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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 DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2010

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© 2010 Todd Anthony Mock

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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.

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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.

I‟d also like to acknowledge Tucker 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.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF TABLES ............................................................................................................ 7

LIST OF FIGURES .......................................................................................................... 8

ABSTRACT ................................................................................................................... 11

CHAPTER

1 INTRODUCTION .................................................................................................... 13

2 A PRIMER ON POLYMER DEGRADATION .......................................................... 16

Discussion Of Arrhenius Parameters ...................................................................... 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 Properties .................................................................................................. 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 ....................................................................... 52

Solving The Heat Conduction Equation .................................................................. 53

Explicit Finite Difference ................................................................................... 54 Stability ............................................................................................................. 59 Crank-Nicolson ................................................................................................. 60 Equation Solution Method ................................................................................ 62

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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

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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 Constants 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

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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

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5-7 Temperature profile for beryllium wall with 001.0t s and 08.0x cm. .......... 63

5-9 Polymer mass fraction being converted to char. ................................................. 68

5-10 Specific heat as a function of char 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 “Out.put” output file for DEFFCON ..................................................................... 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 fluxes for 1 kT and 708 m from the source ................................................................................................... 81

6-1 Bahramian experimental and Bahramian theoretical data for composite ............ 84

6-2 DEFFCON theory overlaid with Bahramian experimental data ........................... 84

6-3 Orientation and dimensions of erosion sample ................................................... 88

6-4 DEFFCON results for erosion experiment of composite material ....................... 89

6-5 Sample dimensions for surface temperature model ........................................... 90

6-6 Surface temperature calculations results for composite material ........................ 91

6-7 Theoretical fit to literary model thermal conductivity ........................................... 92

6-8 Surface temperature numerical model comparison with adjusted DEFFCON with 329 W/cm2 ................................................................................................... 93

6-9 Erosion results for Bahramian numerical model, adjusted DEFFCON with 329 W/cm2 and original DEFFCON with 799 W/cm2(DEFF 100X) ..................... 93

6-10 Basecoast temperature with clearcoat absorption of 0.1 and mica content variation .............................................................................................................. 96

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6-11 Basecoast temperature with 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 polymers with yields (data points left to right) 1, 5, 10, 15, and 20 kT ................................................. 99

6-15 Flow chart for “inverse yield problem” ............................................................... 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

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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 Major: 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 balance of the material before and after

a heat fluence is applied. It is the task of this research to determine the heat fluence

from an assumed improvised nuclear device (IND) incident on a material surface that

has undergone thermal degradation.

We developed DEFFCON (Degradation Effects From Flux CONduction), a two

dimensional, transient heat transfer algorithm, for this research to characterize the mass

loss of paint coated systems. When used in tandem with another code, CANYON

(Stachitas, 2009), which solves the transport of thermal energy through a street canyon,

the “inverse yield problem” is solved; whereby knowing the attributed material mass loss

one can directly determine the thermal fluence attributable to a weapon yield.

This document analyzes 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. Code accuracy is validated using analytical solutions to a typical heat

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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 these 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 systems can be expected. It is determined that a sample of

automotive paint, with 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/cm2 of material. This mass loss can acceptably be measured

using standard laboratory techniques.

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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 for the development of the

improvised nuclear device (IND). One such task will be for National Technical Nuclear

Forensics (NTNF) to determine the yield of the device (“inverse yield problem”). A

proposed method here is to analyze painted surfaces exposed to the thermal radiation,

such as automobile paint, in order to determine total thermal energy radiated from the

detonation. Automobiles offer an excellent “before and after” picture of the event due

to their shape; one side faces the radiation, while the backside is shielded from it.

Radiative heat will propagate 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 nuclear detonation will shadow

other objects from the weapon‟s fireball. Figure 1-1a and Figure 1-1b show these

shadowing effects. The wooden poles (Figure 1-1a) were 1.17 miles from ground zero

at Nagasaki and experienced 5 to 6 cal/cm2. Charring 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 1-1b scorching of paint

on a gas container was observed 1.33 miles from ground zero at Hiroshima except

where protection 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.

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A) B) Figure 1-1. A) shadowing 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 would

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

face the source. The work presented assumes thermal radiation is normal to the

surface and represents the worst case scenario for damage.

Figure 1-2. Shadowing effects for automobile.

A computer 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

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set of user inputs. 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 polymer.

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 CANYON,

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 either 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 demonstrate 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 details the development of DEFFCON: its validation,

finite differencing equations, application of theory and how simulations are conducted.

Case studies and analysis of results are presented in Chapter 6, while Chapter 7

provides concluding statements.

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CHAPTER 2 A PRIMER ON POLYMER DEGRADATION

Polymers are constructed from a series of atoms most frequently carbon,

hydrogen, nitrogen and oxygen. These atoms are held together by electrostatic forces.

Stability and strength of the bond depends on the atoms involve along with their

independent electric charge. Carbon-carbon bonds are particularly stable and are

present 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 from 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 would 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 individual molecules ranging from hundreds

to thousands.

Figure 2-1. Molecular structure of ethylene.

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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 reducing 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 characterized using rate kinetic equations,

which provides the opportunity for 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 radicals. 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

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Where;

ik = rate constant for production of reactive intermediate (1/time)

rk = rate constant for recombination (1/time)

gk = rate constant for production of vapor (1/time)

ck = 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,

nk , can be represented by an equation of the form:

RTE

naAek

(2-1)

Where; A = pre-exponential factor or frequency factor (1/time), usually empirical

aE = activation energy (J/mol)

R = gas constant (J/mol/K) T = 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 temperature ranges [3].

Discussion Of Arrhenius Parameters

The energy required to distort and break these atomic bonds can be linked with

the activation energy, aE , 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].

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Figure 2-3. Activation energy

The exponential term, RTEae , represents the probability that an interaction will have

energy above the activation energy. Therefore the pre-exponential factor, A , 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 individual 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 material 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 heat 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:

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cg

cc

kk

k

m

mTY

0

(2-2)

Where;

m = mass at infinite time (g)

0m = initial mass (g)

TYc = char yield at temperature T , and ck , gk are constants as defined in Figure 2-2.

As time approaches infinity this model predicts a finite char yield when 0ck . While

there are empirical formulas to determine the char 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 the temperature.

Correlations to predict these properties based 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 contributions if the

composition of the polymer is known. To 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]:

g

g

g TTT

TT

22.0

(2-3)

g

g

g TTT

TT

2.02.1 (2-4)

Where;

T , gT = Absolute and glass transition temperature respectively (K)

gT = Thermal conductivity at glass transition temperature (W/(cm K)).

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The relation for the density and heat capacity are:

0

0

11TTB

(2-5)

Txcc 3

0 106.114

3 (2-6)

Where;

0 = density at temperature KT 2980 , in (g/cm3) 41025 xB = volume thermal expansivity per unit mass cm3/(g K)

0c = heat capacity at temperature KT 2980 , in (J/(g K)).

Derivation Of Simple Degradation Model

To obtain a relation for the mass of the polymer as a function 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 intermediate 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]:

IkPkdt

dPri (2-7)

IkkkPkdt

dIcgri (2-8)

Ikdt

dGg (2-9)

Ikdt

dCc (2-10)

Where;

ik = rate constant for production of reactive intermediate (1/time)

rk = rate constant for recombination (1/time)

gk = rate constant for production of vapor (1/time)

ck = rate constant for production of char (1/time)

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dt

dP= rate of change of polymer loss (mass/time)

dt

dI= rate of change of reaction intermediate (mass/time)

dt

dG= rate of change of gas formation (mass/time)

dt

dC= rate of change of char formation (mass/time).

All of these equations are solved for instantaneous amount of each species. It is

important to account for these reactions because in total they represent the amount of

solid material. It can be shown that the overall rate constant for pyrolysis is:

rip Kkkk . (2-11)

Here;

cgi

i

kkk

kK

(2-12)

This equation 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 fraction can be presented as [5],

0

1P

TPTYTY

dt

dmcc , (2-13)

where 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, 0PTP , is related to the pyrolysis reaction rate, and is

obtained by integration while considering non-isothermal conditions

P

P

T

T

a

t

p dtRT

EAdtk

P

dP

0 0

exp0

(2-14)

Where;

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= constant heating rate (K/min)

0TT at 0t .

The introduction 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]:

RTE

RTk

RT

E

RTE

ARTdt

RT

EA

a

pa

a

T

T

a

2exp

2exp

22

0

. (2-15)

Where

RT

EAk a

p exp

And the value of is defines a:

RTE

RTk

a

p

2

2

. (2-16)

Equation 2-14, the relative polymer fraction remaining from thermal irradiation,

becomes:

eP

TP

0

. (2-17)

The final mass fraction equation for non-isothermal heating (constant heating rate)

based on relative char formation is:

exp1

2exp1

2

0

cc

a

p

cc YYRTE

RTkYY

m

Tm (2-18)

Where;

Tm = mass at temperature absolute temperature T , in (g)

0m = initial mass (g)

cY = temperature independent char yield

= heating rate (K/min)

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pk = pyrolysis rate constant (1/min)

R = gas constant (J/mol/K)

aE = 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 Esperanza [6] was considered 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. The 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. Polyurethane waste varnish degradation [6].

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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 model 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 material degradations with input restricted 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 to as the default model. For 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.

Chapter 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 will 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 established based on theory presented earlier in this section. Figure 2-5 shows

the result of this fit.

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Figure 2-5. Theory and experimental fit.

The theoretical fit has a mean relative 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

Ea 89.5 kJ/mol

A 2.00E+07 1/min

R 8.30 J/mol/K

β 20 K/min

Yc 0.1 g/g

These values are optimized to ensure minimum error for early degradation versus

temperature behavior. Because 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:

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Table 2-2. PMMA Arrhenius parameters

Constant Value Units

Ea 160 kJ/mol

A 9.00E+09 1/min

R 8.30 J/mol/K

β 20 K/min

Yc 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 activation 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 degradation compared to measured varnish degradation

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CHAPTER 3 SOURCE TERM EFFECTS AND THERMAL RADIATION

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

consideration 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 delivered 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 extent it

is important to be concerned with the blast wave produced and its radius of destruction.

Thermal Radiation

The fraction of the total yield released as thermal energy is known as the thermal

partition. For air bursts below 15000 ft this value is experimentally shown to be 0.35 for

all yields from 1kT to 10MT [1]. The effective 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]:

4

2

max1

2

PtP . (3-1)

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Where tP and maxP have units of J/s, maxP takes the form:

56.03

max 1033.1 KTYxP (3-2)

The parameter is normalized time, maxtt , where maxt , has units of seconds;

44.0

max 0417.0 KTYt (3-3)

These equations all depend on the yield of the weapon represented as KTY , which has

units of kilotons, referring to the equivalent quantity of TNT (trinitrotoluene).

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/cm2. 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:

24 r

tPF

(3-4)

Where; F = power delivered to receiver (W/cm2) r = slant range (cm) = atmospheric transmittance

= material absorption 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 weapon has detonated 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. The material absorption factor is

representative of how well the receiver will accept the thermal radiation. An absorption

factor of 1.0 is equivalent to 100% acceptance.

30

Figure 3-1. Slant distance illustration

These equations make up what is considered 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 Radiant Heat Transfer

CANYON is another computer code used in this research for the analysis of the

effects of nuclear 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 geometry; 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 delivered energy by approximately 25% compared

to a source with no channeling effect. The term “average” refers to a 1 mile long street,

31

with parallel buildings separated by a street 40 m wide. The buildings themselves are

100 m tall.

Surface Re-Radiation

When 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 will 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/cm2. 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 one that absorbs all radiant energy and reflects

none.

1 (3-5)

Where; = fraction absorbed (equal to 1 if black body)

= fraction of thermal energy emitted from surface (equal to 1 if black body).

The absorption and emissivity terms can 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 the 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.

4Tq (3-6)

32

Here; q = power emitted (W/cm2)

1210676.5 x , Stefan-Boltzmann constant (W/cm2/K4)

T = absolute surface temperature (K). In practical applications it is necessary to include the receiving surface which in this

case would be the ambient air; this then makes Equation 3-7:

44

TTq (3-7)

Where;

T = absolute temperature of the ambient air (K)

The implementation of this equation is shown in the code development section of this

document.

Blast Wave 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. Pressures 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 from 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 damaged as this one. The structural integrity of the object

determines whether or not 5-6 psi overpressure is a good estimation of where to look for

33

samples. As a point of reference for the severity of pressure from an explosion,

consider the ear drum that bursts 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

found 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 obtain this kind of data; tables were

consulted for the numbers presented in this document [1].

34

Figure 3-3. Peak overpressure from a 1-kiloton free air burst from sea-level ambient

conditions [1].

35

CHAPTER 4 MODEL CONCEPTUALIZATION

A typical automotive paint scheme consists of multiple layers 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 substrate represents the

body panel. The material properties are discussed in detail in the following sections.

Figure 4-1. Paint layers and relative dimensions.

Material Properties

Automotive paint schemes are extremely diverse and information about their

chemical and physical properties is hard to come by. Initial efforts were made to collect

data 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 this reason some

assumptions are made in an attempt to simplify and generalize the model. A good

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 κ (W/cm/K) ρ (g/cm3) c (J/g/K) α

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 Phosphate 0.0052 4.00 0.13 1.0

Steel 0.5400 7.80 0.49 1.0

* assumed **depends on weight percent pigment

(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

eventual 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 transformation 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.

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 chain 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, and 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 % transmittance in the near

infrared region (700 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 materials is the active assumption. This assumption is validated

by examination of 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 encompass the infrared region.

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 Chemicals [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 the 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 energy 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

used for optimum protection.

39

Figure 4-4. Keto-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

xIdx

dI (4-1)

Where; = absorption constant (1/cm)

xI = energy intensity as a function of distance into the material.

Integration will give the intensity, thus assuming an initial intensity at the surface of the

material, 00 II . The solution obtained, known as Beer‟s Law is:

xeIxI 0 . (4-2)

40

The ratio of the initial surface intensity to the intensity at the end of the transparent

material, Lx , is the transmitted fraction.

LeI

LI 0

. (4-3)

Assuming no reflection off the surface of the material transmittance and absorption, ,

are related as follows:

1 . (4-4)

This absorption, , depends on how much UVA is present in the material. The 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/clearcoat 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:

UVAIk

dt

UVAd (4-5)

AI

A

303.2

101 (4-6)

Where; k = loss rate (1/time) A = absorbance

UVA = concentration of UVA (particles/volume)

I = light intensity. And since absorbance is proportional to the concentration of the UVA;

Akdt

dA 101 (4-7)

Thus by integration;

110101log 0 ktktAtA . (4-8)

41

Relations 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

protection; 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 exception 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) may contain anywhere

from 1-50% pigment by weight. The overall size of the metal oxide/mica particle ranges

from a fraction of 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). This 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

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 heterogeneous polymer system (polymers

with pigments) will have an effect on thermal stability of the system. It may become

more difficult to vaporize the material because of the limited ability of the polymer bonds

to obtain enough vibrational energy to break.

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 not 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 that 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

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 material 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.

For any polymeric material the char yield is considered to be constant with respect

to temperature. As time progresses, however, the amount of polymer left on the sample

will vary as the char “grows-in”. This will have an effect on the mass of the sample as

well as the thermal conductivity, specific heat and density. To account for these

changes a simple linear combination of the temperature dependent properties is

proposed [22]. These relationships operate under the assumption that the polymer and

char are thoroughly mixed, and that no new material with unique properties is produced.

Thus for the thermal conductivity, specific heat and density:

cv FF 1 (4-9)

cv cFFcc 1 (4-10)

cv FF 1 (4-11)

Where the subscript v and c denote the virgin polymer and char properties

respectively. The coefficient, F , represents the fraction of the material that is still

polymer and is given by:

c

cf

Y

YmF

1. (4-12)

44

Here;

fm = polymer mass fraction

And cY 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 relations for the char material is given as:

3

4

2

321 TTTc (4-13)

Tccc 21 . (4-14)

Where T 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

κ1 0.955 W/m/K

κ2 8.42E-04 W/m/K2

κ3 -4.07E-06 W/m/K3

κ4 5.32E-09 W/m/K4

c1 0.870 J/g/K

c2 1.02E-03 J/g/K2

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 density of the char is found to be 80% of the virgin polymer. It is therefore

assumed 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.

45

Boundary Layer Theory

In heat transfer calculations it is necessary to consider 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 discussion 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 surface. Consider a vertical

surface as depicted in Figure 4-6, as air moves across this surface at a velocity of U a

velocity gradient will be created due to this “sticky” condition. The neighboring particles

will 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 ( U ). The distance where the velocity of the

particles is near that of the free stream, about U99.0 , is considered the boundary layer

thickness .

The thermal boundary layer is present due to the differences in the temperatures

of the surface and the surrounding air. Similarly the thickness of the layer, T , is the

distance where the temperature of the fluid is about T99.0 . More appropriately the

term H in Figure 4-7 should be zero. Here T is the temperature of the ambient air.

Heat transfer through the boundary layer is considered to be conduction rather than

convection. These two boundary layers are approximately equal for hot gases, i.e.

T .

46

Figure 4-6. Thermal and velocity boundary layer

Xy

u

x

u

x

p

y

u

x

uu

2

2

2

2

(4-15a)

Yyxy

p

yxu

2

2

2

2

(4-15b)

Where; = fluid density inside boundary layer

y

u

, = y-component velocity and gradient

x

uu

, = x-component velocity and gradient

x

p

= x-direction pressure gradient

2

2

2

2

,,y

u

x

u

= viscosity and Laplacian

X = body force Equation 4-15b will not be considered due to the velocity boundary layer

approximations:

47

u and yxx

u

y

u

,, .

Simplifying assumptions can be made to Equation 4-15a so that it is easier to

manipulate. 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:

gX (4-16)

2

21

y

ug

x

p

y

u

x

uu

. (4-17)

Where the kinematic viscosity now appears,

, along with gravity, g . 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 gradient of the outside

region where 0u :

gx

p

. (4-18)

Here;

= fluid density outside boundary layer

Substituting Equation 4-18 into 4-17 the following is observed:

2

2

y

ug

y

u

x

uu

(4-19)

The first term on the right hand side of the equation can be related to the fluid property

known as the volumetric thermal expansion coefficient.

48

pT

1 (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 presented in the following approximate form:

TT

1. (4-21)

Substituting in Equation 4-19 yields

2

2

y

uTTg

y

u

x

uu

. (4-22)

However it should be noted that if the fluid is considered an ideal gas, which is the case

for the purposes of this research, beta will simply become; T

1 .

Mass (assumed constant density):

0

yx

u (4-23)

Energy:

2

2

y

T

y

T

x

Tu t

(4-24)

Where t is thermal diffusivity and has units of (m2/s). Further, if nondimensionalizing

parameters are introduced to Equation 4-22 the Grashof number is formed, which

represents the ratio of buoyancy to viscous forces of a fluid.

L

xx * (4-25a)

L

yy * (4-25b)

49

0

*

u

uu (4-25c)

0

*

u

(4-25d)

TT

TTT

s

* (4-25e)

Here L is a characteristic length (overall length of surface) and 0u is a characteristic

velocity. This transforms the momentum equation (4-22) into:

2

*2*

2

0

**

**

Re

1

y

uT

u

LTTg

y

u

x

uu

L

s

(4-26)

Where;

LuL

0Re . (4-27)

To retrieve the Grashof number the first term on the right hand side must be multiplied

by the square of the Reynolds number ( LRe ) to remove the velocity component giving:

2

3

LTTgGr s

L

. (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;

0y : 0u sTT

50

y : 0u TT .

To aid in solving these equations a similarity parameter is introduced as well as a

stream function to handle the velocity components

4/1

4

xGr

x

y (4-29)

4/1

44, xGr

fyx (4-30)

With the help of these relations the three partial differential equations can be reduced to

the following two ordinary differential equations:

023 *2 Tffff (4-31)

0Pr3 **

TfT . (4-32)

Where;

d

dff

Pr = Prandtl number =

.

After transformation of the boundary conditions a numerical solution is possible, the

results of which are presented in Figure 4-7 [23].

The appropriate boundary condition transformations are:

0 : 0 ff 1* T

: 0f 0T

This figure shows the 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 boundary layers.

51

Choosing a value of 6 for (to ensure temperature is equal to that of ambient) the

thermal boundary layer is then:

4/1

4

6

L

T

Gr

Ly (4-33)

Where L is the height of the object that is vertically 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.

Figure 4-7. Numerical solutions for velocity (left) and thermal boundary layer (right).

52

CHAPTER 5 DEFFCON CODE DEVELOPMENT

A two dimensional transient heat conduction code was developed 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 material 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 operates. 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 vertically 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 geometry. 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 experimental

findings explained in depth in Chapter 6. The more complex multi-layer model is also

explored in Chapter 6.

53

Figure 5-1. Vertical orientation of mono-layer polymer system

Solving The Heat Conduction Equation

The main task of DEFFCON is to solve the two dimensional time dependent heat

transfer equation:

q

y

T

x

T

t

T

t

2

2

2

21 (5-1)

Where;

t = thermal diffusivity (cm2/s)

= thermal conductivity (W/cm/K) T = absolute temperature (K)

yx, = horizontal and vertical position respectively (cm)

q = volumetric heat generation (W/cm3).

This equation can be solved numerically using a variety of methods. The methods

described in this research are the explicit and implicit finite differencing methods.

54

Explicit Finite Difference

The finite differencing schemes are broken up into three general equations;

surface, corner and interior. 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 explicit equations are given below along with a nodal representation. The

equations are extended to three points to preserve second order accuracy in space.

The derivations of the finite differencing equations are done using Schmidt‟s method

which requires a Taylor series expansion.

The Taylor series formula:

2

00

000!2

xxxf

xxxfxfcf (5-2)

The derivation is done by subdividing the system into nodes surrounded by control

volumes; these control volumes represent local conservation of energies. The

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 equations 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.

55

The derivation is as follows with the system of equations with weighting coefficients A ,

B , C :

!4!3!2

4

0

4

43

0

3

32

0

2

2

0

0

x

x

Tx

x

Tx

x

Tx

x

TTTA E (5-3a)

!4

16

!3

8

!2

42

4

0

4

43

0

3

32

0

2

2

0

0

x

x

Tx

x

Tx

x

Tx

x

TTTB EE (5-3b)

!4

81

!3

27

!2

93

4

0

4

43

0

3

32

0

2

2

0

0

x

x

Tx

x

Tx

x

Tx

x

TTTC EEE . (5-3c)

The summations to determine the weighting coefficients are:

0

0x

TCoeff (5-4a)

1

0

2

2

x

TCoeff (5-4b)

0

0

3

3

x

TCoeff . (5-4c)

These summations lead to the following equations in their respective order:

032 xCxBxA (5-5a)

1!2

9

!2

4

!2

222

x

Cx

Bx

A (5-5b)

0!3

27

!3

8

!3

333

x

Cx

Bx

A . (5-5c)

Simultaneous solution to this system of equations will give the individual weighting

coefficients as such:

2

5

xA

,

2

4

xB

,

2

1

xA

.

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 along with the

associated error term. This is done by summing all the terms in these equations;

0

4

42

0

2

2

2 12

11245

x

Tx

x

T

x

TTTT oEEEEEE

. (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 accuracy. 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:

0

2

2

4

42

2 12

2

y

T

y

Tx

y

TTT oSN

. (5-7)

Figure 5-3. Vertical nodes with surrounding control volume

The left hand side of Equation 5-1 can be written as:

t

TT

dt

dTnn

tt

0

1

011

(5-8)

Where; 1

0

nT = temperature of “0” position node at time tt

nT0 = temperature of “0” position node at time t .

Pulling all these equations together the equivalent explicit finite differencing formula for

Equation 5-1 becomes:

57

q

y

TTT

x

TTTT

t

TT nn

S

n

N

nn

EEE

n

EE

n

E

nn

t

2

0

2

00

1

0 22451 (5-9)

In this equation the current temperatures (superscript n ) are known, the new

temperature (superscript 1n ) is the one that will 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. However 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:

x

Tq

(5-10)

Where; q = surface flux (W/cm2).

The net flux on the surface is determined to be:

convapplied qqq (5-11)

With the following equalities:

24 r

tPFqapplied

, TThq surconv .

The applied heat flux is energy (from weapon) incident on the surface; the convective

term is a loss representing energy removal from the surface. Here surT is the material‟s

surface temperature.

58

To preserve second order accuracy for the boundary conditions only three nodes

are needed including the node of interest “0”. Thus referring to Figure 5-1 and only

considering the temperatures 0T , ET , EET , and the ambient temperature known T 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 0T , Equations 5-5a and 5-5b become:

12 xBxA (5-12)

0!2

4

!2

22

x

Bx

A . (5-13)

Therefore the weighting coefficients are:

xA

2,

xB

2

1.

This then leads to the derivation of the term for the first derivative of the temperature

with respect to the x-direction:

0

3

32

0 3

1

2

34

x

Tx

x

T

x

TTT oEEE

(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:

TThF

x

TTT n

EEE0

0

2

34 . (5-15)

Finally solving for nT0 in Equation 5-15 and 1

0

nT in Equation 5-9 the following are

formed:

32

2420

xh

TxFTxhTT

n

EE

n

En (5-15)

n

t

nn

S

n

N

nn

EEE

n

EE

n

En Ttq

y

TTT

x

TTTTT 02

0

2

01

0

2245

. (5-16)

59

Similarly an energy balance can be used to incorporate the boundary conditions. This

is done with the Crank-Nicolson equations presented later.

Stability

An instability 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 ( Fo )

expressed in Equation 5-17 [18].

25.0

2

x

tFo t . (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. If the user‟s inputs violate 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 given in Figure 5-4.

This figure is a snapshot from a Microsoft Excel solver spreadsheet, which is developed

for the purpose of determining the time step necessary for the polymer system to

converge given its thermophysical properties and overall dimensions. The figure

represents a multi-layered system much like the paint system presented earlier. The

number that drives the stability is that of the thermal diffusivity of the aluminum

substrate.

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 the 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 has 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:

q

x

TTT

x

TTT

t

TT

B

n

E

nn

W

A

n

E

nn

W

t

nn

2

11

0

1

2

00

1

0 22

2

1. (5-19)

Figure 5-5. Central node.

61

The averaging of the forward ( A ) and backward ( B ) differencing can be seen clearly.

The inclusion of the backward Euler method is what renders Crank-Nicolson an

unconditionally stable numerical approach. For the same physical position in the model

represented by the explicit Equations 5-15 and 5-16, the implicit Crank-Nicolson

equation becomes:

44

00

2

11

0

1

0

2

1

0

1

00

1

0

1

22

2

1

n

film

nn

n

S

nn

N

n

S

nn

N

nn

E

nn

E

t

nn

TTx

TTx

h

x

Fq

y

TTTTTT

x

TTTT

t

TT

(5-20)

Where the applied radiation (x

F

), the convection (

TT

x

h n

0 ) and surface re-radiation

terms ( 44

0

n

film

n TTx

) are all included by conservation of energy. Here the film

temperature ( filmT ) corresponds to the average between 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

( T ). 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

subtracted from it. This definition makes the inclusion of the energy terms into the finite

differencing equations quite simple.

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:

BTA

(5-21)

Where A

is the coefficient matrix of 1nT , and B

is the vector of known nT

temperatures. Gauss-Seidel iteration is used to obtain the values of the new

temperatures, 1nT . This iteration method requires an initial guess for the temperatures

and then solves each equation individually until the ith iterated solution is within a certain

tolerance of the i-1th solution, i.e. the infinity norm is minimized.

Code Validation

DEFFCON is validated against an analytical solution 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:

t

xerfc

q

t

xtqTtxT

tt

t

i

24exp

2, 0

221

0 (5-22)

Where;

63

erfc = complementary error function

0q = surface flux (W/cm2)

x = depth into surface (cm).

The surface considered for the validation study is a beryllium wall that is 25 cm in depth.

The height can be considered infinite due to the fact that heat transfer is not varying in

the vertical 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/cm2. Table 5-1. Beryllium

thermophysical values gives the thermophysical values of beryllium used.

Table 5-1. Beryllium thermophysical values

Constant Value Units

κ 1.998 W/cm/K

ρ 1.85 g/cm3

c 1.824 J/g/K

Figure 5-7. Temperature profile for beryllium wall with 001.0t s and 08.0x cm.

64

This figure shows the overlay of the implicit Crank-Nicolson and explicit numerical

models with the analytical solution. In DEFFCON both numerical 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 time 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. Results of validation study for 40 sec simulation

Method Δt (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 practical 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.

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 differential that exists. It can be shown by analyzing this equation that as

the surface temperature increases the thickness of the layer decreases. Thus at every

time step a boundary layer and film temperature must be calculated by monitoring the

surface temperature of the material.

The thermal 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 the 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. The air layer

always has the same nodal dimensions as the material, meaning that the small x and

y are the same for both. This eliminates the need for a non-uniform meshing

algorithm when considering the finite differencing equations.

The kinematic viscosity, beta term, and any other thermodynamic quantities of the

air layer are calculated at the film temperature filmT . These thermodynamic quantities

have been fitted using data from [18] and are available in the appendix along with their

respective plots.

66

Figure 5-8. Polymer model with inclusion of the boundary layer.

Practical Clearcoat

Chapter 4 shows that the clearcoat will propagate energy according to Beer‟s

law. The important parameter for the clearcoat calculations in DEFFCON is the

material‟s transmittance, which dictates how much energy reaches basecoat. The

literature is vague on the transmittance specific value, therefore for most calculations a

value of 0.7 is used. The effect that this assumption has on the basecoat temperature

is quantified in Chapter 6.

Once a transmittance is chosen DEFFCON will back-calculate for the material‟s

absorption parameter using Equation 4-2. This is then used in Equation 4-1 to

determine how much energy will be delivered to the clearcoat as a function of distance

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:

1 (5-13)

xf exp1 (5-14)

Where; = absorption constant (1/cm) f = absorbed fraction

x = 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:

fQQQ sumsur *0 (5-15)

Where;

0Q = energy available to current node (W/cm3)

surQ = energy on surface node (W/cm3)

sumQ = total energy absorbed by all nodes preceding current node (W/cm3).

Whatever is left over is then deposited at the clearcoat/basecoat interface as another

volumetric absorption based on the basecoat‟s absorptivity. Thus if it is a black coating

all of the energy would be deposited ( 1 ), 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 discarded as a loss to the surroundings.

Practical Char

It is necessary to “grow-in” the material‟s char for the purposes of accurate heat

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

68

the concept of polymer fraction, F , given in Chapter 4 a picture of the char history can

be formed. Figure 5-9 shows this.

Figure 5-9. Polymer mass fraction being converted to char.

Since the polymer fraction and the char fraction must sum to 1, the char fraction can be

plotted as F1 . 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

combination of the polymer‟s and char‟s thermophysical properties can be used to

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.

69

Figure 5-10. Specific heat as a function of char fraction.

Figure 5-11. Thermal conductivity as a function of char.

70

Figure 5-12. Density as a function of char.

In the above figures the term “mix” refers to the combination of char and polymer, thus it

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 relatively 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 are grouped under different “blocks” in the input

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

“Blk” (blocks) follow.

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 illustration 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).

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. This means that

where one material ends, the other begins.

Also contained in the file is the option to choose a differencing model (diffmod).

This allows 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 the 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),

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 the 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 Equation 5-16:

timeruntotalMaxtmsp det* (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 (discussed in Chapter 3) are contained in Block 4, while

the Arrhenius parameters are contained in Block 5. The units of the parameters are in

the block‟s heading. Block 5 also allows for the user to input whether the material is a

polymer (P) or a metal (M) which 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 coefficients if they exist. The Tinf refers to the ambient air

temperature surrounding the model at different locations (1-4) which are the same

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 there 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 file the user has chosen to heat one

end of the material through normal heat transfer until it reaches a desired value of 180

Celsius.

The final block (Blk11) refers to DEFFCON‟s ability to back calculate from a

mass loss in milligrams to fluence. The user initializes the calculation with calc=1, POI

refers to the “point of interest” which is the observed mass loss of the sample in

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 (surface.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.

75

Figure 5-15 is the general output file; out.put. This file contains 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/cm2). 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 the 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 starting with the time elapsed and

then from left to right:

i) Nodal position (ix, jy)

ii) Specific location in centimeters (xo, yo)

iii) Material number (mat num)

iv) Temperatures from the previous (T-old) and current iteration (T-new) in celsius

v) Volumetric absorption in W/cm3

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 that 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.

76

Figure 5-15. “Out.put” output file for DEFFCON

Figure 5-16. DEFFCON a) mass loss output file and b) surface temperature output file

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, CANYON‟s input parameters need to be understood. Figure 5-17 shows

a snapshot 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. These values appear under the “dy”

heading and are for “sfc” 3-6. 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 the event that there is fog or dust in the

surrounding air. In the input file for CANYON the “atmospheric conditions” alters this

attenuation. The “iVis[km]” parameter refers to how much visibility is present with

regards to the air, a visibility of 999 km 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.

78

Figure 5-17. CANYON input file

79

Figure 5-18. Snapshot of CANYON output file CANout.put

80

Figure 5-18 shows the portion of CANYON‟s output file that is to be used in

DEFFCON. In this figure the relevant data starts after the heading “summary results”.

Everything before this heading is an echo of the input file. DEFFCON reads the list

under the heading “FluxRow1Navg” and for the units of “[W/cm^2]” as well as the

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 DEFFCON input file are not always the same a

linear interpolation is initiated. Thus in the event that DEFFCON is at a time step where

a value for the flux does not exist in CANYON‟s output file, DEFFCON will linearly

interpolate using the next flux in CANYON‟s output. 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 points plotted between the two codes can be attributed to the

time report section of DEFFCON‟s input file (Blk3: report); recall that not all times and

values are printed in the output files as specified by the report value.

81

Figure 5-19. Overlay of CANYON output to DEFFCON input fluxes for 1 kT and 708 m

from the source

82

CHAPTER 6 CASE STUDY AND ANALYSIS

Few articles exist on computer modeling that includes the coupling of

degradation of a material with a heat source, and 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 Arrhenius parameters and

thermophysical properties exquisitely; it was written by Bahramian, et al [26] and will be

referred to as Bahramian.

Atmospheric Re-Entry

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

degrade. 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 temperature.

The results 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

this 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

83

the char yield fraction is presented as a function of temperature. The equations are as

follows:

1

exp1

RT

EE

A

ATY

cg

c

g

c (6-1)

RTE

ccaeTT

ATYTY

m

Tm0

0

exp1

(6-2)

Where the new constants are;

gcA , = pre-exponential factor for char and gas formation (1/time)

gcE , = activation energy for char and gas formation (J/mol)

0m = initial mass (g)

Tm = mass at temperature absolute temperature T , in (g)

TYc = char yield at temperature T

R = gas constant (J/mol/K) T = absolute temperature (K). Table 6-1 shows the reported values for these new constants and Figure 6-1 shows the

theoretical as well as the observed degradation as a function of temperature for this

material. It should be noted that the values in the table do not exactly match those

reported in the literature. This is due to what is believed to be an error on the order of

magnitude, if the exact values are used the degradation curves do not match the curves

that are reported.

Table 6-1. Constants for degradation parameters

Constant Value Units

Ea 95.7 kJ/mol

Eg-Ec 10.1 kJ/mol

A 5.29E+06 1/min

Ag/Ac 4 βarticle 10 K/min

βDEFFCON 20 K/min

Yc 0.47

84

Figure 6-1. Bahramian experimental 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

85

Notice that there is no longer a discrepancy between the theoretical and experimental at

the higher temperatures. This can be attributed to the constant char yield assumption,

thus validating this assumption. DEFFCON‟s degradation model is better for this

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. The 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 effort 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.

3

1i

iiv (6-3)

3

1

3

1

i ii

i iii

v

CvC

(6-4)

32

5.0

0

0

2

0 vnvT

T

vk

(6-5)

3

1

41i

ivv (6-6)

Where;

iv = volume fraction of particular material (polymer, char, gas)

iC = specific heats of materials (polymer, char, etc.)

0 = initial thermal conductivity (W/m/K)

kn = experimentally derived coefficient

4v = porosity.

86

Much like the equations presented in Chapter 2, an effort is made to weight these

parameters for the 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 eliminates the need for another density in the function. The coefficient,

kn , was experimentally determined and is specific to the composite material, thus

utilizing Equation 6-5 would be impractical for the purposes of this research.

Assumptions

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 diffusion.

ii. Movement of the liquid is assumed negligible compared to pyrolysis gases.

iii. Pyrolysis gases may be considered „ideal gases‟ but their properties remain constant.

iv. Volatiles formed from the polymer escape from the solid as soon as they are formed.

v. The instantaneous 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

fractions 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

reaction.

87

There are a couple of differences between these assumptions and those of DEFFCON:

1. No effort is made to track the pryrolysis gases and their properties.

2. The densities are averaged if “fillers” are present such as fibrous material or paint

pigments.

All other assumptions overlap with those of DEFFCON. One in particular may be

invalid. Assumption “iv” is a reasonable assumption for a material that absorbs all

incoming energy at the surface, however, for a binary system of clearcoat on top of

basecoat the majority of the energy will be absorbed beneath the surface. This may

allow for pockets of gas to form inside 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

matter 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 vaporization. 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.

2

24

d

mloss

(6-7)

88

Where;

loss = material removed (mm)

m = mass loss as reported by DEFFCON (g)

d = diameter of sample (cm).

Composite Experimental 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 thermophysical 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 height 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

Constant Value Units

flux 799 W/cm2

ρcomposite 1.45 g/cm3

κcomposite 0.005 W/cm/K

ccomposite 1.27 J/g/K

Tg 250* °C

ρAl 2.78** g/cm3

κAl 1.21** W/cm/K

cAl 0.875** J/g/K

*Assumed glass transition temperature for epoxy [27] **Assumed Aluminum 2024-T3

89

The experiment was conducted under 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‟s legend 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 sensitivity 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 which provides for a relative error

of 19%.

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 the

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/cm2. No experimental results are

supplied for the actual surface temperature of this sample. What is given is the

aluminum substrates temperature which was measured by an inserted probe. This

temperature is used along with the progressive heating option in the DEFFCON

simulation. Since DEFFCON doesn‟t model rectangular structures the sample has been

adjusted to give 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 numerical model presented in the literature

alongside the DEFFCON prediction. The legend follows the same format explained

earlier.

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. Upon reviewing Bahramian‟s figures it is apparent that the research

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

main difference when analyzing Bahramian‟s physics lies with the thermal conductivity

and the boundary conditions; particularly the mass blow-off relationship.

The major dissimilarities between the boundary conditions used in the literature

and the conditions used in DEFFCON are the convective heat boundary, and the

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/cm2. 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 be further from Bahramian‟s numerical model for

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 shows this analysis

respectively. The legend title “Adjusted 329” refers to DEFFCON‟s adjusted simulation.

93

Figure 6-8. Surface temperature numerical model comparison with adjusted DEFFCON

with 329 W/cm2

Figure 6-9. Erosion results for Bahramian numerical model, adjusted DEFFCON with

329 W/cm2 and original DEFFCON with 799 W/cm2(DEFF 100X)

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 underestimation but there are

no experimental results to verify. However the main point is that DEFFCON‟s thermal

boundary layer condition, coupled with the original thermal conductivity relation

(Equations 2-3 & 2-4), adequately compensate for the removal of energy from 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 material 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 with 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 transmitted

through the clearcoat. It is apparent in other literature [28] that different amounts of

degradation will be observed depending on whether the paint is black or white. The

black paint absorbs the most energy and will therefore see the most mass loss. These

two colors represent the extremes of absorption. The only thing considered here is the

absorption. Upon examination of automotive paint, many different additives for

95

aesthetic appeal are discovered. If, for instance, the paint had metal flakes 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

κ (W/cm/K) ρ (g/cc) c (J/g/K) Mica (wt %) Polyurethane (wt %) αt (cm2/s)

0.0071 0.986 0.5 100 0 0.0144

0.0036 1.206 1.41 30 70 0.0021

0.0041 1.174 1.28 40 60 0.0027

0.0046 1.143 1.15 50 50 0.0035

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

absorption. They were all subject to the same incident heat flux of 45 W/cm2. As can

be seen in the figures, the less the clearcoat absorbs the less linear the profile. This

observation can be attributed to the fact that the clearcoat does not have enough

energy to impact the temperature of the basecoat; it instead serves as more of heat 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.

96

Figure 6-10. Basecoast temperature with clearcoat absorption of 0.1 and mica content

variation

Figure 6-11. Basecoast temperature with clearcoat absorption of 0.3 and mica content

variation

97

Figure 6-12. Basecoast temperature 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 source term. One very important component in

solving the “inverse yield problem” is CANYON‟s ability to determine the magnitude of

the weapon (yield in kT) from minimal user input; dimensions of the “canyon” and

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]:

B

kTAY (6-8)

Where; = fluence (cal/cm2)

BA, = Constants determined through least squares power 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

98

distances are included with the fluence calculations for 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

Procedure

The procedure to back-calculate the yield from a fluence using CANYON is

simple. In the input file, Figure 6-16, at the very bottom there is a section with a label

“Target Fluence”; this is the user observed fluence. The next entries are problem

bounds seen as “Yield Low” and “Yield High”, a least squares power fit requires at least

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 power fit. The yield for the “Target Fluence” lies somewhere on this fit and

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

99

way as the yield is calculated using CANYON. Thus knowing the fluence calculated by

DEFFCON, the yield of the weapon and the unknown mass loss can be determined

using CANYON and the previously outlined steps, completing the “inverse yield

problem”.

Through a series of runs on several material cases, it is determined that polymers

degrade along a 2nd order polynomial:

ABC 2 (6-9)

Where; = fluence (cal/cm2) = mass loss of polymer (mg)

CBA ,, = 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 monolayer 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 polymer ( ) 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

100

A 2nd order polynomial least squares fit requires three data points to determine all

three constants. Thus three simulations using CANYON must 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 necessary to determine the three polynomial coefficients. Since

there are three unknowns a system of three linear equations must be solved to

determine these coefficients.

n

i

i

n

i

i

n

i

n

i

i CBA1

2

111

1 (6-10)

n

i

i

n

i

i

n

i

i

n

i

ii CBA1

3

1

2

11

(6-11)

n

i

i

n

i

i

n

i

i

n

i

ii CBA1

4

1

3

1

2

1

2 . (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 the procedure for the “inverse yield problem”. 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 simulations significantly.

101

Figure 6-15. Flow chart for “inverse yield problem”

Example Using Automotive Paint Damage

Finally, a study is presented to demonstrate DEFFCON‟s ability to determine

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 visibility to the car is 999 km (100% transmission 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 the 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.

102

Figure 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

Table 6-4. Thickness of layers in automotive sample

Constant Thickness (µm)

Clearcoat 45

Basecoat 20

Primer 23

Zinc phosphate 1.5

Aluminum 1016

Figure 6-17. Degradation of automobile 708 m away from source with yields (data points left to right) 1, 5, 10, 15, and 20 kT

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/cm2

for an automotive paint sample. The three points under “known parameters”

correspond to the first, second, and last points on Figure 6-17, which was produced with

the aid of CANYON. Using DEFFCON to back calculate with the “poi” chosen, a

fluence of 5.25 cal/cm2 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 distance 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.

104

The further one gets from the source the less the 2nd order polynomial fits the

degradation profile. This fit, however, is still within 20% of the data points. It is also

important to notice that at this distance significant damage is not experienced until

around 10‟s of kilotons (third data point). This can be attributed to the shielding

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) Φ (cal/cm2) loss (mg) pulse time (s) comp time (min)

1 1.72 0.0006 0.86 322.2

5 2.96 0.0140 1.74 652.2

10 3.74 0.0534 2.35 940.1

15 4.29 0.1120 2.80 1125.5

20 4.72 0.1837 3.20 1268.3

Table 6-6. Automotive simulation 708 m from source

yield (kT) Φ (cal/cm2) 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 2.80 1188.2

20 11.52 7.965 3.20 1335.3

105

CHAPTER 7 CONCLUSIONS AND FUTURE WORK

The research contained in these pages produced a program called DEFFCON

for the modeling of thermal degradation 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 range of polymers as well as to, somewhat, ease

computational efforts. Validation of these assumptions is shown 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 materials. 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.

Minimum Arrhenius parameters are required and simplifying assumptions, such as

constant char yield, are 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 is the ability to

back-calculate to a weapon yield when delivered fluence is known. Thus by developing

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

discussions with laboratory technicians dictate that bubbles 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 formation 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, further 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 ( ik ) to ensure that reaction time is not an issue. Studies should also be

conducted on variation of assumed parameters, such as the type of aluminum used in

the Bahramian case studies.

107

APPENDIX A AIR THERMOPHYSICAL PROPERTIES

108

109

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Prentice Hall, 2003.

[3] Fogler, H. Scott. Elements of Chemical Reaction Engineering. Upper Saddle

River : Prentice 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. :

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s.l. : University of Florida, 2009.

[9] Geankopolis, Christie John. Transport Processes and Separation Process

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[10] Cameron, John R., Skofronick, James G. and Grant, Roderick M. Physics of the

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Engineers Handbook-7th ed. New York : McGraw-Hill, 1997.

[12] Gokel, George W. Dean's Handbook of Organic Chemistry 2nd Ed. New York :

McGraw-Hill, 2004.

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 Degradation and Stability 2000; 69:1-9.

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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]:51-53.

[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.

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http://www.steelforge.com/metalmeltingrange.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 Free-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.

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[25] Sjoden, Glenn E. Foundatoins in Applied Nuclear Engineering Analysis. s.l. : World Scientific 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 Nuclear 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. [Cited: 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.

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

degree in chemical engineering with a specialization in process engineering from the

University of Florida in 2008. His hobbies include “cracking the wise”, sailing while his

brother yells at him for sailing incorrectly, and target shooting.