.0003 811 FAST 113 0.35 0.03 0.13 0.07 811

C-11 Absorption and Scattering Efficiencies In Rayleigh Scattering Model

The absorption and scattering efficiencies of small particles x ≪1 can be

computed from the following equations [31].

2휋푅 The size parameter 푥 = (C.36) 휆

Where R is the particle radius and λ is wavelength.

2 8 푚2−1 푄 = | | 푥4 (C.37) 푠푐푎 3 푚2+2

푚2−1 푄 = −4 퐼푚 { } 푥 ≈ 푄 (C.38) 푒푥푡 푚2+2 푎푏푠

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Table (2-E) provide the absorption and scattering efficiencies at dg = 0.015 µm and d50 = 0.01 µm.

Table 13. Absorption and scattering efficiencies.

x 푄푠푐푎 푄푎푏푠

0.0785 0.000041469 0.0052

0.0785 0.00020987 0.0078

C-12 The Absorption Coefficient In Rayleigh Model

The absorption coefficient for small particle –size distribution can be calculated from [31]:

푚2−1 6휋푓 푘 = −퐼푚 { } 푣 (C.39) 휆 푚2+2 휆

The complex refractive index of UO2 is 2.51 – 0.076 i at λ = 500 µm or 2.42- 0.009 i at λ = 600 µm [65], and the thermal wavelength of UO2 was 0.4 µm [4]. The volume fraction 푓푣 formula is defined as the ratio of resultant particle volume inside the bubble to the total bubble volume.

The resultant particle volume 푓 = (C.40) 푣 bubble volume

푚 휌 The resultant particle volume of UO2 = (C.41) 푏푢푏푏푙푒 푣표푙푢푚푒

The amount of UO2 was 1 gram in many of the CDV- argon tests [50], and for

Rayleigh scattering model the geometric mean diameter dg was selected as 0.015 µm based on the particle size distribution in reference [50]. The liquid density of UO2 is 8125.7

Kg/m3[36]. Based on this information and according to Equation (3-E), the volume fraction

−5 fv is 2.938 × × 10 at R = 0.1 m from FAST 107. [4]

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Upon substituting into Equation (1-E) the absorption coefficient of UO2 can be

calculated.

(2.51 − 0.076푖)2 − 1 6휋 × 2.938 × 10−5 푘 = −퐼푚 [ ] 휆 (2.51 − 0.076푖)2 + 2 0.4 × 10−6

6휋×2.938× 10−5 푘 = −퐼푚 [0.6391 − 0.0166푖] = 22.9853 m-1 휆 0.4× 10−6

Table (1-E) provide the absorption coefficient of UO2 at dg = 0.015 µm and d50 = 0.01 µm

Table 14. The absorption coefficient at different bubble radius.

Maximum Absorption coefficient (k) at Absorption coefficient (k) at Radius (R) m n = 2.42- 0.009i [34] n = 2.51 – 0.076i [64]

m-1 m-1

0.05 23.45 183.8825

0.06 13.57 106.4135

0.08 5.725 44.8932

0.09 4.02 31.5299

0.1 2.9312 22.9853

0.11 2.2 17.2692

0.13 1.3 10.4621

1.72212[52] 177.0247 1388.1

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