The Pennsylvania State University

The Graduate School

College of Engineering

DSMC ANALYSIS OF FRACTAL-LIKE AGGREGATES OF SPORES

IN THE SEMI-RAREFIED FLOW REGIME

A Thesis inAerospace Engineering

byTong Zhu

Submitted in Partial Fulfillmentof the Requirementsfor the Degree of

Master of Science

December 2012

The thesis of Tong Zhu was reviewed and approved* by the following:

Deborah A. LevinProfessor of Aerospace EngineeringThesis Adviser

Michael M. MicciProfessor of Aerospace Engineering

George A. LesieutreProfessor of Aerospace EngineeringHead of the Department of Aerospace

*Signatures are on file in the Graduate School.

Abstract

The work presented in this paper is a continuation of our efforts to develop a realistic

bio-engineering model for predicting the survivability of anthrax spores subjected to high

temperature gas environment. One of the major mechanisms of deactivating spores is to

expose them to elevated temperatures and careful, exposure-tube experiments have been

carried out to ascertain the deactivation mechanism. Spores typically exist in nature as

aggregates, but, simulating the heat transfer to clumps of spores is difficult because of the

highly irregular geometry of spore clumps. In this work we implement the tunable particle-

cluster and cluster-cluster algorithms to generate fractal-like spore aggregates. We compare

the algorithm output as a function of algorithm parameter input with a typical spore clump

image. Since the spore aggregate size is small and is on the order of the mean free path

even at atmospheric conditions, we use the direct simulation Monte Carlo method to model

the heat transfer to each of the spores in the aggregate. We study the shielding effect of

aggregate spores on a single spore in the clump, finding that for aggregates on the order of

100 that the shielding effect is about 25%∼45% compared to a single, isolated spore.

iii

Contents

List of Figures v

List of Tables vii

List of Symbols viii

Acknowledgments ix

Chapter 1 INTRODUCTION 1

Chapter 2 FRACTAL-LIKE AGGREGATES 5

Chapter 3 GENERATION OF FRACTAL-LIKE AGGREGATES 7

3.1 Tunable Particle-Cluster Aggregation (PCA) Algorithm . . . . . . . . . . . 7

3.2 Tunable Cluster-Cluster Aggregation (CCA) Algorithm . . . . . . . . . . . 9

Chapter 4 DSMC IMPLEMENTATIONOF FRACTAL-LIKE AGGREGATES

OF SPORES 11

Chapter 5 RESULTS AND DISCUSSION 14

5.1 Simple Geometry Configurations . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2 Cases of fractal-like aggregates of spores . . . . . . . . . . . . . . . . . . . . 15

Chapter 6 CONCLUSIONS AND FUTURE WORK 18

Appendix A 23

iv

List of Figures

A.1 Randomization of the positioning procedure in PCA and CCA (LHS) and

rotating procedure in CCA (RHS). . . . . . . . . . . . . . . . . . . . . . . . 24

A.2 Electron micrograph of Bacillus anthracis spores from the Sterne strain (un-

der a high magnification of 12,483X). . . . . . . . . . . . . . . . . . . . . . . 25

A.3 (LHS): Translational temperature contours for a single spore; (RHS): Trans-

lational temperature contours for five spores. . . . . . . . . . . . . . . . . . 25

A.4 Radial translational temperature profile comparison for a single and five spore

cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A.5 (LHS): Surface heat flux and (RHS): Normalized heat rate for five spore

co-linear case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A.6 Surface heat flux for an ellipsoid of B.anthracis Sterne spore shape. . . . . . 27

A.7 Illustration of 20-spore fractal-like aggregates of kf=1.1 and Df=2.0. . . . . 27

A.8 Illustration of 20-spore fractal-like aggregates of kf=1.7 and Df=2.0. . . . . 28

A.9 Illustration of 100-spore fractal-like aggregates of kf=1.1 and Df=2.0. . . . 29

A.10 Illustration of 100-spore fractal-like aggregates of kf=1.7 and Df=2.0. . . . 30

A.11 (LHS): Surface heat flux for 20-spore fractal-like aggregate with complete

accommodation; (RHS): Surface heat flux for 20-spore fractal-like aggregate

with αE=0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.12 (LHS): Normalized heat rate for 20-spore fractal-like aggregate with com-

plete accommodation; (RHS): Normalized heat rate for 20-spore fractal-like

aggregate with αE=0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

A.13 (LHS): Surface heat flux for 100-spore fractal-like aggregate with complete

accommodation; (RHS): Surface heat flux for 100-spore fractal-like aggregate

with αE=0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.14 ((LHS): Normalized heat rate for 100-spore fractal-like aggregate with com-

plete accommodation; (RHS): Normalized heat rate for 100-spore fractal-like

aggregate with αE=0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

A.15 Four different 100-spore CCA aggregates with prefactor kf=1.7 and fractal

dimension Df=2.0 under full accommodation and the distribution of normal-

ized heat rate over each spore in the aggregate. . . . . . . . . . . . . . . . . 33

A.16 Distribution of normalized heat rate. . . . . . . . . . . . . . . . . . . . . . . 34

vi

List of Tables

A.1 Geometry and DSMC Casesa . . . . . . . . . . . . . . . . . . . . . . . . . . 23

vii

List of Symbols

N Number of particles in the aggregate

kf Fractal prefactor

Rg Radius of gyration of the aggregate

Df Fractal dimension

a Mean radius of primary particle

ri Position of the ith particle

d12 Distance between the geometric centers of two aggregates

xRND Random position for the x coordinate

yRND Random position for the y coordinate

zRND Random position for the z coordinate

θxRNDEuler angle of rotation about x axis

θyRND Euler angle of rotation about y′ axis

θzRND Euler angle of rotation about z′′ axis

viii

Acknowledgments

The research being performed at the Pennsylvania State University is supported by the

Defense Threat Reduction Agency, Scientific Research Grant No. HDTRA1-07-1-0014,

technical monitor, Dr. Suhithi Peiris, whose support is gratefully acknowledged. We would

like to thank Professor M. Ivanov of the Institute of Theoretical and Applied Mechanics,

Russia, for the use of the original SMILE code.

ix

Chapter 1

INTRODUCTION

The threat posed by deadly spores such as Bacillus anthracis (anthrax) to society is now

well known, deserving the effort needed to minimize the damage in the event of their release.

One of the requirements of such efforts is that the spore annihilation process must be fast,

reliable and containable. Among several possible ways of spore annihilation, one way is

to expose them to a high temperature gas environment, the subject of the present study.

The modeling and simulation of heat transfer in two phase flows is an active research area

including studies of the fundamental processes related to momentum, energy, and mass

exchange between solid, liquid, and gas phases of a vaporizing slurry droplet,[1] Direct

Simulation Monte Carlo (DSMC) methods used to estimate Martian-dust heating,[2] and

transient particle-fluid heat transfer in Brownian motion.[3]

The work presented in this paper is a continuation of our efforts to develop a realistic

bio-engineering model for predicting the survivability of spores subjected to the high tem-

perature gas environment. The model is intended to simulate the experiments carried out

in a thermal exposure chamber to measure the survivability of spores in a high temperature

gas environment. The exposure time in such an experimental set-up varies approximately

from 25 to 75 ms and can be simulated with different working gases. However, the modeling

of the survivability of biological organisms is challenging and multi-facetted. For example,

the concept of a “single-spore” is an idealization from the usual configuration that anthrax

spores are known to exist. The surfaces of natural spore particles have electrostatic charges

1

2

that cause individual spore particles to clump together. The specific size of the clumps

depends on many factors such as the relative importance of van der Waals, electrostatic, or

hydrophobic microscopic interaction forces or the ionic strength of the suspending solution

and the surface chemistry of the interacting surfaces.[4] In this work we analyze one com-

ponent of spore survivability in a clump, resistance to heat exposure. As the fidelity of our

single spore modeling improves we will then incorporate advanced models into spore-clump

configurations.

In our previous work [5], we developed a thermo-structural model incorporating the

dynamic response of the elastic spore wall to the high temperature gas surrounding a single

spore. The mode of heat transfer from the gas to the spore was heat conduction that takes

place in the transitional Knudsen number regime owing to the small size of spore. The

heat transfer rate from the gas to the spore wall was estimated by standard heat transfer

correlations [6] and verified by a Direct Simulation Monte Carlo (DSMC) analysis [5]. DSMC

methods have been used in modeling heat conduction in high speed flows [2] and have also

been used to study heat conduction in the viscous flows of MEMS micro-nozzle devices[7] as

well as near-zero relative velocity gas-particle flows.[8] In our previous work,[5] it was found

that it took just ∼0.5 ms, in comparison to actual ∼50 ms exposure time, for the spore inner

core to reach a steady state at the gas temperature in a pure thermal response analysis,

ignoring its structural response. Thus it was found that the time constant associated with

the heat transfer was much smaller than the exposure and spore survivability time constant.

The presence of large spore clumps in the experiment was considered as a possible factor in

explaining this discrepancy. However, spore clusters on the order of 100 or greater would

be required to rectify the factor of 100 difference in the spore survivability. Separate SEM

tests of tested spores were collected that demonstrated that the special single spore aspirator

procedure was largely successful and occasionally a cluster of size less than four might be

observed.[9] Subsequent inclusion of the structural response of the spore inner core wall

due to fracture was also considered. The spore core has a small amount of water that

when heated and expanded could exert excessive stress, thereby causing fracture. Using a

wall fracture limit of 300 MPa[10], we predicted the fracture to occur even faster than the

3

time it took the spore to reach the gas steady state temperature. Inclusion of a membrane

porosity model in the dynamic thermo-structural response model of 10% lengthened the

predicted spore survivability lifetime for exposure tube gas temperatures of less than 400 K,

as was observed experimentally.[5]. However, recently obtained tests of Setlow[9] have shown

conclusively that the lack of Dipicolinic Acid (DPA) release from heat exposed spores means

that the disruption of the spore’s inner membrane permeability barrier is not the cause of

high-temperature gas exposure spore kill. More likely, the reason for spore death is due to

either protein damage or some damage to the spore’s inner membrane. The time constants

for these processes are much slower than those for heat transfer and are under investigation

in our single spore modeling.

In the current work, we extended the DSMC analysis of heat conduction from a high

temperature gas to the fractal-like aggregates consisting of a number of spores. Based

on an analysis of the spatially varying shielding effect experienced by different spores in an

aggregate we can determine the resistance posed by the aggregate to the heat that reaches a

given spore. Therefore the survivability of a spore will in turn depend on its spatial location

and the extent to which it is shielded by other spores. Similar studies were carried out by Liu

et al. [11, 8], in which heat conduction was studied between fractal-like aggregates and the

surrounding gas in the transition regime using the DSMC method. The work involved the

numerical generation of fractal aggregates following the above-mentioned algorithms and a

detailed heat transfer analysis between a hot fractal aggregate and the cooler surrounding

gas using the DSMC method. Such an analysis was used for soot particle sizing and an

accurate prediction of soot volume fraction in laser-induced incandescence techniques. The

objective of that study was to establish the characteristic heat transfer length scale of

aggregates or to simply determine the size of the heat transfer equivalent sphere for fractal-

like aggregates. In our current work, we are more interested in determining the spore to

spore variation of the heat conduction rate.

The outline of the paper is as follows: In Sec. 2, we introduce the basic statistical

laws that are satisfied by fractal-like aggregates. In Sec. 3, we present the two methods of

aggregate generation, viz., the tunable particle-cluster aggregation (PCA) algorithm and

4

tunable cluster-cluster aggregation (CCA) algorithm. We then discuss the DSMC modeling

and required numerical parameters of a high temperature gas over aggregates of spores in

Sec. 4. In Sec. 5 we present the results of the combined CCA/PCA algorithm generation

of cluster sizes of 20 and 100 and show their dependency on the fractal prefactor. We elect

to study cluster sizes of 20 and 100 to bracket the general clump sizes of interest. Clumps

with an an aerodynamic diameter less than 10 µ are respirable and thus pose an immediate

health threat by aerosolized BW agents and clumps of approximately 30 µ remain airborne

for long times and can eventually be re-fragmented to yield smaller, thus dangerous, clumps.

The heat transfer and shielding effects of individual spores in an aggregate are simulated

using DSMC and presented in Sec. 5 and in Sec. 6, we briefly review the results and present

conclusions and future work.

Chapter 2

FRACTAL-LIKE AGGREGATES

We assume that clumps of spores form complex geometries, similar to those formed by

aerosols or combustion generated soot comprised of a number of constituting primary par-

ticles. The mean diameter of a single spore is approximately the same as that of a primary

particle of aerosols or soot, i.e., ∼1µm, and these aerosols or combustion soot aggregates are

usually found to consist of a number of primary spherical particles which join together to

form fractal-like aggregates. A number of studies [12, 13, 11, 14] related to these fractal-like

aggregates [12, 13, 11, 14] were conducted and published in the literature. The complex ge-

ometries of these aggregates cannot be successfully approximated as dense or porous spheres

or other simple shapes [12, 13]. Some additional parameters are required to characterize

the structure of such fractal-like aggregates. The transfer of energy between such aggre-

gates and a gaseous environment, and their transport properties are subjects of interest in a

number of applications. Various ultrafine powders, such as carbon black, titania and silica

are produced by vapor-condensation combustion synthesis processes in industries. These

particles show desirable optical, electric and mechanical properties with large surface to

volume ratio and hence are of technological importance [13]. The physical properties of

these aggregates are in turn dependent on the structure of these aggregates.

Forrest and Witten [15] proposed the following statistical scaling law connecting the size

5

6

of the aggregate, Rg, and the number of primary particles,

N = kf

(

Rg

a

)Df

(2.1)

where N is the number of primary particles in the aggregate, a is the mean radius of the

primary particle, kf is the fractal prefactor, Rg is the radius of gyration of the aggregate and

Df is the fractal dimension. The fractal dimension represents the extent of the aggregate

density. As the fractal dimension increases the aggregate size decreases and hence its density

increases, as can be seen from Eq. 2.1. The radius of gyration is defined by the mean square

distance between the geometric center of the aggregate, r0, and all points on the primary

particle surfaces:

R2

g =1

N

N∑

i=1

(ri − r0)2 + a2 (2.2)

r0 =1

N(ri) (2.3)

where ri represents the position of the ith primary particle.

Examination of the literature shows that the fractal dimension for spores of Bacillus

anthracis ranges from 1.1 to 3.0 under different aggregation scenarios [16], and a nominal

value of 2.0 was assumed in our work. Studies conducted by Carrera et al. [17] show that

the Bacillus anthracis spores are typically ellipsoidal in shape with a diameter of ∼0.85µm

with an aspect ratio ranging from 1.51 to 2.00 depending on the species. Most of the cases

modelled here assumed the spores to be spheres with the same radius of 1µm. However an

ellipsoid geometry of a length of 1.75µm and a diameter of 1.00µm is considered as well,

reflecting the aspect ratio of the B. anthracis Sterne (according to Carrera et al. [17]).

Chapter 3

GENERATION OF

FRACTAL-LIKE AGGREGATES

The direct experimental determination of fractal-like aggregate structures is difficult and

involves an interpretation of 2-D images, subject to uncertainties [13]. Therefore a different

approach, which is more feasible and efficient, is generally used by many researchers. In this

approach, fractal-like structures are generated numerically using algorithms which mimic

the formation of fractal geometries in nature [12, 13, 11, 14]. These algorithms usually allow

fine tuning of fractal parameters such as fractal dimension and prefactor. There are two

such algorithms used by a number of authors to create an aggregate numerically, viz., the

particle-cluster aggregation[18] and cluster-cluster aggregation algorithms[19].

3.1 Tunable Particle-Cluster Aggregation (PCA) Algorithm

In this tunable aggregation algorithm, the general procedure for creating an aggregate is by

adding primary spherical particles one by one. The procedure starts with an assumption of

fractal dimension and prefactor values. Primary particles are then added such that there is

at least one point of contact and no overlapping with the existing aggregate particles [18].

In order to get an expression for the position of the newly-added particle, we need to first

consider an arbitrary division of an aggregate of N particles into two basic aggregates of

7

8

each N1 and N2 particles. The gyration radius of the aggregate can be expressed by Eq. 3.1

(N1 +N2)R2

g = N1R2

1 +N2R2

2 +N1N2

N1 +N2

d212 (3.1)

in which R1 and R2 are the gyration radii of the two basic aggregates and Rg is the gyration

radius of the combined aggregate. d12 represents the distance between the geometric centers

of the two aggregates. In the specific case, where N2 = 1, N1 = N − 1 and R1 = a, Eq. 3.1

can be rewritten as

d212

=N

N − 1

(

NRg2−R2

2

)

−NR12 (3.2)

which gives the distance between the geometric center of the N − 1 particle aggregate and

the position of the newly added particle at its geometric center.

In the PCA method the size of the aggregate increases by adding one particle at a time.

Therefore, after substituting the scaling law given by Eq. 2.1 into Eq. 3.2, we obtain the

position of the newly added N th particle, rN , which is situated on the surface of a sphere

determined by Eq. 3.3.

(rN − r0N−1)2 =

N2a2

N − 1

(

N

kf

)

2/Df

−

Na2

N − 1−Na2

(

N − 1

kf

)

2/Df

(3.3)

where r0N−1is the center of mass of the first N − 1 primary particles forming the aggregate.

The new particle is placed randomly on the surface of a particle with the added restrictions of

no overlapping and at least one point of contact with other particles. An iterative procedure

is followed in which particles are repeatedly moved so as to satisfy the constraints [13].

We implemented the PCA algorithm using the following specific steps. The first three

particles used to create the aggregate were combined in a way that satisfies only the con-

tact restriction because the initial aggregate size of three is too small to show the fractal

characteristic expressed by the scaling law given by Eq. 2.1. For simplicity, we assigned

the first three particles to be positioned in a regular triangle shape to form a “base” for all

the subsequent PCA steps. The geometrical center of this “base” was calculated for use in

Eq. 3.3 and is designated as C1, as is shown on the LHS of Fig. A.1. For the fourth and

9

subsequent particles, the newly-added particle was positioned randomly over a sphere of

radius R = rN −r0N−1that was calculated based on the specified kf and Df , by successively

specifying the three coordinates xRND,yRND and zRND of its center C2 (also shown on

the LHS of Fig. A.1). The restrictions of no overlapping and at least one point of contact

with other particles were checked to obtain a suitable site for the newly-added particle. A

distance of less than 5% of the radius of a primary particle between any two particles was

defined as “contact”, while the no-overlapping restriction was strictly satisfied. The three

coordinates of the center of the newly added particle were randomly generated in sequence

to simultaneously fulfill Eq. 3.3.

3.2 Tunable Cluster-Cluster Aggregation (CCA) Algorithm

This algorithm is based on a hierarchical scheme of aggregation at each step resulting in

fractal-like aggregates satisfying the scaling law [19]. The algorithm starts with the smaller

aggregates formed by the use of the particle-cluster aggregation algorithm with prescribed

values of fractal dimension and prefactor. These basic aggregates are then combined pairwise

and we are not restricted to adding one particle at a time. As is discussed in the PCA

algorithm, if N1 and N2 are the sizes of two basic aggregates with gyration radii of R1 and

R2, then the radius of gyration of the combined aggregate, Rg, is given by Eq. 3.1. After

substituting the scaling law given by Eq. 2.1 into Eq. 3.1, we obtain

d212

=a2(N1 +N2)

N1N2

(

N1 +N2

kf

)

2/Df

−

N1 +N2

N2

R2

1−

N1 +N2

N1

R2

2(3.4)

where d12 represents the relative position of the geometric center of the newly added N2

particle aggregate from the geometric center of the N1 particle aggregate.

In each step, it is ensured that the combined aggregates have at least one point of con-

tact and no overlapping. After repeating this procedure for all clusters, a larger cluster

is obtained satisfying the scaling law exactly [19]. In the CCA process, one of the two

aggregates is considered as fixed at its geometrical center C1 and the geometrical center

of the other aggregate is randomly positioned relative to the fixed one following Eq. 3.4.

10

Correspondingly, in Fig. A.1 R = d12 and C2 is the geometrical center of the second aggre-

gate. Also a random rotation is performed that consists of three successive rotations about

the axises of a reference frame centered at C2 with three random angles θxRND, θyRND

and

θzRND(known as Euler’s angles, shown on the RHS of Fig. A.1), of the second aggregate

until there is at least one point of contact with a particle in the first aggregate. “Contact”

here is defined as a distance of less than 1% of the radius of a primary particle between any

two particles. If the random orientation procedure does not generate a case that satisfies the

two restrictions, then the random process of positioning the center of the second aggregate

is repeated until the restrictions are satisfied.

We will be discussing the variation of heat transfer rates among spores in a cluster of

two sizes, 20 and 100 in Sec. 5. To obtain an aggregate of size 20, we take two aggregates

of size 5 to form an aggregate of size 10, and then two of size 10 to form one of size 20.

In this manner, we can proceed to form an aggregate of size 80. Finally, we combine this

aggregate of size 80 with one of size 20 to obtain an aggregate of size 100.

Chapter 4

DSMC IMPLEMENTATION OF

FRACTAL-LIKE AGGREGATES

OF SPORES

Table A.1 summarizes the cases that will be considered, the specific DSMC numerical

parameters used, and the primary geometric shape and resolution chosen. The spherical

or ellipsoidal geometries were approximated by triangular panels and the total number of

vertices and surface panels are shown in Table A.1. The ambient gas was assumed to be

molecular nitrogen at atmospheric pressure and at a temperature of 500 K, the working gas

in the past exposure tube measurements [5]. The Knudsen number (defined as the ratio of

the mean free path of the molecular nitrogen to the radius (or the half minor axis) of the

spherical (or ellipsoidal) spore), for these free stream conditions is 0.103. The spore surface

temperature was assumed to be 300 K. DSMC numerical parameters were varied to establish

that the simulation results were independent of the numerical parameters. For example, the

number of particles per cell was varied from 300 to 100, and then to 50 with the simulation

results unaltered. Similar tests were performed on the cell size (shown in Table A.1) and

the time step (taken to be 1 × 10−9s for all cases). The number of startup and sampling

steps were both set to be 2000 and the total steps varied from 15,000 to 60,000 across the

cases and it was verified that the collision number and the particle number in a cell reached

11

12

steady state. We used the variable hard sphere (VHS) interaction model to obtain the

total collision cross section [20]and the viscosity-temperature dependence index was taken

to be 0.75. Translational as well as internal temperatures were modeled. The rotational

collision number calculated by Parker’s equation,[21] was 5.625. Because of the high collision

rate, the molecular nitrogen translational, rotational, and vibrational temperatures were

found to be equal throughout the domain. Diffuse reflection from the spore surface with

full thermal accommodation was assumed and a free boundary condition was assumed for

the computational domain. The choice of computational domain guaranteed that it was

always large enough to avoid the influence of heat transfer on boundary surfaces. Three

types of geometries will be used in this paper: (1) ellipsoidal, (2) simple, attached sphere

configurations, and (3) aggregates of spheres. All surface geometries were created in the

xSmile system [22] and Table A.1 provides information about the type of primitive, its

number of vertices and the number of surface triangular panels. For multiple spore cases

(i.e., the five co-linear and the fractal cases) the spherical 3D primitive was duplicated to

form the geometry for either the exact case or the complex fractal-like aggregate using the

position vector relationships discussed in Sec. 3.

In our previous work [5], we carried out a similar heat conduction analysis on a sin-

gle spherical spore particle exposed to a high temperature quiescent gas using the DSMC

method to verify the heat transfer coefficient obtained from the empirical correlations [6].

The heat transfer coefficient was found to be in good agreement with the empirical corre-

lations [5].

One of the challenges associated with the DSMC analysis is the modeling of the thermal

accommodation at the spore surfaces. In the proposed work, the effect of an arbitrary

accommodation coefficient of the particle was studied by using the Maxwell model for gas-

surface collisions. The gas molecule is diffusely scattered with a probability α or specularly

reflected with a probability 1−α, where α is the thermal accommodation coefficient. Since

the accommodation coefficient is uniform for a spore, a range of values from unity to 0.1

was assumed. Accommodation coefficients have been determined to be close to unity for

spacecraft materials, but, the specific values for biological systems are simply unknown.

13

The normal and tangential momentum accommodation coefficients were assumed to be

unity in all cases. Another challenge is the proper treatment of boundary conditions at

the boundaries of the computational domain. Due to the heat transfer between the high

temperature quiescent gas and the spore aggregate, the temperature of the boundary will

not be at the initial gas temperature of 500K and will be at a temperature lower than that

of the gas. In the work of Liu et al., a uniform temperature was assumed on all outer

boundaries of the computational domain, and then the domain boundary temperature was

updated by following the energy conservation principle. The current way of solving this

issue is by choosing a computational domain large enough to ensure that the boundaries

are at the same temperature as that of hot gas. In practice, the domain sizes were decided

by trial and error so that the error of the calculated boundary temperatures were less than

2%.

Chapter 5

RESULTS AND DISCUSSION

In order to develop an intuition about the shielding effect, we start with a few “simple”

geometrical configurations and then we will proceed to the fractal geometries which are more

applicable for the modeling of the spore conglomerates shown in Fig. A.2. Unless otherwise

noted, all of the cases correspond to diffuse reflection with full thermal accommodation. The

surface heat flux is calculated for all the surface panels of each spore and the normalized

heat rate is the total heat rate of each spore in the aggregate normalized by the total heat

rate of a single spore with the same thermal accommodation coefficient.

5.1 Simple Geometry Configurations

The five spore co-linear simple case was chosen to show the intuitive effects of shielding from

other spores before considering the more general case of fractal-like aggregates. Figure A.3

shows the translational temperature contours of the single spore and the five co-linear

spore case. Also shown on each of the plots are radial lines starting from the center of the

spore that illustrate the angular position where the temperature radial profile is sampled.

Figure A.4 shows the temperature radial profile both in the front (horizontal radial lines)

for the “edge spore” in the two cases, and on the side (vertical radial lines, in particular,

for the middle-positioned spore in the five spore case). From Fig. A.4 we can see that for

the single spore case the temperature profile is isotropic, as expected. The shielding effect

14

15

of the side spore compared to the front one is more pronounced in the five spore case and

the gradient of the translational temperature from the surface of the spore into the gas is

smaller from the front spore of the five spore case compared to a single spore because of the

influence from the other spores in the configuration. In addition, the surface temperature

jump is ∼70 K for the single spore case compared with ∼40 K for the side profile in the

five spore case. However, the surface temperature jump is ∼60 K for the front profile in the

five spore case, demonstrating an inherent spatial variation of the shielding effect in this

five spore aggregate.

Figure A.5 presents the surface heat flux and normalized heat rate for the five spore

co-linear geometry.

As shown in Fig. A.2, the true geometry of the Bacillus Anthracis spore is elliptical,

but, generation of fractal aggregates of elliptical shape is considerably more complex than

that for spherical primitive shapes. For this reason we compared the heat flux to a spore

of elliptical shape with that of a sphere for a thermal accommodation of unity in the same

environment. Figure A.6 shows that the surface heat flux at the ends of the major axis of

this spore is 22% larger compared to the middle region near the ends of the minor axis.

Note that the small circumferential variation is due to the small numerical noise in the

DSMC simulation. In addition, the integrated surface heat flux averaged over the surface

area is 9.1 MW/m2, or only 12% less than that of a spherical spore. Therefore from the

point of heat transfer, the assumption of a spherical spore shape is reasonable.

5.2 Cases of fractal-like aggregates of spores

A study of the effect of surface thermal accommodation coefficient, αE, on the heat transfer

was conducted for each of the two aggregate sizes.

Similar to the 20-spore aggregate case, Fig. A.13 shows a comparison of the surface

heat flux distribution between the two thermal accommodation conditions for a 100-spore

aggregate case. We can see that the maximum value for surface heat flux is ∼7.7 and

∼ 1.5 MW/m2 for αE=1.0 and 0.1, respectively. It is interesting to note that the maximum

surface heat flux value remains essentially the same in the 20-spore and 100-spore aggregate

16

cases. Note, however, that compared to the single spore case, the thermal inertia is higher,

thus resulting in a total heat transfer of about one order of magnitude less. Figure A.14

shows a comparison of total heat rate experienced by each spore in the aggregate for two

thermal accommodation cases. This value ranges from 3% to 50% for αE=1.0 and from

37% to 75% for αE=0.1. As mentioned for the 20-spore aggregate case, the maximum value

of normalized heat rate is significantly high for αE=0.1 compared to αE=1.0. Once again

the similarity between the LHS and RHS of the figures is evident due to reasons that were

presented for the 20-spore case.

In general, we found a difference of one order of magnitude (or even larger) in the surface

heat flux and normalized heat rate between these aggregates of spores and single spores for

the two thermal accommodation coefficients our previous single spore case. Thus, a new

thermo-structural analysis is needed for better evaluation of the time constant associated

with heat transfer for fractal-like aggregates of spores.

The 20 and 100-spore aggregates presented in Figs. A.8 and A.10 were generated using

the combined PCA and CCA algorithms presented in Sec. 3. As discussed in that section,

the selection of the newly added particle position relative to the original cluster is obtained

by randomly generating a sequence of Cartesian coordinates that satisfy the algorithm and

the random orientation of the second cluster. In the results that were presented, the ran-

domization process was stopped as soon as the contact criteria was satisfied. The question

may be asked as to whether further randomization will produce significantly different heat-

ing rates. To address this question three additional 100-spore aggregates were generated

by randomizing the array (xRND, yRND, zRND) in the PCA and (θxRND, θyRND

, θzRND) in

the CCA stages. Figure A.15 shows the shapes and distribution of normalized heat rate

for the four different 100-spore CCA aggregates so-obtained with a prefactor of kf=1.7 and

fractal dimension Df=2.0 under full accommodation. Note that Aggregate 1 is the baseline

or original aggregate shown in Fig. A.10. It can be seen that the normalized heat rate

ranges from 3% to 50%, 4% to 43%, 2% to 41%, and 3% to 51% respectively for the four

aggregates, showing little variation among the four cases.

The additional randomization also provides a more complete statistical distribution of

17

the normalized heat rate distribution among spores in an aggregate. Figure A.16 shows

the distribution of spores with a normalized heat rate. The averaged distribution is also

indicated by the dashed line in the figure. We can see from the figure that the distributions

are similar among the four aggregate cases and a general trend can be identified with the

averaged distribution, reflecting a sole dependence on kf , Df and the number of spores in

the aggregate and not the degree of randomization. Based on the range of these values, a

good estimate can be obtained for the normalized heat rate experienced by the individual

spores in an aggregate and a mean value for the normalized heat rate is about 25% to 45%.

Chapter 6

CONCLUSIONS AND FUTURE

WORK

A spore in an aggregate of either a “simple” geometry or in a fractal-like cluster experiences

a shielding effect depending on its location relative to other spores. The spore-to-spore

variation was evaluated by using the PCA/CCA fractal generation approach to generate

realistic aggregate geometries. Both the surface heat flux distribution over the panels and

the normalized heat rate to an individual spore in the aggregate were simulated using

DSMC. A comparison of the normalized heat rate over a given spore in the two clusters

showed that the maximum normalized heat rate was not significantly influenced by the

increase in the total number of spores in the cluster. However, the minimum normalized

heat rate decreases significantly with the increase in the total number of spores in the

aggregate. This is of more interest in our effort to completely annihilate the spores in the

aggregate. The effect of different thermal accommodation coefficient on the shielding effect

was also studied. In the two fractal-like aggregate cases, the relative spatial variation of

the surface heat flux and the normalized heat rate showed little dependence on the thermal

accommodation coefficient. When the fate of spores inside a spore clump needs to be

modeled in highly complex engineering models, the results of Fig. A.16 show that a good

approximation would be to assume that individual spores in a spore clump are exposed

to about 25 ∼ 45% of the heat load compared to a single spore. Combined heat transfer

18

19

and water diffusion simulations will be undertaken to help interpret future spore cluster

survivability measurements to be performed in a high temprature exposure tube.

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Appendix

A

Table A.1: Geometry and DSMC Casesa

Case Computational Cell sizeb Fnumc No. of No. of No. of α

Description Domain (µ) (µ) Particles Vertices Panels values

Single sphere 30× 30× 30 (0.33)3 10,874 36,450,000 338 672 1., 0.1Five, 30× 30× 30 (0.33)3 10,874 36,450,000 338 672 1.

linear sporesSingle ellipsoid 15× 15× 15 (0.16)3 10,874 36,450,000 450 896 1.Fractal-like 30× 30× 30 (0.33)3 10,874 36,450,000 338 672 1., 0.1

20-spore aggregateFractal-like 50× 50× 50 (0.63)3 71,680 25,600,000 338 672 1., 0.1

100-spore aggregateaA DSMC time step of 1× 10−9 s was used in all cases.bAll the calculations were performed on Cartesian grids.cFnum is the ratio of real molecules to simulated particles.

23

24

Figure A.1: Randomization of the positioning procedure in PCA and CCA (LHS) androtating procedure in CCA (RHS).

25

Figure A.2: Electron micrograph of Bacillus anthracis spores from the Sterne strain (undera high magnification of 12,483X).

Figure A.3: (LHS): Translational temperature contours for a single spore; (RHS): Transla-tional temperature contours for five spores.

26

Figure A.4: Radial translational temperature profile comparison for a single and five sporecases.

Figure A.5: (LHS): Surface heat flux and (RHS): Normalized heat rate for five spore co-linear case.

27

X(m)

-1E-06

-5E-07

0

5E-07

1E-06

Y(m)

-1E-06

-5E-07

0

5E-07

1E-06

Z(m

)

-1E-06

-5E-07

0

5E-07

1E-06

X Y

Z

Heat flux(W/m2)

1.02E+071.00E+079.80E+069.60E+069.40E+069.20E+069.00E+068.80E+068.60E+06

Figure A.6: Surface heat flux for an ellipsoid of B.anthracis Sterne spore shape.

Figure A.7: Illustration of 20-spore fractal-like aggregates of kf=1.1 and Df=2.0.

28

Figure A.8: Illustration of 20-spore fractal-like aggregates of kf=1.7 and Df=2.0.

29

Figure A.9: Illustration of 100-spore fractal-like aggregates of kf=1.1 and Df=2.0.

30

Figure A.10: Illustration of 100-spore fractal-like aggregates of kf=1.7 and Df=2.0.

Figure A.11: (LHS): Surface heat flux for 20-spore fractal-like aggregate with completeaccommodation; (RHS): Surface heat flux for 20-spore fractal-like aggregate with αE=0.1.

31

Figure A.12: (LHS): Normalized heat rate for 20-spore fractal-like aggregate with com-plete accommodation; (RHS): Normalized heat rate for 20-spore fractal-like aggregate withαE=0.1.

Figure A.13: (LHS): Surface heat flux for 100-spore fractal-like aggregate with completeaccommodation; (RHS): Surface heat flux for 100-spore fractal-like aggregate with αE=0.1.

32

Figure A.14: ((LHS): Normalized heat rate for 100-spore fractal-like aggregate with com-plete accommodation; (RHS): Normalized heat rate for 100-spore fractal-like aggregate withαE=0.1.

Figure A.15: Four different 100-spore CCA aggregates with prefactor kf=1.7 and fractaldimension Df=2.0 under full accommodation and the distribution of normalized heat rateover each spore in the aggregate.

33

Figure A.16: Distribution of normalized heat rate.