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PAGE 1 1 EVALUATION OF MATERIAL RESPONSE TO THERMAL FLASH By TODD ANTHONY MOCK A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DE GREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Todd Anthony Mock PAGE 3 3 To the gorgeous Michelle Ashley Hipps, and my wonderful family, Travis, Ron and Elaine Mock. Particularly to my mother who dragged me kicking and screaming to achieve my educational goals. PAGE 4 4 ACKNOWLEDGMENTS I thank God, through which all things are possible. Furthermore, I thank my advisor Dr. Glenn Sjoden as well as Dr. James Petrosky for the opportunity of a lifetime. r Stachitas for his contributions to this research, as well as all the students at the Florida Institute for Nuclear Detection and Security for their support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 11 C HAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 2 A PRIMER ON POLYMER DEGRADATION ................................ .......................... 16 Discussion Of Arrhenius Parameter s ................................ ................................ ...... 18 Material Behavior ................................ ................................ ................................ .... 19 Derivation Of Simple Degradation Model ................................ ................................ 21 Polyurethane Waste Varnish ................................ ................................ .................. 24 Fitting The Default Model ................................ ................................ ........................ 25 3 SOURCE TERM EFFECTS AND THERMAL RADIATION ................................ ..... 28 Thermal Radiation ................................ ................................ ................................ .. 28 CANYON: 3 D Radiant Heat Transfer ................................ .............................. 30 Surface Re Radiation ................................ ................................ ....................... 31 Blast Wave Effects ................................ ................................ ................................ .. 32 4 MODEL CONCEPTUALIZATION ................................ ................................ ........... 35 Material Prop erties ................................ ................................ ................................ .. 35 Clearcoat (40 50 microns) ................................ ................................ ................ 36 Basecoat/Primer (~20 microns) ................................ ................................ ........ 41 Zinc Phosphate (1 2 microns) ................................ ................................ .......... 42 Other Physical Considerations ................................ ................................ ................ 42 Considerations Including Char ................................ ................................ ......... 43 Boundary Layer Theory ................................ ................................ .................... 45 5 DEFFCON CODE DEVELOPMENT ................................ ................................ ....... 5 2 Solving The Heat Conduction Equ ation ................................ ................................ .. 53 Explicit Finite Difference ................................ ................................ ................... 54 Stability ................................ ................................ ................................ ............. 59 Crank Nicols on ................................ ................................ ................................ 60 Equation Solution Method ................................ ................................ ................ 62 PAGE 6 6 Code Validation ................................ ................................ ................................ ...... 62 Application Of Thermophysical Parameters ................................ ............................ 64 Practical Boundary Layer ................................ ................................ ................. 65 Practical Clearcoat ................................ ................................ ........................... 66 Practical Char ................................ ................................ ................................ ... 67 DEFFCON Setup ................................ ................................ ................................ .... 70 Input File ................................ ................................ ................................ ........... 70 Output Files ................................ ................................ ................................ ...... 74 CANYON In DEFFCON ................................ ................................ .................... 77 6 CASE STUDY AND ANALYSIS ................................ ................................ .............. 82 Atmospheric Re Entry ................................ ................................ ............................. 82 Degradation Differences ................................ ................................ ................... 82 Thermophysical Relations ................................ ................................ ................ 85 Assumptions ................................ ................................ ................................ ..... 86 Erosion Study ................................ ................................ ................................ ... 87 Composite Experimental Parameters ................................ ............................... 88 Surface Temperature Study ................................ ................................ ............. 90 Further Analysis ................................ ................................ ............................... 91 Automotive Paint Studies ................................ ................................ ........................ 94 Basecoat Surface Temperature ................................ ................................ ........ 94 Back Calculation Using CANYON ................................ ................................ .... 97 Procedure ................................ ................................ ................................ ......... 98 Example Using Automotive Paint Damage ................................ ..................... 101 7 CONCLUSIONS AND FUTURE WORK ................................ ............................... 105 APPENDIX: AIR THERMOPHYSICAL PROPERTIES ................................ ................ 107 LIST OF REFERENCES ................................ ................................ ............................. 109 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 112 PAGE 7 7 LIST OF TABLES Table page 2 1 Theory Arrhenius parameters ................................ ................................ ............. 26 2 2 PMMA Arrhenius parameters ................................ ................................ ............. 27 4 1 Material thermophysical properties, [11] [12] [13] ................................ ............ 36 4 2 Constants for thermophysical equations for char ................................ ................ 44 5 1 Beryllium thermophysical values ................................ ................................ ........ 63 5 2 Results of validation study for 40 sec simulation ................................ ................ 64 6 1 C onstants for degradation parameters ................................ ............................... 83 6 2 Parameters for composite erosion and surface temp experiment ....................... 88 6 3 Basecoat and mica thermal properties ................................ ............................... 95 6 4 Thickness of layers in automotive sample ................................ ........................ 102 6 5 Automotive simulation 1000 m from source ................................ ...................... 104 6 6 Automotive simulation 708 m from source ................................ ........................ 104 PAGE 8 8 LIST OF FIGURES Figure page 1 2 Shadowing effects for automobile. ................................ ................................ ...... 14 2 1 Molecular structure of ethylene. ................................ ................................ .......... 16 2 2 Reaction rates ................................ ................................ ................................ .... 17 2 3 Activation energy ................................ ................................ ................................ 19 2 4 Polyurethane waste varnish degradation [6] ................................ ...................... 24 2 5 Theory and experimental fit. ................................ ................................ ............... 26 3 1 Slant distance illustration ................................ ................................ .................... 30 3 2 Damage to automobile from peak overpressure of 5 psi [1] .............................. 33 3 3 Peak overpressure from a 1 kiloton free air burst from sea level ambient conditions [1] ................................ ................................ ................................ ..... 34 4 1 Paint layers and relative dimensions. ................................ ................................ 35 4 2 Production of Nitroxyl radicals [15] ................................ ................................ ..... 37 4 3 Transmittance of HALS at various wavelengths [16] ................................ ......... 38 4 4 Keto enol tautomerism showing UV excitation returning to ground state by release of heat [17] ................................ ................................ ............................. 39 4 5 Absorption in transparent material ................................ ................................ ...... 39 4 6 Thermal and velocity boundary layer ................................ ................................ .. 46 4 7 Numerical solutions for velocity (left) and thermal boundary layer (right). .......... 51 5 1 Vertical orientation of mono layer polymer system ................................ ............. 53 5 2 Nodes with surrounding control volume. ................................ ............................. 54 5 3 Vertical nodes with surrounding control volume ................................ ................. 56 5 4 Excel solver used for Fourier number analysis. ................................ .................. 60 5 5 Central node. ................................ ................................ ................................ ...... 60 5 6 Energy balance on a surface. ................................ ................................ ............. 62 PAGE 9 9 5 7 Temperature profile for beryllium wall with s and cm. .......... 63 5 9 Polymer mass fraction being converted to char. ................................ ................. 68 5 10 Specific heat as a function of ch ar fraction. ................................ ........................ 69 5 11 Thermal conductivity as a function of char. ................................ ........................ 69 5 12 Density as a function of char. ................................ ................................ ............. 70 5 13 DEFFCON input file that is read by the executable ................................ ............ 71 5 14 Zone and material locations ................................ ................................ ................ 72 5 15 ................................ ................................ ..... 76 5 16 DEFFCON a) mass loss output file and b) surface temperature output file ........ 76 5 17 CANYON input file ................................ ................................ .............................. 78 5 18 Snapshot of CANYON output file CANout.put ................................ .................... 79 5 19 Overlay of CANYON output to DEFFCON input fl uxes for 1 kT and 708 m from the source ................................ ................................ ................................ ... 81 6 1 Bahramian experimental and Bahramian theoretical data for composite ............ 84 6 2 DEFFCO N theory overlaid with Bahramian experimental data ........................... 84 6 3 Orientation and dimensions of erosion sample ................................ ................... 88 6 4 DEFFCON results fo r erosion experiment of composite material ....................... 89 6 5 Sample dimensions for surface temperature model ................................ ........... 90 6 6 Surface temperature cal culations results for composite material ........................ 91 6 7 Theoretical fit to literary model thermal conductivity ................................ ........... 92 6 8 Surface temperatur e numerical model comparison with adjusted DEFFCON with 329 W/cm 2 ................................ ................................ ................................ ... 93 6 9 Erosion results for Bahramian numerical model, adjusted DEFFCON with 329 W/cm 2 and original DEFFCON with 799 W/cm 2 (DEFF 100X) ..................... 93 6 10 Basecoast temperature with clearcoat absorption of 0.1 and mica content variation ................................ ................................ ................................ .............. 96 PAGE 10 10 6 11 Basecoast temperature w ith clearcoat absorption of 0.3 and mica content variation ................................ ................................ ................................ .............. 96 6 12 Basecoast temperature with clearcoat absorption of 0.5 and mica content variation ................................ ................................ ................................ .............. 97 6 13 Power fits for yields (points left to right) 1, 5, 10, 15, and 20 kT and distances down a street canyon ................................ ................................ ......................... 98 6 14 Mass loss using output files from CANYON for two p olymers with yields (data points left to right) 1, 5, 10, 15, and 20 kT ................................ ................. 99 6 15 ................................ ............................... 101 6 16 Automotive paint degradation scenario, 2 recievers 708 m and 1000 m from source in a canyon 40 m wide and 100 m tall ................................ ................... 102 6 17 Degradation of automobile 708 m away from source with yields (data points left to right) 1, 5, 10, 15, and 20 kT ................................ ................................ ... 102 6 18 Automotive simulation at various yields and 1000 m away from source ........... 103 PAGE 11 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EVALUATION OF MATERIAL RESPONSE TO THERMAL FLASH By Todd Anthony Mock December 2010 Chair: Glenn Sjoden Ma jor: Nuclear Engineering Sciences Painted surfaces, like most materials, will degrade under adverse conditions. If enough heat is applied to the material, vaporization will occur. The removal of material by this mechanism can be quantified by a mass bal ance of the material before and after a heat fluence is applied. It is the task of this research to determine the heat fluence from a n assumed improvised nuclear device (IND) incident on a material surface that has undergone thermal degradation. We deve loped DEFFCON (Degradation Effects From Flux CONduction) a two dimensional, transient heat transfer algorithm for this research to characterize the mass loss of paint coated system s When used in tande m with another code, CANYON ( Stachitas 2009), which solves the transport of thermal energy through a street canyon, knowing the attributed material mass loss one can directly determine the thermal fluence attributable to a weapon yield. This document analyze s and discusses the implicit and explicit finite differencing methods for the transient heat solver code based on delivery of a thermal pulse to a painted surface, and the specific conditions necessary to afford a solution using DEFFCON. C ode accuracy is validated using analytical solutions to a typical heat PAGE 12 12 transfer problem and further confirmed through analysis based on data from an article by Bahramian, et al. detailing the degradation of an atmospheric re entry heat shield; DEFFCON is able to match th ese detailed literature results with a max absolute relative error of 21% Finally, a procedure for application and candidate values are outlined to correlate a known mass loss on a car surface as a result of IND thermal radiation delivered down a street canyon, to a yield estimate in an urban setting Vaporization physics of polymers for the IND scenario is explained and an account is given of the techniques used to determine at what distances and weapon yields various mass losses to automobile paint sy stems can be expec ted. It is determined that a sample of automotive paint, wit h an aluminum substrate, 0.7 km away from a 1.9 kT nuclear explosion, with a 35% thermal partition, in an average street canyon will lose approximately 5.1 mg/cm 2 of material This mass loss can acceptably be measured using standard laboratory techniques. PAGE 13 13 CHAPTER 1 INTRODUCTION In the event of a domestic nuclear explosion, a system of tasks will be undertaken in order to assess damage and to determine methods used f or the development o f the improvised nuclear device (IND) One such task will be for National Technical Nuclear Forensics (NTNF) to determine the yield of the device A proposed method here is to analyze painted surfaces exposed to the thermal radi ation, such as automobile paint, in order to determine total thermal energy radiated from the icture of the event due to their shape; one side faces the radiation, while the backside is shiel ded from it Radiative heat will propag ate in a specific way in an urban environment and the work by T. Stachitas on this matter is used in this research It can be shown that objects at the forefront of a n uclear d etonation will shadow other objects f Figure 1 1 a and Figure 1 1 b show these shadowing effects. The wooden poles ( Figure 1 1 a) were 1.17 miles from ground zero at Nagasaki and experienced 5 to 6 cal/cm 2 C harring of the upper part of the poles is observed while the lower part of the poles remained undamaged due to shielding from a fence that was later knocked down by the blast wave. In Figure 1 1 b scorching of p aint on a gas container was observed 1.33 miles from ground zero at Hiroshima except where protect ion was provided by the valve [1] This shadowing is an important effect to consider; shadowing by buildings and other objects will play an important role in assessing damage. The intent is to ultimately find an object that has not been shielded by any surrounding objects, i.e. one that has had direct exposure. PAGE 14 14 A ) B ) Figure 1 1 A ) shadowi ng effects on telephone poles (left) and 1b) valve (right) [1] This can be further extrapolated to the scenario involving an automobile ( Figure 1 2 ) Here the circle depicts the thermal flash from a nuclear weapon; region 1 woul d receive the thermal damage, while region 2 would remain un damaged by the thermal radiation due to the shadowing by the overall structure. This is a three dimensional problem; as such varying degrees of damage will appear on locations of the car that fa ce the source. The work presented assumes thermal radiation is n ormal to the surface and represents the worst case scenario for damage. Figure 1 2 S hadowing effects for automobile A co mputer code is developed here for this analysis. The code, DEFFCON (Degradation Effects From Flux CONduction), is a two dimensional transient heat transfer algorithm that uses implicit or explicit finite differencing methods to analyze a PAGE 15 15 set of user input s. The main task of the program is to assess an amount of damage to a multi layer painted surface exposed to an applied time dependent heat flux, including that from a nuclear weapon. Damage is quantified by mass loss through vaporization of the paint po lymer. The secondary task of the DEFFCON program is to enable the user to back calculate a heat flux (if it is initially unknown) from the mass loss. From this calculated flux, the user can then utilize another code developed by T. Stachitas called CANY ON, which can take the flux and then extrapolate a weapon yield in kilotons which also accounts for urban canyon effects ; i.e. the effects of radiative heat energy channeling down a street with structures on e ither side using a gray diffuse transfer model. These gray diffuse surfaces are hypothetical surfaces that emit equally at all wavelengths and emit reflect and absorb diffusely A complete discussion of this type of transfer is given in the work by T. Stachitas. DEFFCON and CANYON will be used to d emonstrate how a calculation to link a weapon yield to a mass loss in paint is carried out. Chapter 2 is a primer for polymer degradation and details degradation mechanisms. Chapter 3 contains a discussion on source term effects and thermal radiation ; a brief account of CANYON is also given. Chapter 4 describes the model concept which includes material specifics and heat transfer mechanisms. Chapter 5 det ails the development of DEFFCON: its validation, finite differencing equations, application of theo ry and how simulations are conducted. Case studies and analysis of results are presented in Chapter 6, while Chapter 7 provides concluding statements. PAGE 16 16 CHAPTER 2 A PRIMER ON POLYMER DEGRADATION Polymers are con structed from a series of atoms most frequently carb on, hydrogen, nitrogen and oxygen. Th ese atoms are held together by electrostatic forces. Stability and strength of the bond depends on the atoms involve along with their indep endent electric charge. Carbon carbon bonds are particularly stable and are p resent in many natural materials that include cellulose, sugars, and natural rubbers. Of particular interest is the latter of the three which belongs to a group of materials known as polymers. Polymer is a term coined by Jons Jakob Berzelius in 1827 fro m the Greek polys, meaning many, and meros, meaning parts [2] It is used to denote molecular substances of high molecular mass formed by the polymerization or joining together of monomers which are molecules of lower molecular mass. One such example wo uld be the linking of several individual ethylene molecules to create polyethylene by the trading of the double bond between the carbons to a single bond represented in Figure 2 1 Here the letter n refers to a large number of in dividual molecules ranging from hundreds to thousands. Figure 2 1 Molecular structure of ethylene PAGE 17 17 The transformation of the double bond to a single bond in this instance can be done by adding free radicals to the system so that a single bond orientation is favorable. What are of interest, however, are not the production mechanisms of polymers but rather the destruction. Polymers will degrade under various conditions mainly through chemical or physical reactions. Pyrolysis occurs when an organic compound, like a polymer, is subjected to very high temperatures. During this process the bonds between the atoms in a molecule will distort and sometimes break. This may cause some of the solid polymer to vaporize reduc ing its overall mass This mass loss represents a straightforward way to quantify the changes that are happening to the polymer Furthermore this physical change can be characte rized using rate kinetic equations which pro vides the opportunity f or computer model formulation. The degradation route from solid polymer to the vapor phase is illustrated in Figure 2 2 The central term, reactive intermediate, represents temporary products such as free r adicals. The degradation of solid polymer is assumed to occur in a single step involving rapid equilibrium between the polymer and reactive intermediate that simultaneously produces gas and char. Figure 2 2 Reaction rates PAGE 18 18 Where; = rate constant for production of reactive intermediate (1/time) = rate constant for recombination (1/time) = rate constant for produ ction of vapor (1/time) = rate constant for production of char (1/time) These rate constants are part of a chemical reaction kinetics equation that dictates how quickly products will form. Although they are not truly constant they are independent of the concentration of the species involved in the reaction and are almost always strongly dependent on temperature. This temperature dependence of a specific reaction rate can be represented by an equation of t he form: (2 1) Where; = pre exponential factor or frequency factor (1/time), usually empirical = activation energy (J/mol) = gas constant (J/mol/K ) = absolute temperature (K) Equation 2 1 which is known as the Arrhenius equation has been verified empirically to give the temperature behavior of most reaction rate constants within experimental accuracy over fairly large tem perature ranges [3] Discussion Of Arrhenius Parameters The energy required to distort and break these atomic bonds can be linked with the activation energy, associated with the reaction. As depicted in Figure 2 3 the activation energy corresponds to the maximum in potential energy that must be overcome in order for the reaction to produce products. Thus the activation energy is the minimum kinetic energy that reactants must have in order to form products [4] PAGE 19 19 Figure 2 3 Activation energy The exponential term, represents the probability that an interaction will have energy above the activation energy. Therefore t he pre exponential factor, is the frequency of interactions. Thus when the terms are combined the result is the rate at which reactions of interest (those that are above activation energy) occur. Material Behavior The individua l rate constants, as listed above, relate to intermediate reactions that make up the entire pyrolysis process. What are of most interest in the model are the rates at which the volatiles and char are formed. The volatile term refers to the vaporized mate rial that eventually leaves the system, making up the quantifiable mass loss. The char is solid produced by a reaction that competes with the vaporization reaction, and represents the carbonaceous solid that remains on the sample. This char acts as a hea t and mass transfer barrier that lowers the heat release rate thus decreasing the amount of mass leaving the system. The char yield can be calculated using the ratio of the char form rate constants to the char form and gas form rate constants as follows: PAGE 20 20 (2 2) Where; = mass at infinite time (g) = initial mass (g) = char yield at temperature and are constants as defined in Figure 2 2 As time approaches infinity this model predicts a finite char yield when While there are empirical formulas to determine the c har yield of a material [5] for simplicity it will be assumed constant as explained later in this chapter. The thermophysical quantities: density, thermal conductivity, and heat capacity need to be taken into account. These parameters vary depending on t he temperature. Correlations to predict these properties base d on chemical structure alone are scarce; empirical structure property relationships have been developed that allow calculation of thermal properties from additive atomic or chemical group contr ibutions if the composit ion of the polymer is known. T o render this research useful for broad application, an approximation of temperature dependence for polymer thermodynamic quantities will be employed. Thermal conductivity as a function of temperature relative to the value at the glass transition temperature is as follows [5] : (2 3) (2 4) Where; = Absolute and g lass transition temperature respe ctively (K) = Thermal conductivity at glass transition temperature (W/(cm K)) PAGE 21 21 The relation for the density and heat capacity are: (2 5) (2 6) Where; = density at temperature in (g/cm 3 ) = volume thermal expansivity per unit mass cm 3 /(g K) = heat capacity at temperature in (J/(g K)) Derivation Of Sim ple Degradation Model To obtain a relation for the mass of the polymer as a functio n of temperature we must refer to Figure 2 2 Here there exists a rapid equilibrium between the reaction of the polymeric solid to the reactive in termediate and the reversal recombination process while the overall forward reaction to form gas and char materials proceeds slowly. Thus the derived systems of rate equations are [5] : (2 7) (2 8) (2 9) (2 10) Where; = rate constant for production of reactive intermediate (1/time) = rate constant for recombination (1/time) = rate constant for production of vapor (1/time) = rate constant for production of char (1/time) PAGE 22 22 = rate of change of polymer loss (mass/time) = rate of change of reacti on intermediate (mass/time) = rate of change of gas formation (mass/time) = rate of change of char formation (mass/time) All of these equations are solved for instantaneous amount of each species. It is impor tant to account for these reactions because in total they represent the amount of solid material. It can be s hown that the overall rate constant for pyrolysis is: (2 11) Here; (2 12) This e quation shows the balancing of the forward reaction to the fraction that reverses back to the polymer form due to recombination. The overall rate law in terms of instantaneous mass f raction can be presented as [5] (2 13) w here the char yield is held constant. This equation shows, that the mass is a function of the portion of the material that will vaporize and the portion that will not (the char material). The polymer fraction, is related to the p yrolysis reaction rate, and is obtained by integration while considering non isothermal conditions (2 14) Where; PAGE 23 23 = constant heating rate (K/min) at The int roduction of the heating rate transforms the variable of integration from time to temperature. The integration of the right hand side is an exponential integral that can be approximated to yield [5] : (2 15) Where A nd the value of is defines a: (2 16) Equation 2 14, the relative polymer fraction remaining from thermal irradiation, becomes: (2 17) The final mass fract ion equation for non isothermal heating (constant heating rate) based on relative char formation is: (2 18) Where; = mass at temperature absolute temperature in (g) = initial mass (g) = temperature independent char yield = heating rate (K/min) PAGE 24 24 = pyrolysis rate constant (1/min) = gas constant (J/mol/K) = activation energy (J/mol) This final equation will be used to create a polymer degradation model to be implemented in the computational computer code. Polyurethane Waste Varnish One particular study by E speranza [6] was consider ed as an initial model of degradation for testing and validation. In this study, waste varnish based on a polyurethane polymer was subject to four different heating rates in order to obtain a weight fraction versus temperature profile for the material. T he experiments were conducted under inert atmospheric conditions, thus the model does not account for combustion. Figure 2 4 shows this laboratory obtained profile. Figure 2 4 P olyurethane waste varnish degradation [6] PAGE 25 25 An in depth kinetic study is presented in [6] but is not used in the computer model because it is a very specific case in which the degradation involves three different reaction mechanisms. In other words, to mo del the results given, three different mass loss equations are used that have individual reaction orders ( species concentration dependencies ) as well as activation energies and pre exponential factors It is unrealistic to expect that this kind of detail can be obtained without an extensive laboratory examination of the material in question. It is the goal of this research to implement a generic model that will fit, within a reasonable tolerance, a wide range of polymer m aterial degradations with input re stricted to the Arrhenius parameters. It will be shown that use of this model enables us to examine degradation expected in a real world scenario. Fitting The Default Model This model will be referred t o as the default model. F or the automotive paint system being considered, it will be assumed that the degradation will follow the mathematical fit of this curve. This assumption is based on a paint system considered to be composed of polyurethane or materials that behave similar to polyurethane. Chapte r 3 further explains these material approximations. Figure 2 4 shows that as the heating rate increases, the difference between the individual heating rate curves decrease. Therefore the data represented by the 20 C/min rate wil l be fitted as it is taken to be a good approximation of a near infinite heating rate, which is appropriate when taking into account the source term of the computer model; a nuclear thermal flash. Collecting the 20 C/min rate data, a best fit curve is es tablished based on theory presented earlier in this section. Figure 2 5 shows the result of this fit. PAGE 26 26 Figure 2 5 Theory and experimental fit T he theoretical fit has a mean rela tive error of approximately 14% with respect to the laboratory data and has the following Arrhenius and char parameters: Table 2 1 Theory Arrhenius parameters Constant Value Units E a 89.5 kJ/mol A 2.00E+0 7 1/min R 8.30 J/mol/K 20 K/min Y c 0.1 g/g These values are optimized to ensure minimum error for early degradation versus tempe rature behavior. B ecause the area of interest is in the range of 0.9 0.1 mass fraction the fit should be the most accurate around this region. For other materials the only values that will change are the activation energy, pre exponential factor and the char yield. For instance the values for a specific type of poly methyl methacrylate (PMMA) are: PAGE 27 27 Table 2 2 PMMA Arrhenius parameters Constant Value Units E a 160 kJ/mol A 9.00E+09 1/min R 8.30 J/mol/K 20 K/min Y c 0.0 g/g Which will produce a degradation curve comparable to the waste varnish but shifted to the right as can be seen in Figure 2 6 This shift is caused by the combined effects of the pre exponential and the a ctivation energies. In general increasing the activation energy will shift degradation to the right, while the increase in pre exponential will have the opposite impact. Figure 2 6 Model PMMA degradatio n compared to measured varnish degradation PAGE 28 28 CHAPTER 3 SOURCE TERM EFFECTS AND THERMAL RADIATIO N For the purposes of this study it is important to recognize two main events that follow a nuclear detonation. Thermal radiation from a nuclear weapon is the main con sideration concerning the effects of the device. Relative to a conventional explosive, during a nuclear detonation a copious amount of heat is released producing temperatures estimated to be tens of millions of degrees [1] While the rate of the energy d elivered by a nuclear weapon is not constant, it is extremely fast ( < 1 second) and can thus be characterized as a constant. Accurate representation of this energy is necessary to assess effects to the receiver (automobile). It is also necessary for the case study presented later to consider the effects of energy leaving the receiver in the form of thermal radiation due to the intense surface temperatures. To a lesser ext ent it is important to be concerned with the blast wave produced and its radius of d estruction. Thermal Radiation The fraction of the total yield released as thermal energy is known as the thermal partition. For air b ursts below 15000 ft this value is experimentally shown to be 0.35 for all yields from 1kT to 10MT [1] The effecti ve thermal radiation from the weapon is defined as that emitted from the heated air of the fireball within the first minute following the explosion. The amount of thermal energy delivered to an object depends on the height of the burst and total yield, as well as other weapon characteristics [1] It has been determined that the thermal source as a function of time scenario can be represented as [7] : (3 1) PAGE 29 29 Where and have units of J/s takes the form: (3 2) The parameter is normalized time, where has units of seconds; (3 3 ) These equations all depend on the yield of the weapon represented as which has units of kilotons, referring to the equivalent quantity of TNT (trini trotoluene). The power relation (Equation 3 1) must be converted in order for it to be useful in the numerical model. The useful units would be W/cm 2 Since J/s=W all that is left to do is distribute this energy over the surface are of a sphere. Thus the appropriate relation for the model for an open field, unobstructed view to the receiver, is: (3 4) Where; = power delivered to receiver (W/cm 2 ) = slant range (cm) = atmospheric transmittance = material abso rption factor Note that the slant range is not the ground distance from the epicenter to the receiver, but the actual distance to the receiver and varies with height. Figure 3 1 depicts this concept where the weapo n has detonat ed at some height, h The atmospheric transmittance, also depends on the range from the weapon to the receiver, and is affected by particles suspended in the air. T he material absorption factor is representative of how well the r eceiver will accept the thermal radiation. An absorption fac tor of 1.0 is equivalent to 100% acceptance. PAGE 30 30 Figure 3 1 Slant distance illustration These equations make up what is consider ed to be the open field model. This refers to the fact that there are no obstructions between the source and the receiver. CANYON: 3 D R adiant Heat Transfer CANYON is another computer code used in this research for the analysis of the effects of nu clear weapons. Developed by T. Stachitas, CANYON is a code that accounts for the scenario when the receiver is in a narrow passage like a street with buildings on either side of it. This code calculates the channeling of the thermal radiation due to the presence of the surrounding buildings. Under the assumption of gray diffuse surfaces and black body source emission the amount of energy the receiver experiences can be determined. The total power at the exit of the canyon depends mostly on the geometr y; how wide the street is and the height of the buildings. Variation in building materials is also a factor but makes little difference in the end result based on a study provided by Stachitas [8] The average street canyon is shown to amplify the deliv e red energy by approximately 25% compared to a sour ce with no channeling effect PAGE 31 31 with parallel buildings separated by a street 40 m wide. The buildings themselves are 100 m tall. Surface Re Radiation Wh en thermal radiation falls upon a body, part is absorbed by the body in the form of heat, part is reflected back into space, and part may be transmitted through the body. In the case of the automotive paint system surface temperatures of the material wil l not reach the extremes necessary to cause significant re radiation. For example if a surface is 600 K then it will re radiate about 0.74 W/cm 2 However if a surface gets hot enough the re radiation term becomes significant; this will be the case in a study discussed in Chapter 6 For this reason the re radiation is discussed here. Often for the sake of simplicity a material will be classified as a black body radiator. A black body is defined as on e that absorbs all radiant energy and reflects n one. (3 5) Where; = fraction absorbed (equal to 1 if black body) = fraction of thermal energy emitted f rom surface (equal to 1 if black body) The absorption and emissivity terms c an depend on other factors such as wavelength, incident angle material, and temperature. Using a gray diffuse assumption, both are independent of angle and wavelength; furthermore th e emissivity is equal to the absorption [9] Due to the modular nature of the computer model developed in this research, it would be possible to add spectral and angular dependencies with relative ease. The emissive power can therefore be determined by Equation 3 6. (3 6) PAGE 32 32 Here; = power emitted (W/cm 2 ) Stefan Boltzmann constant (W/cm 2 /K 4 ) = absolute surface temperature (K) In practical applications it is necessary to include the receiving surface which in this case wou ld be the ambient air; this then make s Equation 3 7 : (3 7) Where; = absolute temperature of the ambient air (K) The implementation of this equation is shown in the code development section of this do cument. Blast W ave Effects Another minor consideration will be made with respect to the blast radius of the weapon. During an explosion a shockwave will traverse from the center of the event outward that possess a considerable amount of pressure. Press ures exceeding 6 psi (pounds per square inch) will cause a significant amount of damage to an automobile and may make location of the vehicle difficult or impossible depending on whether or not it still exists. Figure 3 2 shows a vehicle that was damaged by a shockwave with a peak overpressure of 5 psi fr om a nuclear weapon As can be seen the vehicle was badly damaged although it remained in running condition [1] Samples for analysis can still be taken from an object as dama ged as this one The stru ctural integrity of the object determines whether or not 5 6 psi overpressure is a good estimation of where to look for PAGE 33 33 samples. As a point of reference for the severity of pressure from an ex plosion, consider the ear drum that b urst s under pressures of approximately 2.5 psi [10] Figure 3 2 Damage to automobile from peak overpressure of 5 psi [1] Determining a minimum forensic distance or the distance at which objects can be f ound and examined will rely on the peak overpressure that the object will experience. This pressure is a function of both the distance from the epicenter, the yield of the device, and also varies with the height of the burst and atmospheric conditions. Figure 3 3 shows the decay of the pressure as a function of the distance from the burst of a 1 kiloton weapon. Scaling laws and relations exist to obt ain this kind of data; tables were consulted for the numbers presented in this document [1] PAGE 34 34 Figure 3 3 Peak overpressure from a 1 kiloton free air burst from sea level ambient conditions [1] PAGE 35 35 CHAPTER 4 MODEL CONCEPTUALIZAT ION A typical automotive paint scheme consists of multiple la yers with varying thicknesses Figure 4 1 represents the paint model considered for degradation analysis. The materials are ordered sequentially with increasing depth i.e. the clearcoat is the material that is exposed to the outside air while the metal substr ate represent s the body panel. The material properties are discussed in detail in the following sections. Figure 4 1 Paint layers and relative dimensions Material Properties Automoti ve paint schemes are extremely diverse and information about their chemical and physical properties is hard to come by. Initial efforts were mad e to collect d ata about the individual layers, including densities, heat capacities and absorption parameters. These values, represented in Table 4 1 were collected from different handbooks [11] [12] [13] and will be used to prepare a temperature dependent heat transfer model that is discussed later. Obtaining the exact composition is challenging since the respective layers are all made according to protected original equipment manufacturer (OEM) specifications. It is impossible to account for the all the variations in chemical make up utilized by the entire automotive industry. For t his reason some assumptions are made in an attempt to simplify and generalize the model. A good PAGE 36 36 approximation that can be made for the purposes of our work is that they are thermoset (heat cured plastics) epoxies and polyurethanes. Table 4 1 Material thermophysical properties [11] [12] [13] Material (W/cm/K) (g/cm 3 ) c (J/g/K) Polyurethane CC 0.0021 1.20 1.80 0.3* Polyurethane BC** 0.0041 1.17 1.28 1.0 Epoxy Primer 0.0024 1.40 1.11 1.0 Zinc Phos phate 0.0052 4.00 0.13 1.0 Steel 0.5400 7.80 0.49 1.0 assumed depends on weight percent p igment (40% in this instance) Clearcoat (40 50 microns) This layer is characterized as a thin transparent material composed of an acrylic melamine [14] that is used primarily to protect the pigmented basecoat. Chemically included in the clearcoat are light stabilizers which come in two basic forms; ultraviolet light absorbers (UVA) or hindered amine light stabilizers (HALS). Without a stabilizer ev entual film failure may be observed. Usually both are included in this layer for optimum defense against weathering. HALS are used to inhibit photo oxidation of the automotive clearcoat. Degradation of the HALS in the coating is inevitable. Mechanisms of removal include volatilization and washing out. Studies [15] have been done to assess the concentration of the active HALS in a weathered paint system at long exposure times; the term active refers to the initial HALS in the paint, including the transfor mation products capable of stalling photo oxidation. According to [15] the chemistry that explains how HALS prevents photo oxidation can be explained through the production of nitroxyl radicals as seen in Figure 4 2 PAGE 37 37 Figure 4 2 Production of Nitroxyl radicals [15] After the formation of the nitroxyl radicals they are then used to scavenge other free radicals that could otherwise propagate free radical ch ain oxidation (labeled products in Figure 4 2 ). The research goes on to conclude that the useful amount of HALS that is in a sample can be reliably determined by the combined observation of steady state nitroxyl concentration, an d residual parent material (active HALS) concentration. The ability to determine the quantity of HALS/UVAs in a paint system is of importance because additives concentration is directly related to the percentage of transmitted energy through the clearcoat to the basecoat. Figure 4 3 shows the transmittance of energy as a function of wavelength for different amounts of HALS in a clearcoat system. From this figure it appears that 100 % tran smittance in the near infrared region (7 00 nm) which is the region of concern, can be expected. However experimental data is unavailable to confirm this hypothesis; thus exponential decay of energy through clear mater ials is the active assumption. This assumption is validated by examination o f the protection mechanisms of the UVA chemical species. Unlike the HALS anti oxidant approach, the UVA mechanism is more of a mechanical protection and serves to block higher wavelengths of light. The protection provided by the UVA species would encompa ss the infrared region. PAGE 38 38 Figure 4 3 Transmittance of HALS at various wavelengths [16] UVAs work a little differently than hindered amine light stabilizers. An article written by Ciba Specialty Chemical s [17] explains that UVA prevents wavelengths of light above 290 nm from reaching the chromophoric, or light absorbing, groups in the polymer. They convert UV energy to heat that is dissipated throughout the coating without affecting the polymer. Then th e UVA is able to return back to its ground state without adverse affects to its own chemical bond. This process is known as Keto enol tautomerism and is presented in Figure 4 4 After an extended period of exposure from an energ y source the mechanism will become less efficient. In other words, the UVAs will no longer be able to convert the UV energy into heat and will produce free radicals. For this reason, in a clearcoat system both chemical species (UVA and HALS) would be use d f or optimum protection. PAGE 39 39 Figure 4 4 K eto enol tautomerism showing UV excitation returning to ground state by release of heat [17] Absorption of energy in a transparent medium, Figure 4 5 containing a chemical species like UVA follows an exponential decay which is derived as follows [18] : Figure 4 5 Absorption in transparent material (4 1) Where; = absorption constant (1/cm) = energy intensity as a function of distance into the material Integration will give the intensity, thus assuming an initial intensity a t the surface of the material, T : (4 2) PAGE 40 40 The ratio of the initial surface intensity to the intensity at the end of the transparent material, is the transmitted fraction. (4 3) A ssuming no reflection off the surface of the material transmittance and absorption, are related as follows : (4 4) T his absorption, depends on how much UVA i s present in the material. T he more UVA is present ,m the less energy will reach the underlying basecoat. So unlike HALS, UVA compounds will self shield each other closer to the basecoat/cl earcoat interface. Development of a relation for the absorption as a function of the life of the material is given in [19] and is derived as follows: (4 5) (4 6) Where; = loss rate (1/time) = absorbance = concentration of UVA (particles /volume) = light intensity And since absorbance is proportional to the concentration of the UVA; (4 7) Thus by integration; (4 8) PAGE 41 41 R elations such as this one are important to assess initial parameters to input into the computer model. It is unlikely that an object at the scene of the event would be un weathered; knowing the absorbance of the automotive clearcoat at its current point in its lifetime will increase the accuracy in the result obtain by the computer simulation. It should be noted that most clearcoat systems will employ both methods of prot ection; the UVAs can limit the production of free radicals by incoming energy, those that are produced can be controlled by HALS reactions. In the research presented here only the UVAs ability to exponentially decrease the energy will be considered in the computer model. Basecoat/Primer (~20 microns) The basecoat is the layer directly underneath the clearcoat and contains the pigmentation that gives the paint its color. It is assumed to have nearly the same composition as the clearcoat with the excepti on of the pigment particles. These particles can alter the thermophysical properties of the material that will in turn change the heat transfer and the temperature profile. According to a patent on automotive paint [20] pigment particles may be composed of metal oxide encapsulated mica. This particular patent states that the thermosetting polymer (paint) ma y contain anywhere from 1 50% pigment by weight. The overall size of the metal oxide/mica particle ranges from a fraction o f a micron to a micron. As is shown in Table 4 1 the pigments will slightly increase the materials ability to transfer heat (thermal conductivity) T his can be attributed to the metal oxide component of the particle. Depending on the weight percent of metal oxides in the system this value can increase significantly. This metal oxide will vary depending on color; titanium dioxide is usually associated with a white paint while iron oxide will give a more red color. For the purpose of this research only PAGE 42 42 the end effect of the pigments is taken into account, i.e. the changing of the thermophysical properties. No attempt is made to include some type of geometric representation of the particle in the computer model. The primer is similar to the basecoat in as also contains pigment particles. However it will be considered to be an epoxy resin thermoset [13] Again only the change in the thermophysical properties will be considered in the computer model. It should be noted that in general, creating a het erogeneous polymer system (polymers with pigments) will have an effect on thermal st ability of the system. I t may become more difficult to vaporize the material because of the limited ability of the polymer bonds to obtain enough vibrational energy to bre ak. Zinc Phosphate (1 2 microns) The last material that is in direct contact with the substrate is a layer of zinc phosphate. This phosphate layer acts to protect the metallic substrate from corrosion. This layer as well as the metallic substrate will n ot be allowed to degrade. The interest in this research is to degrade enough polymeric material as to determine incident heat flux. In an instance where all polymers are removed gives a large margin of error when deducing maximum heat due to the fact tha t it will require a much more extreme temperature to degrade the phosphate or substrate layers. Aluminum alloys have a melting range from 620 to 800 C while steels melt at approximately 1250 C [21] ; these temperatures are outside the scope of interest. Other Physical Considerations In order to accurately model any situation, it is necessary to consider as many physical processes as is practical keeping in mind which of these has the greatest effect on the outcome. There are two other main principles that need to be accounted for in PAGE 43 43 this model, one is material specific, while the other is a fundamental heat transfer principle. Considerations Including Char As shown in Chapter 2 as the material undergoes degradation it produces a carbonaceous materia l known as char. This conversion of polymer to char will change the way that heat is transferred due to the variation in thermophysical properties between the two materials. F or any polymeric material the char yield is considered to be constant with res pect to temperature. As time progresses, however, the amount of polymer left on the sample well as the thermal conductivity, specific heat and density. To account fo r these changes a simple linear combination of the temperature dependent properties is proposed [22] These relations hips operate under the assumption that the polymer and char are thoroughly mixed and that no new material with unique properties is produ ced Thus for the thermal conductivity, specific heat and density: (4 9) (4 10) (4 11) Where the subscript and denote the virgin polymer and char properties respectively. The coefficient, represents the fraction of the material that is still polymer and is given by: (4 12) PAGE 44 44 Here; = p olymer mass fraction And is the previously discussed char yield; a constant specific to the material. The relations for the virgin polymer thermophysical properties are given in an earlier section (eqns 2 3 to 2 6) while the relat ions for the char material is given as: (4 13) (4 14) Where is the absolute temperature and the numbered coefficients are constants given in [22] and are presented in Table 4 1. Table 4 2 Constants for thermophysical equations for char Constant Value Units 1 0.955 W/m/K 2 8.42E 04 W/m/K 2 3 4.07E 06 W/m/K 3 4 5.32E 09 W/m/K 4 c 1 0.870 J/g/K c 2 1.02E 03 J/g/K 2 A temperature dependent relationship for the char density is not given, and is assumed to be constant. It should be noted that in more than one instance in the literature the densit y of the char is found to be 80% of the virgin polymer. It is therefore as sumed that this is the case for all polymeric materials in the numerical model known as DEFFCON. The further assumption is made that the thermal conductivity as well as specific heat relations will not change from polymer to polymer. The ramification of these assumptions is outlined in the results and analysis section. Details on the implementation of these equations are given in the code development section of this document. PAGE 45 45 Boundary Layer Theory In heat transfer calculations it is necessary to conside r a concept known as boundary layer theory. There are different types of boundary layers such as velocity, thermal and in the case of mass transfer, concentration. The focus of this section is the velocity and thermal boundary layers where the discussio n was taken from [18] The velocity boundary layer develops because of a no slip condition that air particles have with the heated surface. This no slip condition means that particles that are in contact with the surface essentially stick to the surfac e. Consider a vertical surface as depicted in Figure 4 6 as air moves across this surface at a velocity of a wil l also be affected by this condition because of the friction forces between them. As the distance from the surface increases the particles velocities get closer to matching the velocity of the free flowing stream ( ). The distance where the velocity of the particles is near that of the free stream, about is considered the boundary layer thickness The thermal boundary layer is present due to the differences in the temperatures of th e surface and the surrounding air. Similar ly the thickness of the layer, is the distance where the temperature of the fluid is abou t More appropriately the term in Figure 4 7 shoul d be zero. Here is the temperature of the ambient air. Heat transfer through the boundary layer is considered to be conduction rather than convection. T hese two boundary layers are approxima tely equal for hot gases, i.e. PAGE 46 46 Figure 4 6 Thermal and velocity boundary layer (4 15a) (4 15b) Where; = fluid density ins ide boundary layer = y component velocity and gradient = x component velocity and gradient = x direction pressure gradient = viscosity and Laplacian = body force Equation 4 15b will not be considered due to the velocity boundary layer approximations: PAGE 47 47 and Simplifying assumptions can be made to Equation 4 15a so that it is easier to man ipulate. For instance, the body force is known to be that of gravity in this case and viscosity will not be considered in the x direction. Dividing through by density and adding the body force term gives: (4 16) (4 17) Where the ki nematic viscosity now appears, along with gravity, Further simplification of Equation 4 17 can be achieved by assuming that the region of space outside the boundary layer is quiescent; not moving. Therefore the x pressure gradient at any point in the boundary layer must equal the pressure gradie nt of the outside region where : (4 18) Here; = fluid density outside boundary layer Substituting Equation 4 1 8 into 4 17 the following is observed: (4 19) The first term on the right hand side of the equation can be related to the fluid property known as the volumetr ic thermal expansion coefficient. PAGE 48 48 (4 20) This property of the fluid provides a measure of the amount that the density will change in response to a change in the temperature at a constant pressure. Equation 4 20 can be prese nted in the following approximate form: (4 21) Substituting in Equation 4 19 yields (4 22) However it should be noted that if the fluid is considered an ideal gas, which is the case for the pu rposes of this research, beta will simply become; Mass (assumed constant density): (4 23) Energy: (4 24) Where is thermal diffusivity and has units of (m 2 /s). Further if nondi mensionalizing parameters are introduced to Equation 4 22 the Grashof number is formed, which represents the ratio of buoyancy to viscous forces of a fluid. (4 25a) (4 25b) PAGE 49 49 (4 25c) (4 25d) (4 25e) Here is a characteristic length (overall length of surface) and is a c haracteristic velocity. This transf orms the momentum equation (4 22 ) into: (4 26) Where; (4 27) To retrieve the Grashof number the first term on the right hand side must be multiplied by the sq uare of the Reynolds number ( ) to remove the velocity component giving: (4 28) In order to solve Equations 4 21 to 4 23 some boundary conditions must be established. For instance at the surface of the material the temperature is that of the material and both velocities are equal to zero. At an infinite distance from the surface the temperature approaches the ambient air temperature and the x directional velocity equals zero; : PAGE 50 50 : To aid in solving these equations a similarity parameter is introduced as well as a stream function to handle the velocity comp onents (4 29) (4 30) With the help of these relations the three partial differential equations can be reduced to the following two ordinary differential equations: (4 31) (4 32) Where; = Prandtl number = After transformation of the boundary conditions a numerical solution is possible the results of which are p resented in Figure 4 7 [23] The appropriate b ou ndary condition transform ations are : : : This figure sho ws th e approximate equality of for air velocity of zero (quiescent) and a temperature close to that of the ambient air. For a particular value of the relation can be rearranged to provide the thickness of the bou ndary layers. PAGE 51 51 Choosing a value of 6 for (to ensure temperature is equal to that of ambient) the thermal boundary layer is then: (4 33) Where is the height of the object that i s ver tically oriented. Equation 4 33 is directly used in DEFFCON to calculate the thickness of the thermal boundary layer. This is essential because the boundary layer has a significant effect on the heat removal Fig ure 4 7 Numerical solutions for velocity (left) and thermal boundary layer (right). PAGE 52 52 CHAPTER 5 DEFFCON CODE DEVELOP MENT A two dimensional transient heat conduction code was dev eloped using Fortran 90 to assess the damage to a paint system subject to an incident heat flux. DEFFCON employs the degradation model from Chapter 2 as well as the weapon source equation from Chapter 3 loaded into a series of finite differencing equations with the end goal of removing mater ial to achieve an appropriate mass loss. DEFFCON can be operated using an explicit or implicit finite differencing algorithm as designated by the user. Choosing an appropriate algorithm is not intuitive and requires some knowledge of how each method oper ates. The time and physical scale of the case as well as the material properties play a significant role in determining which algorithm will provide an accurate result while also minimizing computational time. The geometry is considered to be vertica lly oriented as can be seen in Figure 5 1 The numbers 1 4 on the figure serve as location identification utilized by the code. The locations will also be identified by cardinal directions i.e. position 1 is west, position 4 is north etc. The bottom left corner serves as the origin of the model. Therefore nodal positions increase from left to right and from bottom to top. As can be seen in the figure the incident flux is applied to the zero centimeter x position of the geometr y. This diagram represents a simplified version of the paint system known as the block model. The block model contains only one layer of polymer adhered to the aluminum substrate. The block model serves as a starting point for code verification with exp erimental findings explained in depth in Chapter 6. The more complex multi layer model is also explored in Chapter 6. PAGE 53 53 Figure 5 1 Vertical orientation of mono layer polymer system Solvi ng The Heat Conduction Equation T he main task of DEFFCON is to solve the two dimensional time dependent heat transfer equation: (5 1) Where; = thermal diffusivity (cm 2 /s) = th ermal conductivity (W/cm/K) = absolute temperature (K) = horizontal and vertical position respectively (cm) = volumetric heat generation (W/cm 3 ) This equation can be solved numerically u sing a variety of methods. The methods described in this research are the explicit and implicit finite differencing methods. PAGE 54 54 Explicit Finite Difference The finite differencing schemes are broken up into three general equations; surface, corner and inte rior. The surface equations are reserved for the individual surfaces 1 4, the corner equations are for the intersections of the surfaces such as 1 and 3 while the interior equations are contained within the boundaries of the geometry. The general explici t equations are given below along with a nodal representation. The equations are extended to three points to preserve second order accuracy in space which requires a Ta ylor series expansion. The Taylor series formula: (5 2) The derivation is done by subdividing the system into nodes surrounded by control volumes ; these control volumes represent local conservation of energies. T he equations for the surrounding nodes are developed using Equation 5 1. Weighting coefficients are then introduced in each direction for a given step size. The weighting coefficients are solved based on which partial differential equation is required and then the e quations are summed to yield differencing equations with their respective truncation errors [24] Initially considering the x direction only Figure 5 2 shows the nodal orientation Figure 5 2 Nodes with surrounding control volume PAGE 55 55 The derivation is as follows with the system of equations with weighting coefficients : (5 3a) (5 3b) (5 3c) The summations to determine the weighting coefficients are: (5 4a) (5 4b) (5 4c) These summations lead to th e following equations in their respective order: (5 5a) (5 5b) (5 5c) Simultaneous solution to this system of equations will give the individual weighting coeffi cients as such: PAGE 56 56 These coefficients are used along with the Equations 5 3a through c to obtain the second partial derivative of the temperature with respect to the x position al ong with the associated error term. This is done by summing all the terms in these equations; (5 6) The error term is the second term in the right hand side of Equation 5 6 and shows that this relation is has second order accu racy. Similarly a relation is derived for the vertical component of the heat equation represented by the nodes in Figure 5 3 This produces the second partial of the temperature with respect to the vertical direction giving: (5 7) Figure 5 3 Vertical nodes with surrounding control volume The le ft hand side of Equation 5 1 ca n be written as: (5 8) Wh ere; Pulling all these equations together the equivalent explicit finite differencing formula for Equation 5 1 becomes: PAGE 57 57 (5 9) In this equation the cur rent temperatures (superscript ) are known t he new temperature (superscript ) is the one that wil l be solved for. From Chapter 3 the thermal radiation from the source, and the convection term must be addressed This convection term plays no role in the simulations because it is replaced by the thermal boundary layer which is a conductive layer. H owever it is still present in the finite differencing equations and is zeroed out using the input file, discussed later To develop a relation that includes the boundary conditions the following surface flux relation is required: (5 10) Where; = surface flux (W/cm 2 ) The net flux on the surface is determined to be: (5 11) With the following equalities: The applie d heat flux is energy (from weapon) incident on the surface; the convective term is a loss representing energy removal from the surface. Here surface temperatur e PAGE 58 58 To preserve second order accuracy for the boundary conditions only three nodes Figure 5 1 and on ly considering the temperatures and th e amb ient temperature known as the derivation for Equation 5 9 can be repeated. Equations 5 3a and 5 3b are needed because there are two points to the left of Equations 5 5a and 5 5b become: (5 12) (5 13) Therefore the weighting coefficients are: This then leads to the derivation of the term for the first derivative of the temperature with respect t o the x direction: (5 14) Thus preserving second order accuracy. The completed relation can be obtained by combining the numerical solution (5 14) with Equations 5 10 and 5 11: (5 15) Finally s olving for in Equation 5 15 and in Equation 5 9 the following are formed : (5 15) (5 16) PAGE 59 59 Similarly an energy balance can be used to incorporate the boun dary conditions. This is done with the Crank Nicolson equations presented later. Stability A n in stability condition exists with an explicit system of equations. This instability means that the solution will oscillate about the true solution depending on the input parameters; specifically the time and position steps. For the heat transfer Equations previously derived the instability condition is referred to as the Fourier number ( ) expressed in Equation 5 17 [18] (5 17) Thus for a particular thermal diffusivity ( ), which is the driving mechanism in this formula, certain conditions must be satisfied in order to obtain a solution. If this condition is not met DEFFCON will not converge I f the user the stability condition, DEFFCON will compensate by using an appropriate time step for the amount of x meshes and thermal diffusivity that are given. An example of the solution to Fourier number instability is giv en in Figure 5 4 This figure is a snap shot from a Microsoft E xcel solver spreadsheet which is developed for the purpose of determining the time step necessary for the polymer system to converge given its thermophysical properti es and overall dimensions. The figure represents a multi layered system much like the paint system presented earlier. T he number that drives the stability is that of the thermal diffusivity of the aluminum substrate. PAGE 60 60 Figure 5 4 Excel solver used for Fourier number analysis For a system with 100 x meshes and an overall x dimension of 1 mm the necessary time step (det) should be no more than 0.4 microseconds. Crank Nicolson Since th e explicit (forward Euler) algorithm has a stability condition, the Fourier number, it is more practical to utilize a method that is unconditionally stable. Unconditionally stable refers to the fact that no matter what the time or meshing step size is, a solution is possible. The validity of this solution however may be questionable due to propagation of error. The Crank Nicolson (C N) finite differencing scheme is an average of both forward Euler and backward Euler methods. For a central node that h as no boundary conditions this averaging is obvious. Consider Equation 5 18 which only considers one dimensional x direction heat transfer represented by Figure 5 5 : (5 19) Figure 5 5 Central node PAGE 61 61 The averaging of the forward ( ) and backward ( ) differencing can be seen clearly The inclusion of the backward Euler method is what renders Cr ank Nicolson an unconditionally stable numerical approach. For the same physical position in the model represented by the ex plicit Equations 5 15 and 5 16, the implicit Crank Nicolson equation becomes: (5 20) Where the applied radi ation ( ), the convection ( ) and surface re radiation term s ( ) are all included by conservation of energy. Here the film temperature ( ) corresponds to the average b etween the ambient air temperature and the surface temperature. This particular temperature exists due to the thermal boundary layer that is formed. In the event that the user wants to use a known convection coefficient the film temperature is set equal to the ambient air temperature ( ). Thus the film temperature is used as a substitute for the ambient air temperature when a boundary layer is considered. All fluid (air) properties are evaluated at the film temperature. Figure 5 6 shows the energy balance on the material surface. The positive sense is considered to be the addition of energy onto the material. Thus the applied flux is a positive term and the remaining convective and radiation terms are su btracted from it. This definition makes the inclusion of the energy terms into the finite differencing equations quite simple. PAGE 62 62 Figure 5 6 Energy balance on a surface Equation Solution Method The results presented in this research are produced using the implicit set of nodal equations. The set can be formed as a linear system: (5 21) Where is the coefficient matrix of and is the vector of known temperatures. Gauss Seidel iteration is used to obtain the values of the new temperatures, This iteration method requires an initi al guess for the temperatures and then solves each equation individually until the i th iterated solution is within a certain tolerance of the i 1 th solution i.e. the infinity norm is minimized Code Validat io n DEFFCON is validated against an analytical s olution to a common heat transfer problem. For a constant surface heat flux incident on a semi infinite solid the analytical solution to the transient temperature profile is: (5 22) Where; PAGE 63 63 = complementa ry error function = surface flux (W/cm 2 ) = depth into surface (cm) The surface considered for the validation study is a beryll ium wall that is 25 cm in depth T he height can be considered infinite due to the fact that heat transfer is not varying in the vert ical dimension. As this is semi infinite no heat is lost on the backside of the wall at the 25 cm point. Figure 5 7 shows the temperature profile into the wall after it has had 40 seconds of exposure to a flux of 10 W/cm 2 Table 5 1 Beryllium thermophysical values gives the thermophysi cal values of beryllium used Table 5 1 Beryllium thermophysical values Constant Value Units 1.998 W/cm/K 1.85 g/cm 3 c 1.824 J/g/K Figure 5 7 Temperature profile for beryllium wall with s and cm PAGE 64 64 This figure shows the overlay of the implicit Crank Nicolson and explicit numerical models with the analytical solution. In DEFFCON both numeri cal models are within a 5% absolute relative error with the true result. The more interesting result is that the implicit routine achieved these results in less than half the time that the explicit values are reached; this is due to the fact that the time step for Crank Nicolson (C N) is stable. Table 5 2 shows the relative errors along with runtimes and t ime step of these numerical models. Notice that the time step of C N is 100 times larger than the value for the explicit model. With such a simple simulation if the time steps are equal the explicit routine will run much faster, this can be attributed to the fact that C N must go through several iterations to achieve a result. Table 5 2 Re s ults of validation study for 40 sec simulation Method t (sec) Error (%) Runtime (s) C N 0.1 2.01 28.16 C N 0.001 1.01 151.2 Explicit 0.001 0.41 71.68 Application Of Thermophysical Parameters Modeling the physical world using computation is a daunting task and requires some abstract analysis. The pr actical approach to three particular theories is discussed below and includes the thermal boundary layer, clearcoat absorption and char production. Since the theory behind these mechanisms is presented in previous sections all that is left to do is apply them. Luckily the physics behind these mechanisms are well documented in other texts so the foundation is strong. None the less some care is taken in explaining the details of their application. PAGE 65 65 Practical Boundary Layer As discussed in Chapter 4 the heat difference between a surface and its surrounding air will create a thermal boundary layer Equation 4 32 describes the thickness of this layer which depends on surface length of the material and the temperature differe ntial that exists. It can be sh own by analyzing this equation that as the surface temperature increases the thickness of the layer decreases. Thus at ever y time step a boundary layer and film temperature must be calculated by monitoring the surface temperature of the material. T he t hermal boundary layer is considered to be a purely conducting layer; therefore it is as if another material were simply placed in front of the sample. Figure 5 8 is a modified version of Figure 5 1 that represents the position of the layer of air. The air transmits all of the incident flux; it does not absorb any energy. As the surface of the material is heated the air layer will conduct heat away from the surface as it is always cooler. The zero x position is still considered to be the front of th e polymer and for the purposes of visualization can be considered to occupy the area on the negative x axis. This is done because degradation of the boundary layer is not tracked. T he air layer always has the same nodal dimensions as the ma terial meaning that the small and are the same for both. This eliminates the need for a non uniform meshing algorithm when considering the finite differencing equations. The kinematic viscosity, be ta term and any other thermodynamic quantities of the air layer are calc ulated at the film temperature These thermodynamic quantities have been fitted using data from [18] and are available in the appendix along with their respec tive plots. PAGE 66 66 Figure 5 8 Polymer model with inclusion of the boundary layer Practical Clearcoat Chapter 4 shows that the clearcoat will propagate energy according t law. The impo rtant parameter for the clearcoat calculations in DEFFCON is the w h i c h dictates how much energy reaches basecoat. The literature is vague on the transmittance specific value therefore for most calculation s a value of 0.7 is used The e ffect that this assumption has on the basecoat temperature is quantified in Chapter 6. Once a transmittance is chosen DEFFCON will back absorption parameter using Equation 4 2. This is then u sed in Equation 4 1 to determine how much energy will be delivered to the clearcoat as a function of distance PAGE 67 67 toward the basecoat. Rearranging Equation 4 3 can produce the absorption; utilizing Equation 4 2 will provide the absorbed fraction of energy in a node of the clearcoat: (5 13) (5 14) Where; = absorption constant (1/cm) = absorbed fraction = width of an x node Therefore as the energy progresses further into the clearcoat each node will experience less energy than the preceding one. The energy available at any one node is represented by Equation 5 15: (5 15) Where; = energy available to current node (W/cm 3 ) = energy on surface node (W/cm 3 ) = total energy absorbed by all nodes preceding current node (W/cm 3 ) Whatever is left over is then deposited at th e clearcoat/basecoat interface as another volumetric absorption b Thus if it is a black coating all of the energy would be deposited ( ), if the coating is not black then it is assumed that it will absorb a fraction of the left over energy. In reality the un absorbed energy would reflect off of the surface, however in this research reflectance is not considered so the remainder of this energy is di scarded as a l oss to the surroundings Practic al Char I transfer. For the purposes of illustration consider an arbitrary polymer with a char yield of 0.1, knowing the mass fraction as a function of temperature of this polymer and using PAGE 68 68 the concept of polymer fraction, given in Chapter 4 a picture of the char history can be formed. Figure 5 9 shows this Figure 5 9 Pol ymer mass fraction being converted to char Since the polymer fraction and the char fraction must sum to 1 the c har fraction can be plotted as As the polymer degrades the char fraction increases until it reaches the char yield of the material which is 0.1. At this point no polymer exists and as is shown in the figure the material is completely (mass fraction 1) char. This char fraction is used in the computer model to provide a mechanism for including the presence of increasing char in the material. From Chapter 4 it is known that a simple linear develop an overall equation to account for the chemical conversion. Figure 5 10 through 5 12 show how these changes affect heat transfer properties of another polymer that will be later used in a case study discussed in the analysis section. PAGE 69 69 Figure 5 10 Specific heat as a functio n of char fraction Figure 5 11 Thermal conductivity as a function of char PAGE 70 70 Figure 5 12 Density as a function of char is representative of the actual physical material present. DEFFCON Setup DEFFCON is a fairly complex code which requires multiple user inputs to operate. It also requires that the user be relativ ely familiar with the numerical limits of the differencing model chosen (implicit or explicit) and when each model is appropriate. The details given above should help in the sorting out of most of these issues. Input File The inputs required by the user file, reviewing the input file will aid in the instruction of how a problem must be set up as well as give insight into how the code was developed. The explanation of the different PAGE 71 71 Figure 5 13 DEFFCON input file that is read by the executable Block 1 is the general parameters block. It contains general limits of the code such as the number of materials (maxmatls) and the number of max nodes in either direction (maxxnode, maxynode). The first entry is the number of zones (maxzones). This entry tells the code how many material zones to expect. A zone refers to the location and orientation of a material, an illustr ation is given in Figure 5 14 This parameter also works with Block 7; notice that when maxmatls equals 2 there are total of two zoned materials (zoneid). PAGE 72 72 Figure 5 14 Zone and material locations As shown in Figure 5 14 the x and y dimensions (xhi1, xlow1, etc) refer to the bounds of the zones. Notice that two dimension indices (xhi1(zone1), xlow1(zone2), ylow1(zone 1&2), yhi1(zone1&2)) can share the same boundary T his means that where one material ends the other begins. Also contained in the file is the option to choose a differencing model (diffmod). This a llows the user to decide whether to use the implicit or explicit model, if the implicit model is chosen (as is the case in this input file) then the code will look for an iteration tolerance (IMP diffTol) which was explained in an earlier section. The remainder of Block 1 requires the number of polymers (numpoly) contained in th e model so that it can determine how many glass transition temperatures (Tglass) that it will search for in Block 4, as well as whether or not the user would like to use a boundary layer (air layer), PAGE 73 73 whether or not to print the results of the air layer to the output file (print layer) and whether or not DEFFCON will be reading and input file from CANYON (canfact). Block 2 is labeled the geometric bounds because it contains the global bounds of the model, i.e. where the entire thing ends and begins. This section works in the same manner that the material zoning works. Block 3 contains the time bounds and report parameters. It is of particular importance to note the time step parameter (det). This is where the user must be careful to operate within th e limitations of the differencing model that was selected in Block 1. The report parameter chooses how often the user would like data outputted to the output files and the maxtmstp number is how long the simulation run should last which operates by Equati on 5 16: (5 16) The next blocks, Block 4 and 5, provide the material properties. Thermophysical parameters of each material along with the glass transition temperature and the thermal radiation absorption parameter (discuss ed in Chapter 3) are contained in Block 4 while the Arrhenius parameters are contained in Block 5. The units of the parameters are in polymer (P) or a metal (M) whi ch will direct the program to use the appropriate degradation model. The char yield is also present in this block and was discussed in Chapter 4 Block 6 contains the heat boundaries and coefficients (Blk 6) which are the locations of convection coeffi cients if they exist. The Tinf refers to the ambient air temperature surrounding the model at different locations (1 4) which are the same PAGE 74 74 locations given in Figure 5 8 Block 8 works in much the same way as Block 7; it maps the location of a volumetric heat flux (Qoloc) if one exists. Block 9 gives the initial temperature of the material (Tinit), the atmospheric transmittance and flux parameters. The flux parameters refer to the values that actually cause the material to be heated and degrade. If ther e is a constant flux (const flux) then it ignores the other values such as weapon yield (yld) and range (rng). The next block (Blk10) contains the fixed temperature data. This block exists if the maxfixed parameter from Block 1 is not equal to zero. It works the same way as Block 7 in that it require dimensions to map the location of the constant temperature. These entries allow the user to hold a temperature constant (H) or heat a section of the material to a constant value (P). In this particular fil e the user has chosen to heat one end of the material through normal heat transfer until it reaches a desired value of 180 Celsius. mass loss in milligrams to fluence. The use r initializes the calculation with calc=1, POI milligrams. Three sets of points are required for this part of the program to work. This calculation is done using a method outlined in Chapter 6. Output Files DEFFCON outputs three different files at the completion of each simulation. These files consist of a general output file (out.put), a mass loss file (mass.out) and the temperatures of the surface of the material (sur face.out). Each file outputs data according to the report value in the input file. Figures 5 14, 5 15a and 5 15b show snap shots of what the files look like. PAGE 75 75 Figure 5 15 is the general output file; out.put. This file contain s the temperature profile of the entire sample. First the file echoes some of the input parameters such as finite difference model (implicit) and whether or not the user is reading from a canyon file (canyon mode) and is including the air boundary layer. Next the flux applied to the surface can be seen (799.0 W/cm 2 ). If instead a weapon flux was used, the yield and range would be in this location. Then the ouput states how many iterations were required to arrive at the result (iterations), along with t he average temperature of the sample (Tmean), the remaining polymer (actrempoly), the horizontal and vertical stepping (dex, dey respectively) as well as the time step (det) and the current power that is applied. Finally the temp profile is presented star ting with the time elapsed and then from left to right: i) N odal position (ix, jy) ii) S pecific location in centimeters (xo, yo) iii) M aterial number (mat num) iv) T emperatures from the previous (T old) and current iteration (T new) in celsius v) Volumetric absorption in W/c m 3 vi) The remaining mass fraction of the node (FractRemain) Figure 5 16 a and b show the mass loss and surface temperature respectively, along with the time that these values occur. As can be seen at the bottom of Figure 5 15b, each file has a time stamp th at tells the user how long it took for the simulation to run. This simulation time will depend, of course, on the computers particular processor. Simulation times with processor specifications are given in the automotive simulations of Chapter 6. PAGE 76 76 Figure 5 15 Figure 5 16 DEFFCON a) mass loss output file and b) surfa ce temperature output file PAGE 77 77 CANYON In DEFFCON One of the most important features of DEFFCON is its ability to read the output file from CANYON, specifically the part of the file that lists the flux history. To produce these results meters need to be understood. Figure 5 17 shows a snap shot of the input file. Minimal discussion on the input file is given, for a detailed description of each parameter consult [25] and [8] The parameters of interest are the surface dimensions which detail the canyon itself, the weapon yield parameters and the search parameters at the end of the file. In the surface dimensions section the only parameters that are altered are the length of the canyon, or distance to receiver. 6. All other dimensions refer to heights of buildings or widths of the streets. This simulation parameters section provides the yield of the weapon and time variables. The yield is the only concern in this section, varying from 1 20 kT in simulations. The Target fluence parameters are discussed in Chapter 6. Atmospheric conditions play a significant role in how much energy will reach the receiver. The thermal radiation will decrease in t he event that there is fog or dust in the attenuation regards to the air, a visibility of 999 k m corresponds to 100 % transmission of thermal energy. A discussion on how transmission is calculated concerning visibility is presented in research done by Stachitas [8] For the purposes of this research 999 km visibility is always used. PAGE 78 78 Figure 5 17 CANYON input file PAGE 79 79 Figure 5 18 Snap shot of CANYON output file CANout.put PAGE 80 80 Figure 5 18 shows the Everything before this heading is an echo of the input file. DEFFCON reads the list corresponding time in seconds (Time [s]). This is used as a replacement for the weapon and flux parameters that are usually loaded in otherwise. Since the time steps between the CANYON results and the DE FFCON input file are not always the same a linear interpolation is initiated. Thus in the event that DEFFCON is at a time step where ly interpolate using the next flux in CANY Figure 5 19 is an overlay of the pulse produced by CANYON and the pulse that DEFFCON interpolates and implements; as is expected the two overlap without significant error. The discrepancy between the number of point s plotted between the two codes can be attributed to the values are printed in the output files as specified by the report value. PAGE 81 81 Figure 5 19 Overlay of CANYON output to DEFFCON input fluxes for 1 kT and 708 m from the source PAGE 82 82 CHAPTER 6 CASE STUDY AND ANALY SIS Few articles exist on computer modeling that includes the coupling of degradation of a material with a heat source, a nd provide experimental data with enough detail to try and reproduce results. One particular article had enough detail and experimental data available for an in depth analysis. This article documents the degradation mechanism along with all necessary Arr henius parameters and thermophy sical properties exquisitely; it was written by Bahramian, et al [26] and will be referred to as Bahramian. Atmospheric Re E ntry The problem of atmospheric re entry is not a new one but for the purposes of this research is a very important one. As a vehicle is reintroduced to an atmosphere after being in orbit it is met with an onslaught of particles that in turn produce friction that produce an enormous amount of heat. This heat will inevitably cause the material to de grade. The material in question is a laminated composite that is comprised of 50/50 weight percent asbestos cloth mixed with phenolic resin. Two different experiments are run to test the erosion properties of the material as well as the surface temperatu re. The result s of this research show agreement within 20% relative error with the findings in the article using degradation models stated earlier. Degradation Differences The degradation mechanisms by Bahramian [26] are the same as is presented in thi s research. However the final equation for mass loss differs significantly. Activation energies of the char as well as gas formation for the material are given. Furthermore PAGE 83 83 the char yield fraction is presented as a function of temperature. The equation s are as follows: (6 1) (6 2) Where the new constants are; = pre exponential factor for char and gas formation (1/time) = activation energy for char and gas formation (J/mol) = initial mass (g) = mass at temperature absolute temperature in (g) = char yield at temperature = gas constant (J/mol/K ) = absolute temperature (K) Table 6 1 shows the reported values for these new constants and Figure 6 1 shows the theoretical as well as th e observed degradation as a function of temperature for this material. It should be noted t hat the values in the table do not exactly match those reported in the literature. This is d ue to what is believed to be an error on the order of magnitude, if the exact values are used the degradation curves do not m atch the curves that are reported. Table 6 1 Constants for degradation parameters Constant Value Units E a 95.7 kJ/mol E g E c 10.1 kJ/mol A 5.29E+06 1/m in A g /A c 4 article 10 K/min DEFFCON 20 K/min Y c 0.47 PAGE 84 84 Figure 6 1 Bahramian e xperimental and Bahramian theoretical data for composite If Equation 2 17 is used, with a constant char yield, instead of Equations 6 1 and 6 2 better agreement is achieved with the literature data. This is shown in Figure 6 2 with the activation energies and pre exponentials reported in Table 6 1. Figure 6 2 DEFFCON theory overlaid with Bahramian experimental data PAGE 85 85 Notice that there is no longer a discrepancy between the theoretical and experime ntal at the higher temperatures. T his can be attributed to the constant char yield assumption, thus validating thi s assumption. material. Thermophysical Relations Another difference in heat transfer theory lies with the relations used to calculate the values of thermal conductivities, specific heats and densities. Th e equations reported require very specific knowledge about how the material reacts to heat. Some of these issues involve the creation of pores or the changing of volume fractions of materials within the composite. The Equations 2 3 through 2 6 are an eff ort to keep things very general so that DEFFCON will return reasonable results for a broad range of materials. The thermophysical equations of the literature are presented below. (6 3) (6 4 ) (6 5) (6 6) Where; = volume fraction of particular material (polymer, char, gas ) = specific heats of materials (polymer, char, etc.) = initial thermal conductivity (W/m/K) = experimentally derived coefficient = porosity PAGE 86 86 M uch like the equations presented in Chapter 2, an effort is made to weight these parameters for th e developing char material as well as the asbestos fiber. In DEFFCON, however, this fiber would be the equivalent of the pigment particles in the paint; recall that the density of the pigments was added into that of the polymer matrix. This assumption el iminates the need for another density in the function. The coefficient, was experimentally determined and is specific to the composite material, thus utilizing Equation 6 5 would be impractical for the purposes of this research. A ssumptions Several assumptions are outlined and should be reviewed. Some of these assumptions are key in explaining the results of the numerical model and coincide with the assumptions contained in this document. i. No energy is transferred by mass diffus ion. ii. Movement of the liquid is assumed negligible compared to pyrolysis gases. iii. iv. Volatiles formed from the polymer escape from the solid as soon as they are formed. v. The ins tantaneous density of the composite depends on the mass fraction of the polymer remaining in the solid and behavior of thermal degradation of polymeric matrix. vi. The specific heat capacity of the composite is a mass weighted average of the relative mass frac tions of polymer, char, and fiber remaining in the composite. vii. The change of heat conductivity coefficient of the composite depends on temperature change. viii. The decomposition of the polymer (weight loss) occurs in a single step and exhibits a first order reac tion. PAGE 87 87 There are a couple of differences b etween these assumptions and those of DEFFCON: 1. No effort is made to track the pryrolysis gases and their properties 2. pigments All other assumptions overlap with those of DEFFCON. One in particular may be incoming energy at the surface however for a binary system of clearcoat on top of baseco at the majority of the energy will be absorbed beneath the surface. This may allow for pockets of gas to form ins ide the material and be trapped, unable to escape. This will change the manner in which the mass loss of the system progresses. This is a ma tter than needs further reflection and is not considered here. Erosion Study The literature outlines an experiment in which an oxyacetylene flame is incident onto a sample of the composite in order to discover how much material will be removed by vapori zation. Material removal is quantified by a change in the materials overall thickness. Two distances are tracked; the char layer and pyrolysis surface. The comparable distance that DEFFCON can calculate is the pyrolysis surface given by the simple mass density relation Equation 6 6. (6 7 ) PAGE 88 88 Where; = material removed (mm) = mass loss as reported by DEFFCON (g) = diameter of sample (cm) Composite Exp erimental Parameters Contained in Table 6 2 are the necessary parameters that are loaded into DEFFCON consisting of the incident flux (from oxyacetylene flame) and dimensions of the sample to be eroded, as well as the thermophysi cal properties for both material [26] and substrate. Figure 6 3 shows the orientation of the sample with respect to the flame. The composite is on an aluminum substrate 2 mm in thickness while the composite itself is 25 mm in hei ght with a 10 mm diameter. Figure 6 3 Orientation and dimensions of erosion sample Table 6 2 Parameters for composite erosion and surface temp experiment C onstant Value Units flux 799 W/cm 2 composite 1.45 g/cm 3 composite 0.005 W/cm/K c composite 1.27 J/g/K T g 250 C Al 2.78 ** g/cm 3 Al 1.21 ** W/cm/K c Al 0.875 ** J/g/K *Assumed glass transition temperature for epoxy [27] **Assumed Aluminum 2024 T3 PAGE 89 89 The experiment was conducted unde r normal atmospheric conditions ( open air ) The laboratory result for this experiment was a surface reduction of 2.6 mm. The DEFFCON results are presented in Figure 6 4 Figure 6 4 DEFFCON results for erosion experiment of composite material In this figure the title DEFF 100x refer to the DEFFCON results using a total meshing of 100 x nodes. As can be seen the model shows little sensiti vity to meshing. The y nodes were kept constant at a total of 15; since no variation in temperature distribution exists in the y direction, nodal analysis was not conducted. Figure 6 4 also shows an end result of about 2.1 mm wh ich pro vides for a relative error of 19% PAGE 90 90 Surface Temperature Study A second experiment was conducted in which the numerical model from [26] was used to determine the surface temperature of another sample. The dimensions of th e sample are given in Figure 6 5 This sample had the same material properties given in Table 6 2 as well as the same incident flux of 799 W/cm 2 No experimental results are supplied for the actual surface temperature of this sample. What is given is th e aluminum substrates temperature which was measured by an inserted probe. This temperature is used along with the progressive heating option in the DEFFCON simulation. adjusted to gi ve equivalent surface area, i.e. diameter of 11.284 cm. Figure 6 5 Sample dimensions for surface temperature model Figure 6 6 shows the results of both the numer ical model presented in the literature alongside the DEFFCON prediction. The legend follows the same format explained earlier. PAGE 91 91 Figure 6 6 Surface temperature calculations results for composite material The relative error between the numerical solutions is approximately 17%. Some further analysis is presented on the differences in shape. Further Analysis There exist some inconsistencies between the two numerical models that require some discussion. presented here does a significantly better job of predicting surface erosion. There are differences in the underlying theory when compared with the literature so it is necessary to alter the DEFFCON model in order to surmise where variation is the greatest. The ity and the boundary conditions; particularly the mass blow off relationship. The major dissi milarities between the boundary conditions used in the literature and the conditions used in DEFFCON are the convective heat boundary, and the PAGE 92 92 energy released when pyrolysis gas comes off the sample. The conditions significantly reduce the net flux to the system from 799 to 329 W/cm 2 Adjusting the flux in the DEFFCON model as well as using a different fit for the thermal conductivity produces results closer to the reported experimental value of surface erosion. However, this alteration causes DEFFCON to surface temperature. Figure 6 7 shows an adjustment made to roughly fit the reported theoretical thermal conductivity model using Equation 6 5. Figure 6 7 Theoretical fit to literary model thermal conductivity This rough approximation produces some interesting results when applied to the surface temperature and erosion experiments, Figure 6 8 and 6 9 sho ws this analysis PAGE 93 93 Figure 6 8 Surface temperature numerical model comparison with adjusted DEFFCON with 329 W/cm 2 Figure 6 9 Erosion results for Bahramian numerical model, adjusted DEFFCON with 329 W/cm 2 and original DEFFCON with 799 W/cm 2 (DEFF 100X) PAGE 94 94 The figure above shows the numerical results reported by Bahramian for the erosion of the sample, as can be seen the surface loss of 1.8 mm falls significantly short of the reported 2.6 mm. Thus it is may be appropriate to assume that the numerical model for the surface temperature, Figure 6 8, is also an underestimatio n but there are boundary layer condition, coupled with the original thermal conductivity relation (Equations 2 3 & 2 4), adequately compensate for the removal of energy fr om the surface of the material. This analysis shows that it is important to have accurate thermophysical relations, specifically the thermal conductivity as it is the driving parameter in this simulation. Automotive Paint Studies Since the main materia l of interest for this research is a multi layer paint system, particularly that of an automobile, it is appropriate to use DEFFCON to analyze some degradation and other affects on car paint. The following details a clearcoat transmittance study coupled w ith the results of varying the percent pigmentation in the polymer matrix. Basecoat Surface Temperature Factors that affect the mass loss of the automotive paint system include how much energy is absorbed by the system as well as how much energy is tran smitted through the clearcoat. It is apparent in other literature [28] that different amounts of degradation will be observed depending on whe ther the paint is black or white The black paint absorbs the most energy and will therefore see the most mass l oss. These two colors represen t the extremes of absorption. The only thing considered here is the absorption Upon examination of automotive paint, many different additives for PAGE 95 95 aesthetic appeal are discovered. If, for instance, the paint had metal flak es then reflection would be significant. The following only considers black paint while varying the weight percent of mica in a polyurethane matrix. The transmittance is also varied to determine how the temperature is affected. Table 6 3 shows the change in the density, thermal conductivity, and specific heat capacity with increasing mica content. Table 6 3 Basecoat and mica thermal properties (W/cm/K) (g/cc) c (J/g/K) M ica (wt %) P olyurethane (wt %) t (cm 2 /s) 0.0071 0.986 0.5 100 0 0.0144 0.0036 1.20 6 1.41 30 70 0.0021 0.0041 1.174 1.28 40 60 0.0027 0.0046 1.143 1.15 50 50 0.003 5 Mica properties taken from [29] [30] The first row of this table represents pure mica powder which is considered to be the main component in that pigment material. Every other entry is referred to as basecoat, where its properties are simply a weighted combination in the same manner that the char properties are combined in Equations 4 8 through 4 10. Figure 6 10 to 6 12 show the three different basecoats (labeled BC1, BC2, BC3) with varying clearcoat absor ption They were all subject to the same incident heat flux of 45 W/cm 2 As can be seen in the figures the less the clearcoat absorbs the less linear the profile. This observation can be attributed to t he fact that the clearcoat does not have enough energy to impact the temperature of the basecoat; it instead serves as more of h eat sink to remove energy. As the energy is shared more evenly between the two, the temperature rises more rapidly due to the fact that the clearcoat no longer serves as a sink. PAGE 96 96 Figure 6 10 Basecoast tem perature with clearcoat absorption of 0.1 and mica content variat ion Figure 6 11 Basecoa st temperature with clearcoat absorption of 0.3 and mica content variation PAGE 97 97 Figure 6 12 Basecoast tempe rature with clearcoat absorption of 0.5 and mica content variation Back Calculation Using CANYON CANYON (Stachitas) is a code that has been developed to determine the effects of an urban environment on the weapon s ource term. One very important component in ability to determine the magnitude of observed fluence. It was observed that the fluence from the source as a function of yield can always be fitted by a power function [25] : (6 8 ) Where; = fluence (cal/cm 2 ) = Constants determined through least squares po wer fit Figure 6 13 shows a least squares fit for various distances down a canyon that is 40 m wide and 100 m high that was assumed to be a general street canyon [8] Three PAGE 98 98 distances are included with the fluence calculations f or 1, 5, 10, 15 and 20 kT. The data points of which proceed from left to right. Figure 6 13 Power fits for yields (points left to right) 1, 5, 10, 15, and 20 kT and distances down a street canyon Procedu re The procedure to back calculate the yield f rom a fluence using CANYON is simple. In the input file, Figure 6 16, at the very bottom there is a section with a label boun two data points to determine the constants present in Equation 6 8. CANYON, therefore, takes these two yields and calculates the fluence for both of them and produces a p is calculated by CANYON For a known mass loss of a particular material, steps are taken to determine the fluence that produced this loss with DEFFCON; this is accomplished in much the same PAGE 99 99 way as the yield is calculated using CANYON. Thus knowing the fluence calculated by DEFFCON, the yield of the w eapon and the unknown mass loss can be determined using CANYON and the previously outlined steps p Through a series of runs on several material cases, it is determined that polymers degrade along a 2nd order polynomial: (6 9) Where; = fluence (cal/cm 2 ) = mass lo ss of polymer (mg) = constants determined through least squares 2nd order polynomial fit Figure 6 14 shows the results of these simulations on the composite material (comp) from the case study and a monola yer of black polyurethane (poly) of the same dimensions. The simulations are conducted at a distance of 708 m away from the weapon using the same canyon width and height described earlier. Therefore if the only parameter that is known is a mass loss of p olymer ( ) the fluence can be obtained Figure 6 14 Mass loss (cumulative) using output files from CANYON for two polymers with yields (data points left to right) 1, 5, 10, 15, and 20 kT PAGE 100 100 A 2nd order polynomial least squares fit requires three data points to determine all th ree constants. Thus three simulations using CANYON m ust be completed initially so that they can be used in DEFFCON. The distance must be constant and the yields must vary; the yields should not be close together to insure a wide range of degradations produced by DEFFCON. Three mass losses are then produced by DEFFCON, which can be used with the three fluences outputted by CANYON, to construct the equations neces sary to determine the three polynomial coefficients. Since there are three unknowns a system of three linear equations must be solved to determine these coefficients. (6 10) (6 11) (6 12) Once the coefficients are determined, the fluence is then back calculated using the fitted polynomial and placed into CANYON for the final yield calculation. Figure 6 15 provides a flow chart outlining Once the user is familiar with the sequence of computation, the process can be made more efficient by saving particular CANYON outputs for reuse. This would decrease the computational time between simulation s significantly. PAGE 101 101 Figure 6 15 Example Using Automotive Paint Damage rmine fluence from a specified mass loss. The receiver, the car, is placed a distance of 708 m 1000 m away from a detonation. The paint has properties specified by Table 4 1 and the visib ility to the car is 999 km (100% transmis sion of the source). The canyon is of average width (40 m) and height (100 m). A visual representation of the two scenarios considered is presented in Figure 6 16 Here the weapon is represented by the circle with the star in t he middle. The numbered receivers are placed between two continuous uniform walls. Figure 6 17 shows the result of the mass loss at yields (from left to right) of 1, 5, 10, 15, and 20 kT. The mass losses are from simulations on a sample that is 1.0 cm in diameter and has layer thicknesses given in Table 6 4 PAGE 102 102 Figure 6 16 Automotive paint degradation scenario, 2 recievers 708 m an d 1000 m from source in a canyon 40 m wide and 100 m tall Table 6 4 Thickness of layers in automotive sample Constant Thickness ( m) Clearcoat 45 Basecoat 20 Primer 23 Zinc phosphate 1.5 Aluminum 1016 Figure 6 17 Degradation of automobil e 708 m away from source with yields ( data points left to right) 1, 5, 10, 15, and 20 kT PAGE 103 103 Referring back to ( the last part of the input file ), a point of interest of 3. 895 milligrams is chosen. This works out to be a material loss of approximately 5 mg/cm 2 for an automotive paint sample. correspond to the first, second, and last points on Figure 6 17 which was produced with a fluence of 5.25 cal/cm 2 is calculated This result is completely reasonable when re examining Figure 6 17 This fluence value, when placed into CANYON, produces a yield of 1.9 kT which coincides with Figure 6 13 This distance of 708 meters proves to produce appreciable damage Other distances could be chosen but it is the recommendation based on this research that suggests a dis tance under 1000 m. The distance from the source is of significant importance due to the blast wave effects described earlier. Ground collectors must be able to find a sample that has experienced enough quantifiable damage. Figure 6 18 shows the results of the simulation carried out at 1000 m. Figure 6 18 Automotive simulation at various yields and 1000 m away from source (data points left to right) 1, 5, 10, 15, and 20 kT. PAGE 104 104 The further one gets from the source the less the 2nd order polynomial fits the d egradation profile. This fit however, is still within 20% of the data points. It is also important to notice that at this distance significant dam age is not experienced until behavior of the buildings. Only a fraction of the energy produced by a nuclear detonation will be propagated down the street canyon. and Table 6 6 show the results of the DEFFCON simulations, they include times, yields, fluences and time it took for computation to complete (comp time). The computer used for these simulations contained an Intel Core i7 920, 2.67 Ghz, 4 core and has 8 logical processors. Table 6 5 Automotive simulation 1000 m from source yield (kT) (cal/cm 2 ) loss (mg) pulse time (s) comp time (min) 1 1.72 0.0006 0.86 322.2 5 2.96 0.01 40 1.74 652.2 10 3.74 0.053 4 2.35 940.1 15 4.29 0.112 0 2.8 0 1125.5 20 4.72 0.1837 3.2 0 1268.3 Table 6 6 Automotive simulation 708 m from source yield (kT) (cal/cm 2 ) loss (mg) pulse time (s) comp time (min) 1 4.22 2.139 0.86 366 5 7.24 5.616 1.74 739 10 9.14 6.978 2.35 997.8 15 10.47 7.627 2.8 0 1188.2 20 11.52 7.965 3.2 0 1335.3 PAGE 105 105 CHAPTER 7 CONCLUSIONS AND FUTU RE WORK The research contained in these pages produced a program called DEFFCON for the modeling of thermal deg radation of polymer systems, specifically automotive paint. Furthermore the complexities of handling the multi layer system with one semi transparent layer are discussed. A series of assumptions are made in order to render the program fit for a wide ran ge of polymers as well as to, somewhat, ease computational efforts. Validation of these assumptions is s hown in the analysis of the composite thermal heat shield. Accurate characterization of Arrhenius parameters as well as thermophysical relations is needed to obtain realistic results. Degradation behavior is dependent on these values and the variation is obvious when comparing different ma terials. As is shown in the basecoat/mica case study, minor changes in values such as the conductivity, density and heat capacity have little consequence. Concomitantly, while different degradation theories exist, that which is outlined here proves to be superior. Minim um Arrhenius parameters are required and simplifying assumptions, such as constant char yield, a re shown to be reasonable When used in tandem with the code CANYON, which propagates the radiative energy down the street canyon simulating degradation of the paint system in an urban environment is made possible. It is determined, using CANYON, that locating samples at distances near a half mile of the epicenter of the source is necessary to observe appreciable amounts of damage in an average street canyon. Further explanation of the minimum forensic distance is given in [31] Included in CANYON i s the ability to back calculate to a weapon yield when delivered fluence is known. Thus by developing PAGE 106 106 DEFFCON to calculate a fluence based on a mass loss, using a 2nd order polynomial fit, the ultimate task of this research is obtained. The model, as it currently stands, accounts for a variety of different physical effects. The major question, as stated earlier, is how a system that has a transparent layer on top of an absorbing layer will act under the specified conditions. Recent d iscussions with lab oratory technicians dictate that bubble s will form at the clearcoat/basecoat interface and delay mass loss This delay may occur due to the fact that the vapors produced at this location cannot escape. Once they escape the mass changes drastically, much like if a balloon were to pop; if the gas contained was of significant mass then loss would be considerable over a small time frame. Upon further discussion it is apparent that a threshold temperature, and thus distance from source, exists where bubble f ormation occurs. A way to eliminate the error associated with these bubble would be to determine this threshold distance and only take samples beyond this location. Therefore more laboratory work is very important in verifying the findings of this work. Finally, f urther sensitivity studies should be conducted. Since the pulse of the weapon is very short, less than 1 second for 20 kT, analysis should be done on the rate parameters ( ) to ensure that reaction time is not an issue. S tudies should also be conducted on variation of assumed parameters, such as the type of aluminum used in the Bahramian case studies. PAGE 107 107 APPENDIX A AIR THERMOPHYSICAL P ROPERTIES PAGE 108 108 PAGE 109 109 LIST OF REFERENCES [1] Glasstone, Samuel and Dolan, Philip J. The Effects o f Nuclear Weapons. s.l. : United States Department of Defense, 1977. [2] Brown, Thodore L., et al. Chemistry The Central Science. Upper Saddle River : Prentice Hall, 2003. [3] Fogler, H. Scott. Elements of Chemical Reaction Engineering. Upper Saddle River : Prenti ce Hall PTR, 2006. [4] Atkins, Peter and de Paula, Julio. Physical Chemistry. New York : W.H. Freeman and Company, 2002. [5] Harper, Charles A. Handbook of Building Materials for Fire Protection. New York : McGraw Hill, 2004. [6] Pyrolysis of varnish wastes based on a polyurethane. Esperanza, M. M., et al. 52, Alicante : Journal of Analytical and Applied Pyrolysis, 1999. [7] Bridgman, C. J. Introduction to the Physics of Nuclear Weapons Effects. s.l. : Defense Threat Reduction Agency and Air Force Institute of Technology, 2001. [8] Stachitas, Tucker. Evaluation of 3 D Radiant Heat Transfer in Street Canyons. s.l. : University of Florida, 2009. [9] Geankopolis, Christie John. Transport Processes and Separation Process Principles. s.l. : Prentic Hall, 2003. [10] Cameron, John R., Skofroni ck, James G. and Grant, Roderick M. Physics of the Body. s.l. : Medical Physics Publishing, 1996. [11] Perry, Robert H., Green, Don W. and Maloney, James O. Perry's Chemical Engineers Handbook 7th ed. New York : McGraw Hill, 1997. [12] Gokel, George W. Dean's Handbo ok of Organic Chemistry 2nd Ed. New York : McGraw Hill, 2004. PAGE 110 110 [13] Mark, James E. Polymer Data Handbook. New York : Oxford University Press, 1999. [14] Chemical surface characterizatoin and depth profiling of automotive coating systems. Adamson, K. Philadelphia : Progress in Polymer Science 2000; [15] 25:1363 1409. [16] Determination of active HALS in weahter automotive paint systems I. development of ESR based analytical techniques. Kucherov, A. V., Gerlock, J. L. and Matherson Jr, R. R. s.l. : Polymer Degra dation and Stability 2000; 69:1 9. [17] Ciba Tinuvin 1130. [Online] Ciba. [Cited: May 5, 2008.] http://cibasc.com/tinuvin_1130 2.htm [18] A UVA/HALS Primer: Everything You've Ever Wanted to KNow About Light Stabilizers Part I. s.l. : Metal Finishing 1999;97 [5] :51 5 3. [19] Incropera, Frank P. and Dewitt, David P. Fundamentals of Heat and Mass Transfer. s.l. : John Wiley & Sons, Inc., 2002. [20] Predicting the In Service Weatherability of Automotive Coatings: A New Approach. Bauer, David R. s.l. : Journal of Coatings Technology 1997;69(864):85 95. [21] Panush, Sol. Pearlescent Automotive Paint Composition. 4551491 United States of America, November 5, 1985. material. [22] All Metals & Forge [Online] All Metals & Forge 2009. [Cited: May 29, 2010.] http://www.steelforge.com/metalmelting range.htm [23] Henderson, J. B., Wiebelt, J. A. and Tant, M. R. A Model for the Thermal Response of Polymer Composite Materials with Experimental Verification. Journal of Composite Materials. s.l. : Sage, 1985. Vol. 19. [24] Ostrach, Simon. An analysis of Laminar F ree Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of The Generating Body Force. s.l. : National Advisory Committee for Aeronautics, 1953. PAGE 111 111 [25] Sjoden, Glenn E. Foundatoins in Applied Nuclear Engineering Analysis. s.l. : World Sc ientific Publishing Co. Pte. Ltd., 2009. [26] Stachitas, Tucker. CANYONS Users Manual Version 12.0. 2009. [27] Ablation and Thermal Degradation Behaviour of a Composite Based on Resol Type Phenolic Resin: Process Modeling and Experimental. Bahramian, Ahmad Reza, et al. s.l. : Polymer, 2006, Vol. 47. [28] Prediction of the Glass Transition Temperatures for Epoxy Resins and Blends Using Group Interaction Modelling. Gumen, V. R., Jones, F. R. and Attwood, D. s.l. : Polymer, 2000, Vol. 42. [29] Bauer, William A. Determination of N uclear Yield from Thermal Degradation of Automobile Paint. s.l. : Air Force Institute of Technology, 2010. [30] The Enginnering ToolBox. [Online] [Cited: September 6, 2010.] http://www.engineeringtoolbox.com/ [31] Walker, Roger. simetric.co.uk. [Online] 2009. [Cite d: September 6, 2010.] http://www.simetric.co.uk/si_materials.htm [32] Koehl Michael A. Thermal Flash Simulator. s.l. : Air Force Institute of Technology, 2009. PAGE 112 112 BIOGRAPHICAL SKETCH Todd Anthony Mock was born on an autumn morning in the desert of Tucson, Arizona where he lived for six years before moving to central Florida. He is the son of Ronald and Elaine Mock and brother to Travis Mock. He received a Bachelor of Science d egree in c hemical e n gineering with a specialization in p rocess e ngineering from the University of Florida in 2008. sailing while his brother yells at him for sailing incorrectly, and target shooting. 