patadiyadharmesh.files.wordpress.com · Contents Publications based on this Thesis iv...

Modelling, Simulation, andStatistical Studies of Primary

Fragmentation of Coal ParticlesSubjected to Detonation Wave

A Thesis

Submitted For the Degree of

Doctor of Philosophy

in the Faculty of Engineering

by

Patadiya Dharmeshkumar Makanlal

Department of Aerospace Engineering

Indian Institute of Science

BANGALORE – 560 012

March 2015

Dedicated to

My ParentsMaganbhai & Vijayaben

and my sistersGeeta, Bharati, Taru (લાલો), Nila (ટીના), Kalindee (દકા), Hirva

यदयदाचरतिति शरषठसतिततदवतिरतो जनः | स यतपरमाण करति लोकसतिदनवतिर ति || (गीतिा ३.२१)

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Contents

Publications based on this Thesis iv

Acknowledgments vi

Abstract ix

Nomenclature xi

List of Tables xviii

List of Figures xix

1 Introduction 1

2 Simple Model: Constant Temperature Boundary Condition 82.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Solution by Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Calculation of Thermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Failure Theories and Fragmentation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Maximum Principal Stress Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Maximum Principal Strain Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Maximum Shear Stress Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.4 Maximum Distortion Energy Theory . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.5 Maximum Strain Energy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Model with Convective Boundary Condition 253.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Solution by Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Temperature Obtained from Numerical Solution . . . . . . . . . . . . . . . . . . . . . . 283.4 Calculation of Thermal Stresses Developed . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Radial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Tangential Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 General Model - Numerical Studies 334.1 Governing Equations and Numerical Modules . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Heat Transfer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.2 Volatilization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.3 Solid Mechanics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.4 Developed Stress Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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4.1.5 Weibull’s Weak Link Theory and Fracture Criteria Model . . . . . . . . . . . . . 394.2 Fragmentation due to Thermal Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.4 Fragmentation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Fragmentation due to Volatilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.2 Volatilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.4 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.5 Fragmentation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Weibull Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5.1 Coal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5.2 Preheating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.5.3 Detonation Wave Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Fragmentation Statistics 645.1 Coal Particle Mixture and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Statistical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Time Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4 Volatilization Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.5 Significance of Statistical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Conclusions 79

A Values of Parameters Used 82

B Numerical Code in C Language 84

Bibliography 108

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Publications based on this Thesis

Journals

Published

1. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Detonation Initiated Disinte-

gration of Coal Particle Due to Maximum Strain Energy Theory”, Journal of Coal

Science and Engineering, 19(4):435-440, 2013.

2. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Computational Model for Ther-

mal and Volatilization Induced Spontaneous Fragmentation of Coal Particle”, In-

ternational Journal of Advancements in Mechanical and Aeronautical Engineering,

2(1):161-165, 2015.

Under review

1. Jaisankar S., Patadiya D. M., and Sheshadri T. S. “Shock Wave Induced Thermal

Fragmentation of Coal Particles” in Fuel.

2. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Devolatilization and Statistical

Studies of Coal Particle Mixture Subjected to Plasma Initiated Detonation Using

Weibull Theory” in Combustion Science and Technology.

3. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Statistical Studies on Spon-

taneous Primary Fragmentation of Coal Air Mixture subjected to Detonation” in

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Combustion, Explosions, and Shock Waves.

Conferences and Symposia

Published

1. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Application of Maximum Prin-

cipal Strain Theory for Study of Coal Particle Disintegration when Subjected to

Detonation Wave”, ICCS&T 2013, pp. 603-613, Oct. 2013, University Park, Penn-

sylvania, USA.

2. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Computational Model for Ther-

mal and Volatilization Induced Spontaneous Fragmentation of Coal Particle” Proc.

of the Second Intl. Conference on Advances in Mechanical and Robotics Engineering

- AMRE 2014, pp. 44-48, Oct. 2014, Zurich, Switzerland.

Accepted

1. Patadiya D. M., Jaisankar S., and Sheshadri T. S., “Numerical Model for Thermal

and Volatilization Induced Spontaneous Fragmentation of Coal Particle” accepted

for poster presentation, in Frontiers of Computational Physics 2015, ETH Zurich,

Switzerland.

Acknowledgments

“Ph.D. is not just a degree but it is an entire training program in which you

are supposed to learn the methodology of how to identify, analyse and solve the

problem”

This was the first statement given by my guide Prof. T. S. Sheshadri to me when I

met him first time. I am lucky because I had an opportunity to successfully learn this

methodology from him. Without writing fancy words I want to say he is the best guide

in the world. The other person behind this thesis is Dr. Jaisankar S. whose valuable

input in writing the numerical code is significant. In other words I consider him as my

unofficial co-guide. I thank Kunal Patil for advising me about interview procedure in

Indian Institute of Science (IISc).

While studying PhD, throughout my entire stay in IISc, I was involved in various social

activities when working for groups like IISc Gujarati Parivar, prasthutha, praharshini,

Sanskrut Sangh, Hindi Samiti, Students’ Council (SC) etc. and made many friends.

These activities worked as non scientific problems in my life. It provided me the platform

to discuss and get help on academic and non academic matters through interacting with

many people. We should understand humans because we are surrounded by humans

and throughout our life we have to deal with humans. After all, education will have

true meaning if we are able to apply the above said methodology to practical life and

do something good for the welfare of society. I am writing about each one of them. If I

forget anyone then it is purely unintentional.

vii

I am thankful to my friends from Aeroacoustic and Plasma Dynamic Laboratory

(APDL). Dr. Ramesh Narsimha, Muttanna, Vinod Kumar, Chandan, Shrikant, Amaren-

der, Reginald, Sagish and Moses Apollo. I was not from computer engineering back-

ground but Vinod Kumar was the first person who taught me how to program. Re-

gardless of how much I argued and fought with Muttanna, he was always there to help

me.

I am from Gujarat, India and it is natural that I first came in to contact with IISc

Gujarati Parivar. The group provided me the environment and people from my native

place. I am thankful to all the people involved in this group. To name a few; Kunalbhai,

Siddharth, Swetank, Ruchik, Mustafa, Mehul, Chitrang, Arpit, Nirmit, Yatin, Mahmadi,

Amit, Kalpesh, Pranav and others.

prasthutha & praharshini were the type of groups which my heart wanted to join. It

was the platform which I perceived in my mind from my early life but it came in to my

real life when I joined IISc. Discussions, talks, and many more leadership qualities I

learned from this group. I want to thank my friends with whom I worked. To name a

few Raghvendra, Abhiram, Krishna.

In all the groups I used to contribute as volunteer but Hindi Samiti was the group in

which I both contributed voluntarily and enjoyed as well. It will be my life long memory

to enjoy the cultural events like Holi, Diwali, Ram Navami, Janmashtami etc. organised

by this group. In this group for the first time ever I learned how to recite Ramayan.

The group taught me how to love all Indians. This provided me mental relaxation

throughout my stay in IISc. I am thankful to my friends with whom I worked. To name

a few; Rajnish, Anil Sharma, Nishant, Piyush Kumar, Praveen Kumar, Surendrkumar,

Lokesh.

Specifically I want to thank SC, IISc for giving me responsibility to serve entire IISc

student community as a General Secretary. With this responsibility I learned how to bal-

ance between academic and non academic life. I am thankful to the team members with

viii

whom I worked. Each one of them served the purpose of living teacher to me. I learned

many things from them. To name a few Rishikesh Pandey, Sreevalsan, Pramod Ku-

mar, Javed, Vipin Gupta, Bipin Kumar, Nivesh, Anilprasad, Saurabh Agarwal, Saurabh

Dixit, Pankaj Jain, Sharadaprasad, Swapnil.

Finally I want to tell about people behind me to whom I am too small to thank;

my parents and sisters. I remember my sister Geeta who took care of me when my

mother was at work. Bharti who still once in a year fasts for me and used to help in

my study. Taru who can fight with anybody for me. As a child whenever I had tussle

with classmates or others she used to come and protect me. Nila taught me how to play

cricket and fly kites. In short she was my childhood friend and taught me how to enjoy

childhood. Daxa, I remember woke up at 4:00 early morning for 4 years without failing

a single day and prepared my lunch when I was under graduate student. Hirva, I spent

my childhood playing with her. My each sister is a mother for me.

There is something in the IISc student life and environment which I am unable to

express in words. Students from different regions of India come here and it was great

experience to learn and understand their life, culture, traditions, feelings, sentiments

etc. I am thankful to IISc for providing me such a joyous research environment. I really

enjoyed studying in IISc.

Last but never the least, I thank the almighty for giving me strength for doing my

work.

Abstract

Coal is likely to remain an important energy source for the next several hundred years

and hence advances in coal combustion technologies have major practical impact. Deto-

nation combustion of coal initiated by a plasma cartridge driven detonation wave holds

promise for improving both system and combustion efficiencies. Both fragmentation and

chemical kinetic pathways are qualitatively different in comparison to conventional coal

combustion. The present work is a theoretical investigation of fragmentation due to

detonation wave. The theoretical simulation starts with simple model and progressively

incorporates more realistic analysis such as combined convective and radiative boundary

condition. It studies the passing of detonation wave on coal particles suspended in air.

Concepts of solid mechanics are used in analysing fragmentation of coal particles. A

numerical model is developed which includes stress developed due to both thermal and

volatilization effects. Weibull statistical analysis is used to predict the fracture time and

fracture location resulting from principal stress induced. It is observed that coal parti-

cles fragment within microseconds. Radiation does not have much effect on developed

stress. Volatilization does not have much effect on fragmentation for the particle size

considered in this work and stress due to thermal effect dominated the fragmentation.

Coal size distribution statistics is considered to obtain real regime. Coal is used as mix-

ture of different sized particles in real combustors. Hence it is important to analyse

the effect of detonation wave on mixture of coal particles. Results presented in this

work from simulation run suggest that plasma assisted detonation initiated technology

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can fragment coal particles faster. Average fracture time of mixture of coal particles is

far less than detonation travel time for the detonation tube considered here. Simula-

tion results suggest that almost 90% of coal particles fragment early. Average fracture

time reduces as Mach number increases. Same phenomena can be observed for volatile

matter generated at fracture and flow of volatile matter at fracture. Hence it can be

concluded that plasma assisted detonation combustion leads to different volatilization

and fragmentation pathways.

Nomenclature

Bi Biot number

C2 Arbitrary constant

Cp Specific heat of coal in J/(kg.K)

E Young’s modulus of elasticity in MPa

Ea Activation energy in J/kmol

h Convective heat transfer coefficient in W/(m2.K)

h Specific Enthalpy in J/kg

k Thermal conductivity in W/(m.K)

ko Pre-exponential factor in 1/s

M Mach number

m Weibull modulus

Mvol Molecular weight of volatile in kg/kmol

N Factor of safety

n Reaction order of pyrolysis

Nmol Molar flow rate of volatile in kmol/(m2.s)

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xii

p Pressure in Pa

Pf Weibull failure probability

Ps Weibull survival probability

q Specific energy supplied in J/kg

R Dimensionless radius

r Radius in m

Rf Non-dimensional fracture location

ro Outer radius of coal in m

Ru Universal gas constant in J/(kmol.K)

rpore Average pore radius in m

s Laplace variable

T Temperature in K

t Time in s

tsd Thermal diffusion time of shock in seconds

tsp Travel time of shock over particle in seconds

tst Travel time of shock over tube in seconds

V Actual dimensionless mass of volatile in kg/kg of coal

v Specific volume in m3/kg

V ∗ Maximum dimensionless mass of volatile in kg/kg of coal

Vo Weibull characteristic volume in m3

xiii

x, y, z Arbitrary variables

Greek Symbols

α Thermal diffusivity in m2/s

β Coefficient of thermal expansion in 1/K

ε Voidage fraction of pore

εb Emissivity of body

εmax Maximum strain

γ Specific heat ratio

µ Dynamic viscosity of volatile in Pa.s

µa Statistical mean of variable

ν Poisson’s ratio

ρ Density of coal in kg/m3

ρc Density of char in kg/m3

σ Stress in MPa

σb Stefan-Boltzmann constant in W/(m2.K4)

σo Weibull characteristic strength in MPa

σr Radial stress in MPa

σr,tot Total radial stress in MPa

σd Standard deviation

σt Tangential stress in MPa

xiv

σu Weibull ultimate strength (Threshold stress) in MPa

σuu Ultimate strength of coal in MPa

τ Shear stress in MPa

τ Dimensionless temperature

τmax Maximum shear stress in MPa

τt Tortuosity of pores

θ Dimensionless time

Subscripts

1, 2, 3 Principal directions

d Downstream

f Fracture condition

i Initial condition

∞ Surrounding condition

r, θ, φ Spherical coordinates

u Upstream

Acronyms

CDF Cumulative Distribution Function. 5, 7, 66, 69, 71, 75

CTE Coefficient of Thermal Expansion. 60

CV Coefficient of Variation. 5, 7, 66, 69, 71, 72, 77, 78, 80

DSM Developed Stress Model. 4, 5

FCM Fracture Criteria Model. 4, 5

FEM Finite Element Method. 34

HTM Heat Transfer Model. 4, 5

PDF Probability Density Function. 5, 7, 66, 67, 69, 71, 75, 77, 80

PFM Primary Fragmentation Model. 5

POF Probability of Failure. 40

RDX Research Department eXplosive. 34

SD Standard Deviation. 5, 66, 76

SM Statistical Model. 5

SMM Solid Mechanics Model. 4, 5

xv

Acronyms xvi

SND Standardised Normal Distribution. 69

SRV Standardised Random Variable. 69

VM Volatilization Model. 4, 5

WLT Weibull’s Weakest Link Theory. 3, 4, 6, 34, 39

List of Tables

2 Simple Model: Constant Temperature Boundary Condition

2.1 Time and location where induced thermal stress and strain reach peak

values obtained from Maximum Principal Stress Theory . . . . . . . . . 24

4 General Model - Numerical Studies

4.1 Fracture results for 50 µm size coal particle at different Mach numbers at

β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . 56

5 Fragmentation Statistics

5.1 Different time scales for all sizes of particles at different Mach numbers . 65

5.2 Typical particle size distribution . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Fracture results for different size coal particles at constant values of β =

60× 10−6, V ∗ = 0.3, σo = 11, σu = 8, m = 6 . . . . . . . . . . . . . . . . 68

5.4 Results obtained w.r.t fracture time for coal particle mixture subjected to

detonation waves of different Mach numbers at β = 60× 10−6, V ∗ = 0.3,

σo = 11, σu = 8, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Volatile matter generation statistics at M = 6, β = 60× 10−6, V ∗ = 0.3,

σo = 11, σu = 8, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xvii

LIST OF TABLES xviii

5.6 Volatile matter flow statistics at M = 6, β = 60×10−6, V ∗ = 0.3, σo = 11,

σu = 8, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

List of Figures

1 Introduction

1.1 Research overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Simple Model: Constant Temperature Boundary Condition

2.1 Profile for the non-dimensional temperature varying with non-dimensional

time and radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Stress state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Temperature profiles at different Mach numbers; solid lines for 50 µm size,

dashed lines for 100 µm and dotted lines for 150 µm size coal particles . 15

2.4 σ1 profile at M = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 σ1 profile at M = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 σ1 profile at M = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Profile of resultant stress obtained from Eq. 2.31 at M = 3 . . . . . . . . 17



2.10 Maximum strain distribution at M = 3 . . . . . . . . . . . . . . . . . . . 18



2.13 τmax distribution at M = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 19

xix

LIST OF FIGURES xx









2.22 Comparision of failure theories for 50 µm size coal particle at M = 3 . . 23

3 Model with Convective Boundary Condition

3.1 Discretization regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Comparison between temperatures obtained from analytical and numeri-

cal solutions for size=50 µm . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Comparison between σr obtained from analytical and numerical solutions

at M = 3, β = 40× 10−6, size=50 µm . . . . . . . . . . . . . . . . . . . . 30

3.4 Comparison between σt obtained from analytical and numerical solutions

at M = 3, β = 40× 10−6, size=50 µm . . . . . . . . . . . . . . . . . . . . 31

4 General Model - Numerical Studies

4.1 Evolution of temperature in time (in µs) for 50 µm size coal particle at

M = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Evolution of stresses in time (in µs) for 50 µm size coal particle at M = 7,

β = 40× 10−6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Evolution of variables in time (in µs) for 50 µm size coal particle till

fracture at 42 µs for M = 5, β = 40× 10−6, σo = 12, σu = 10, m = 6 . . 44

LIST OF FIGURES xxi

4.4 Evolution of variables in time (in µs) for 10 µm size coal particle till

fracture at 10 µs for M = 4, β = 40× 10−6, σo = 12, σu = 4, m = 4 . . . 45

4.5 Failure characteristics of coal particles of different sizes at different Mach

number when β = 40× 10−6, σo = 9, σu = 6, m = 6 . . . . . . . . . . . . 46

4.6 Variation of temperature (in K) on different size coal particles when M =

4, β = 40× 10−6, σo = 12, σu = 4, m = 4 . . . . . . . . . . . . . . . . . 47

4.7 Temperature plots for M = 6, size = 50µm, β = 40 × 10−6, V ∗ = 0.3,

σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.8 Volatile matter variables at time 1.38 ms, M = 6, size = 50µm, β =

40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . . . . 50

4.9 Volatile matter variables at longer time scale for time 20 ms, M = 6,

size = 50µm, β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6 . . . . . . 51

4.10 Stresses at time 1.38 ms, M = 6, size = 50µm, β = 40× 10−6, V ∗ = 0.3,

σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.11 Principal stresses at time 1.38 ms, M = 6, size = 50µm, β = 40 × 10−6,

V ∗ = 0.3, σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . . . . . . . . . . 52

4.12 Comparison of σr,tot with relation to volatile matter content at 634.8 µs,

M = 6, size 50 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.13 Local failure probability for M = 6, size = 50µm, β = 40×10−6, V ∗ = 0.3,

σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.14 Comparison of failure probability for different initial volatile matter con-

tent at M = 6, size = 50µm, β = 40 × 10−6, σo = 12, σu = 6, m = 6,

R = 0.09 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.15 Temperature and stress gradients at time 1.38 ms, M = 6, size = 50µm,

β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6 . . . . . . . . . . . . . 55

4.16 Effect of Weibull parameters on coal fracture time at different m . . . . 57

4.17 Effect of CTE and Young’s Modulus at M = 4 . . . . . . . . . . . . . . . 60

LIST OF FIGURES xxii

4.18 Effect of preheating at different Mach numbers . . . . . . . . . . . . . . 61

4.19 Failure characteristics of coal particles of different sizes at different Mach

number and m = 6, σo = 9 . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Fragmentation Statistics

5.1 PDF and CDF plots of fracture time at M = 6, β = 60× 10−6, V ∗ = 0.3,

σo = 11, σu = 8, m = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 PDF and CDF plots of volatile matter generated at fracture at M = 6,

β = 60× 10−6, V ∗ = 0.3, σo = 11, σu = 8, m = 6 . . . . . . . . . . . . . . 73

5.3 PDF and CDF plots of volatile matter flow at fracture at M = 6, β =

60× 10−6, V ∗ = 0.3, σo = 11, σu = 8, m = 6 . . . . . . . . . . . . . . . . 74

5.4 Comparison of average fracture time and coefficient of variation with dif-

ferent Mach numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 1

Introduction

Coal is the major fossil fuel, easily accessible and likely to remain as a major energy

source for long time to come in the future. The applications range from household to

industrial purposes such as metal castings, thermal power plants etc. in which coal is

burnt at moderate temperatures. Coal is likely to remain an important energy source for

several hundred years and hence improvements in combustion and thermodynamic cycle

efficiencies have immediate practical impact. Detonation combustion of coal particles us-

ing plasma initiated detonation wave can enhance the combustion and thermodynamic

cycle efficiencies of the combustion process. Detonation mimics constant volume combus-

tion which is thermodynamically more efficient. The sudden exposure of coal particles

to high temperature detonation/plasma shock waves are known to trigger spontaneous

pulverization of coal particles. Primary fragmentation of coal in the combustion process

is known to substantially impact the combustion efficiency.

The usual steps of coal combustion in power plants include thermal heating leading

to primary pulverization and devolatilization, pyrolysis, and ignition/combustion of py-

rolysis products. Several attempts have been made and reported in improving each of

these steps for a more efficient burning [1]. Among the different steps of combustion,

the heterogeneous steps of devolatilization and char burning are the slowest steps which

when improved can greatly contribute to overall efficiency. Primary fragmentation of

1

Chapter 1. Introduction 2

coal associated with devolatilization is considered as a key step affecting both hydro-

dynamics and combustion characteristics as well as char distribution in a burner [2].

Several studies highlight the role of high temperature operations in the acceleration of

primary fragmentation of coal particles [3] and complete burnout. Plasma-assisted pul-

verized coal combustion is one such technology operating at high temperatures which

has the potential to promote spontaneous fragmentation, early ignition, and enhanced

stabilization of pulverized coal. In addition to enhancing the combustion efficiency of the

flame, it reduces harmful emissions from power coals of all ranks (brown, bituminous,

anthracite and their mixtures) [4, 5].

Coal particle fragmentation may happen due to thermal stress or excessive pressure

caused by volatile matter with microscopic cause being excess energy resulting in break-

ing of chemical bonds or growth of Griffith cracks [6]. Some authors have given priority to

the mechanical stress generated by the pressure of volatiles [7] whereas others prioritised

the thermal stress [8]. Several attempts made to understand the influence of heating

temperature and the size of coal particles highlights the role of thermal stress [8].

Some reports are available on primary fragmentation of coal when fed to burners/-

furnaces. Coal combustion studied by Chirone et al. [9] in a fluidized bed with each of

the thermal effect, volatile transport and volatile release constituting a submodel indi-

cated that coal particles break in hemispherical shape and parallel to its bedding plane.

Fragmentation behaviour of coal particle in a non slagging cyclone combustor account-

ing both thermal stresses and Knudsen pressure due to volatiles was studied by No et

al.[14] which indicated a loosely fragmented core region. Coal particle combustion in

drop tube furnace was experimentally studied by Dacombe et al. [3] and a fragmenta-

tion pattern was presented. It was reported by Crandall et al. [10] and Hasselman [11]

that component is assumed to be failed when stress developed at a point exceeds the

tensile strength of the body. Enkhzhargal et al. [12] simulated coal particle failure as the

surpassing of induced thermal stresses above the ultimate tensile strength and observed


that coarser particles (d > 10 mm) failed at medium temperature (about 1000 ◦C) while

finer ones up to 100 µm remained intact up to 1500 ◦C. A study on primary coal particle

fragmentation under high heating rate and high temperature condition by Senneca et

al. [13] show origination of small fragments from the outer shell of the particles due to

thermal stress and larger fragments from the inner core as a result of internal stresses

due to volatile release. A semidetailed fragmentation model was proposed by Senneca et

al. [14] and simulated to understand the effect of temperature, heating rate and initial

particle size on the primary fragmentation of coal.

The work presented here reports on numerical simulation of coal fragmentation when

subjected to a detonation wave as in a detonation combustor. Focus here is on improving

fragmentation of coal particles when subjected to detonation waves which can typically

be a plasma shock wave. Direct initiation of detonation may be brought about typically

by an electric plasma cartridge at one end of the detonation tube. Spherical shape of

the coal particle is assumed. The governing differential equations are discretized and

solved numerically using finite volume procedure and stresses induced due to thermal

and volatilization are calculated. Several theories have been applied to understand the

fracture of brittle material in the literature [3,10,15]. Among such models, the Weibull’s

Weakest Link Theory (WLT) [16] used to study detonation induced brittle thermal

fracture [17] is found superior to the critical stress theory for coal fragmentation [18]

and is applied here, to study the coal particle fragmentation subjected to detonation

waves.

Research Pathway

1. Analytical solution for single coal particle for simple boundary conditions.

2. Analytical solution for single particle for convective boundary condition.


3. Numerical solutions for single particle with radiation boundary condition and volatiliza-

tion included. Numerical method validated by comparison with analytical methods

given in No. 1 and No. 2 above.

4. Inclusion of irregularities in coal particle interior through Weibull parameters.

5. Statistics of mixture of particles of different sizes for statistical fragmentation anal-

ysis.

Overview of Present Research

1. First step of the solution approach is developing Heat Transfer Model (HTM) of

simple constant temperature boundary condition using analytical methods. Particle

fracture time and fracture location are predicted due to thermal stress only using

failure criteria suggested by various failure theories.

2. This model is applied to solve more realistic boundary condition; convective bound-

ary conditions.

3. Numerical model for solution of more complex problem is sought after successfully

developing HTM analytically.

4. Numerical code is developed to solve convective and radiative boundary condition

and validated by comparing with analytical solutions.

5. After developing HTM numerically Volatilization Model (VM) of volatile matter is

developed.

6. HTM and VM linked with Solid Mechanics Model (SMM) is used to obtain principal

stresses developed in coal particle. This gives Developed Stress Model (DSM).

7. Fracture Criteria Model (FCM) using WLT is developed after successfully devel-

oping DSM. This model is capable of providing fracture time, fracture location,


temperature at the time of fracture, volatilization matter present at fracture, flow

of volatile at fracture, pressure at fracture.

8. This completes Primary Fragmentation Model (PFM) for single coal particle only

which includes HTM, VM, SMM, DSM and FCM.

9. Statistical Model (SM) is developed for mixture of coal particles having different size

and properties. Model is capable of providing average, Standard Deviation (SD),

Probability Density Function (PDF), Cumulative Distribution Function (CDF), Co-

efficient of Variation (CV) of fracture time, volatile matter at fracture and flow of

volatile at fracture for mixtures of different size coal particles.

FCM

HTM VM

THERMAL STRESS PRESSURE

SMM

DSM

PFM

SM

Figure 1.1: Research overview


Thesis Layout

Present work is development of numerical model to solve primary fragmentation of coal

particles subjected to detonation. Each chapter presents development of more robust

solution than the solution presented in previous chapter. Chapters are development

phases studied in deriving the evolution of research methodology of starting with simpler

and developing to more realistic conditions.

Chapter 2 presents analytical studies on coal fragmentation when subjected to a det-

onation wave as in a detonation combustor. Simple constant temperature boundary

condition is assumed. Spherical shape of the coal particle is assumed. Governing differ-

ential equations and boundary conditions are nondimensionalised and solved. Analytical

solution of the heat transfer process in the coal particle has been obtained. Induced

thermal stresses were calculated analytically. Fragmentation behavior is predicted by

various failure theories. Three dimensional plots are presented showing developed ther-

mal stresses with respect to time and location on the coal particle.

More general analytical solution of the problem is given in Chapter 3. Convective

boundary condition is used to solve the governing differential equation unlike constant

temperature boundary condition used in chapter 2. Numerical solution developed and

validated by comparing with analytical solution.

Chapter 4 gives general numerical solution technique to solve this problem. Volatiliza-

tion is considered in this chapter to make the obtained solution more realistic. Volatiliza-

tion is considered as simple first order reaction. To start with simplistic approach flow

of volatile is considered as viscous or bulk continuum flow [19]. Failure criteria is defined

based on probabilistic approach suggested by Weibull. Three parameter WLT is used to

calculate failure probability. Some aspects of Weibull parameters are discussed. Effect of

preheating, coal properties and detonation strength on coal fragmentation are discussed.


Previous chapters deals with primary fragmentation of single coal particle while Chap-

ter 5 is concerned with study of primary fragmentation of mixture of different size coal

particles subjected to detonation. Fragmentation parameters like time, volatile matter

generation and flow of volatile are calculated and normalized to know how fracture is

distributed. Prediction is made more reliable through including rigorous statistical pa-

rameters like PDF, CDF and CV. Average fracture time of the coal particle mixture is

compared with the time traveled by single detonation wave from one end of detonation

tube to the other.

Chapter 2

Simple Model: Constant

Temperature Boundary Condition

Coal particle is subjected to high temperature shock wave. Heat transfer inside the

particle is considered only radially varying as time scale of shock propagation across

particle is far lower than thermal diffusion time inside particle. This gives rise to the

following description.

2.1 Governing Equations and Boundary Conditions

Unsteady heat conduction equation with spherical symmetry is [20]

1

r2

∂

∂r

(kr2∂T (r, t)

∂r

)= ρCp

∂T (r, t)

∂t(2.1)

Considering negligible spatial variation in thermal conductivity

1

r2

∂

∂r

(r2∂T (r, t)

∂r

)=ρCpk

∂T (r, t)

∂t(2.2)

Variation in thermal conductivity as reported by Honda et al. [21] is marginal in the

temperature range of 1000 ◦C to 2100 ◦C. Most of the studies done in this research are

8

Chapter 2. Simple Model: Constant Temperature Boundary Condition 9

in the Mach number range 3 to 6 which correspond up to 2200 ◦C. Also, temperature

profile is more sensitive to thermal diffusivity (α = k/ρcp) than k as inferred from Eq. 2.2.

Gu [22] reported that there is little change in α with change in temperature. Moreover,

Maloney et al. [23] reported tenfold variateion in k hardly changing the average particle

temperature by 5% for heating temperature range from 300 K to 2100 K. Hence, any

variation in k should hae minimal influence on particle temperature.

And the boundary conditions are:

T (ro, t) = T∞ (2.3)

where T∞ is the skin temperature induced by the detonation wave and can be obtained

from Rankine-Hugoniot relations. The skin temperature T∞ which corresponds to normal

shock is taken constant as time scale of interest is far lesser than time for external

conditions to change due to shock propagation.

From symmetry

∂T

∂r

∣∣∣∣r=0

= 0 (2.4)

The initial condition is:

T (r, 0) = Ti (2.5)

Eq. 2.1 simplifies to

∂2T

∂r2+

2

r

∂T

∂r=

1

α

∂T

∂t(2.6)


Introducing dimensionless variables

τ(R, θ) =T (r, t)− T∞Ti − T∞

, R =r

ro, θ =

αt

r2o

(2.7)

Eq. 2.6 becomes

∂2τ(R, θ)

∂2R+

2

R

∂τ(R, θ)

∂R=∂τ(R, θ)

∂θ(2.8)

with boundary and initial conditions

τ(1, θ) = 0 (2.9)

∂τ

∂R

∣∣∣∣R=0

= 0 (2.10)

And,

τ(R, 0) = 1 (2.11)

2.2 Solution by Laplace Transform

Taking Laplace transform of Eqs. 2.8 to 2.11,

L[∂2τ(R, θ)

∂2R+

2

R

∂τ(R, θ)

∂R

]= L

[∂τ(R, θ)

∂θ

](2.12)

And the boundary and initial conditions become

L [τ(1, θ)] = 0⇒ τ(1, s) = 0 (2.13)

L[∂τ

∂R

∣∣∣∣R=0

]= 0⇒ dτ(R, s)

dR

∣∣∣∣R=0

= 0 (2.14)

and


L [τ(R, 0)] = 1⇒ τ(R, 0) = 1 (2.15)

From Eq. 2.12

d2τ

dR2+

2

R

dτ

dR= sτ(R, s)− τ(R, 0) (2.16)

This is the Bessel’s differential equation and its solution after simplification is [24, 25]

τ(R, s) =C2 sinh(

√sR)

R+

1

s(2.17)

Applying boundary condition of Eq. 2.13

τ(1, s) =C2 sinh(

√sR)

R+

1

s= 0 (2.18)

Therefore

C2 =−1

s sinh(√s)

(2.19)

The complete transformed solution is

τ(R, s) =− sinh(

√sR)

R s sinh√s

+1

s(2.20)

The final solution as a function of R and θ can be found by taking Inverse Laplace

Transform of Eq. 2.20.

τ(R, θ) = L−1 [τ(R, s)] = L−1

[− sinh(

√sR)

R s sinh√s

]+ L−1

[1

s

](2.21)

The inverse of the above equation can be found by Bromwich’s Contour Integral method

[26]. The method can be given as

if,

∫ ∞0

f(t) = F (s) (2.22)


then,

f(t) =1

2πi

∮F (z)etz︸︷︷︸f(z)

dz = Sum of Residues (2.23)

And residues are given by,

R[f(z); zj] =1

(p− 1)!limz→zj

dp−1

dzp−1[(z − zj)f(z)] (2.24)

Using the Bromwich’s Contour Integral method and simplifying, the inverse of the Eq.

2.21 will give the final solution as below,

τ(R, θ) =T (r, t)− T∞Ti − T∞

=∞∑m=1

−2(−1)m

λmRsin(λmR)e−λ

2mθ (2.25)

Where eigen condition is sinh(√z)=0. The eigen values are the poles of the function

f(z), which are, z = −λ2m = −m2π2

Profile for the non-dimensional temperature varying with non-dimensional time and

non-dimensional radius is given in Fig. 2.1.

00.2

0.40.6

0.81

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

θ

M=3

R

τ

(a) M = 3

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0

0.5

1

θ

M=5

R

τ

(b) M = 5

00.2

0.40.6

0.81

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

θ

M=7

R

τ

(c) M = 7

Figure 2.1: Profile for the non-dimensional temperature varying with non-dimensional

time and radius

2.3 Calculation of Thermal Stresses

It is necessary to calculate thermal stresses to understand the fracture pattern of coal

particle.


Radial Stresses

Radial stress [27] is given by

σr =2βE

(1− ν)

[1

r3o

∫ ro

0

Tr2dr − 1

r3

∫ r

0

Tr2dr

](2.26)

After substituting the value of temperature T from Eq. 2.25 and simplifying,

σr =4βE(Ti − T∞)

(1− ν)

∞∑m=1

(−1)me−λ2mθ

λm [cosλmλm

+1

R3

(−R cos(λmR)

λm+

sin(λmR)

λ2m

)](2.27)

Tangential Stresses

Tangential stress [27] is given by

σt =βE

(1− ν)

[2

r3o

∫ ro

0

Tr2dr +1

r3

∫ r

0

Tr2dr − T]

(2.28)


σt =2βE(Ti − T∞)

(1− ν)

∞∑m=1

(−1)me−λ2mθ

λm[2 cosλmλm

+sin(λmR)

R− 1

R3

(−R cos(λmR)

λm+

sin(λmR)

λ2m

)](2.29)

2.4 Failure Theories and Fragmentation Analysis

Numerical results were obtained from the analytical solution by writing a code in C

language. Typical values for various parameters are given in Appendix A. The initial


coal temperature is taken as Ti = 300 K. Three typical sizes of coal particles are taken as

50 µm, 100 µm, and 150 µm. The Mach number of detonation waves are taken as 3, 5,

and 7. Developed stresses are compared with ultimate strength plane (in black colour)

in plots.

For simplicity, spherically symmetric coal particle is considered. Hence stress compo-

nents consist of radial stress component σr and tangential stress component σt. Various

failure theories are used to predict fracture of coal particles. Stresses developed due to

failure theories are calculated. Ultimate strength is considered as limiting criteria of

fracture for all the theories. State of stress is given in Fig. 2.2 where radial stress is

considered as tensile and tangential stress considered as shear.

Figure 2.2: Stress state


0

0.5

1

0

0.005

0.01200

400

600

800

Dimensionless radius RTime in Seconds

Te

mp

era

ture

in

Ke

lvin

(a) M = 3

0

0.5

1

0

0.005

0.010

500

1000

1500

2000


Te

mp

era

ture

in

Ke

lvin

(b) M = 5

0

0.5

1

0

0.005

0.010

1000

2000

3000

4000


Te

mp

era

ture

in

Ke

lvin

(c) M = 7

Figure 2.3: Temperature profiles at different Mach numbers; solid lines for 50 µm

size, dashed lines for 100 µm and dotted lines for 150 µm size coal particles

2.4.1 Maximum Principal Stress Theory

This theory, developed by Rankine [28], is based on calculating maximum principal

stresses induced in coal particle and comparing it with ultimate strength. Mathemati-

cally, these maximum and minimum principal stresses are expressed as,

σ1,3 =σr2±√(σr

2

)2

+ σ2t ≥

σuuN

(2.30)

3D plots of maximum principal stress induced inside coal particles are given in figures

2.4 to 2.6.


(a) 50 µm (b) 100 µm (c) 150 µm

Figure 2.4: σ1 profile at M = 3

(a) 50 µm (b) 100 µm (c) 150 µm


(a) 50 µm (b) 100 µm (c) 150 µm


2.4.2 Maximum Principal Strain Theory

This theory was developed by Saint Venant [28]. According to this theory, the failure

or yielding occurs at a point in the member when the maximum principal strain in the


bi-axial stress system reaches the limiting value of strain. Mathematically, it is expressed

as,

σ1 − ν σ3 ≥σuuN

(2.31)

3D plots of resultant stress obtained from Eq. 2.31 inside coal particles are given in

figures 2.7 to 2.9.

(a) 50 µm (b) 100 µm (c) 150 µm

Figure 2.7: Profile of resultant stress obtained from Eq. 2.31 at M = 3

(a) 50 µm (b) 100 µm (c) 150 µm


(a) 50 µm (b) 100 µm (c) 150 µm



Maximum Strain

Maximum strain is given by

εmax =σ1

E− νσ3

E(2.32)

3D plots of maximum strain inside coal particles are given in figures 2.10 to 2.12

(a) 50 µm (b) 100 µm (c) 150 µm

Figure 2.10: Maximum strain distribution at M = 3

(a) 50 µm (b) 100 µm (c) 150 µm


(a) 50 µm (b) 100 µm (c) 150 µm



2.4.3 Maximum Shear Stress Theory

This theory, developed by Guest and Tresca [28], is based on calculating maximum

shear stresses induced in the coal particle and comparing it with ultimate strength.

Mathematically, the maximum shear stress is expressed as,

τmax =

√(σr2

)2

+ σ2t ≥

σuu2×N

(2.33)

3D plots of maximum shear stress induced inside coal particles are given in figures

2.13 to 2.15.

(a) 50 µm (b) 100 µm (c) 150 µm

Figure 2.13: τmax distribution at M = 3

(a) 50 µm (b) 100 µm (c) 150 µm



(a) 50 µm (b) 100 µm (c) 150 µm


2.4.4 Maximum Distortion Energy Theory

This theory was suggested by Hencky and Von Mises [28]. According to this theory

failure or yielding occurs at a point in a member when the distortion strain energy,

also called as shear strain energy, per unit volume in bi-axial stress system reaches the

limiting distortion energy per unit volume. Mathematically, it is expressed as,

√σ2

1 + σ23 − σ1σ3 ≥

σuuN

(2.34)


figures 2.16 to 2.18.

(a) 50 µm (b) 100 µm (c) 150 µm



(a) 50 µm (b) 100 µm (c) 150 µm


(a) 50 µm (b) 100 µm (c) 150 µm


2.4.5 Maximum Strain Energy Theory

This theory was developed by Haigh [28]. According to this theory failure or yielding

occurs at a point in member when the strain energy per unit volume in bi-axial stress

system reaches the limiting strain energy per unit volume. Mathematically, it is expressed

as, √σ2

1 + σ23 − 2νσ1σ3 ≥

σuuN

(2.35)


figures 2.19 to 2.21.


(a) 50 µm (b) 100 µm (c) 150 µm


(a) 50 µm (b) 100 µm (c) 150 µm


(a) 50 µm (b) 100 µm (c) 150 µm


2.5 Results and Discussions

Stress plots obtained from various failure theories looks similar in appearance but are

different in detail. Fracture is observed relatively earlier or delayed depending on the


failure theory used. As shown in Fig. 2.22, fissure in Fig. 2.22(e) is different than in Fig.

2.22(a) while it looks similar to Fig. 2.22(d). Reason for this difference is in the fact

that the fundamental concepts of both theories are different. That is why stresses nearby

fissure are different in those figures. Maximum Principal Stress Theory assumes fracture

when developed stress exceeds ultimate stress while Maximum Distortion Energy Theory

assumes fracture when first distortion observed in particle. Distortion in material, usually

occurs when material starts to yield, and is followed by ultimate stress as typically

observed in stress-strain relationship plot. Due to this Maximum Distortion Energy

Theory predicts earlier fracture with lesser stress than Maximum Principal Stress Theory.

Maximum Principal Strain Theory predicts fracture based on strain and the stress is not

considered to predict fracture. Similarly Maximum Strain Energy Theory and Maximum

Distortion Energy Theories predicts fracture based on energy supplied to particle unlike

the theories which predicts fracture based on limiting stress and limiting strains.

(a) Rankine’s Theory (b) Saint Venant’s Theory (c) Guest’s Theory

(d) Henckey’s Theory (e) Haigh’s Theory

Figure 2.22: Comparision of failure theories for 50 µm size coal particle at M = 3


One set of fragmentation results obtained from Maximum Principal Stress Theory are

presented in Table 2.1.

Table 2.1: Time and location where induced thermal stress and strain reach peak

values obtained from Maximum Principal Stress Theory

Mach No. Size (µm) Exposure time (ms) Location

3 1.38 0.1

5 50 1.38 0.1

7 1.38 0.1

3 2.76 0.1

5 100 4.41 and 0.85 0.1 and 0.95

7 4.41 and 0.85 0.1 and 0.95

3 2 and 8.82 0.95 and 0.1

5 150 5.521 and 1.38 0.2 and 0.95

7 6.902 and 0.85 0.2 and 0.95

2.6 Summary

Coal particles subjected to a detonation wave experience highly stressed and strained

inner and outer regions. Three different regimes emerge in coal particle based on the dif-

ferent particle sizes when the coal particle is subjected to temperature shock. The largest

particles explode into smaller fragments as break up develops throughout the coal parti-

cle. The medium particles fragment in the outer region and left over surviving fraction

of same particles then fragment in the interior. The smallest particles fragment in the

interior. As the Mach number increases the entire process rapidly speeds up. This sug-

gests that coal particle under the effect of detonation wave is highly stressed and strained

and that detonation combustion of coal is qualitatively different from conventional coal

combustion. The theories studied in this chapter do not account for irregularities and

randomness in the properties of coal particles and a more realistic Weibull theory is used

later in Chapter 4.

Chapter 3

Model with Convective Boundary

Condition

3.1 Governing Equations

Unsteady heat conduction equation with spherical symmetry is [20]

1

r2

∂

∂r

(kr2∂T (r, t)

∂r

)= ρCp

∂T (r, t)

∂t(3.1)

And the boundary conditions are:

−k∂T∂r

∣∣∣∣r=ro

= h(T − T∞) (3.2)

where T∞ is the surrounding gas temperature induced by the detonation wave. And

∂T

∂r

∣∣∣∣r=0

= 0 (3.3)


T (r, 0) = Ti (3.4)

25

Chapter 3. Model with Convective Boundary Condition 26

Assuming constant thermal conductivity Eq. 3.1 simplifies to

∂2T

∂r2+

2

r

∂T

∂r=

1

α

∂T

∂t(3.5)

Introducing dimensionless variables

τ(R, θ) =T (r, t)− T∞Ti − T∞

, R =r

ro, θ =

αt

r2o

, Bi =hrok

(3.6)

Eq. 3.5 becomes

∂2τ(R, θ)

∂2R+

2

R

∂τ(R, θ)

∂R=∂τ(R, θ)

∂θ(3.7)

with boundary and initial conditions

∂τ

∂R

∣∣∣∣R=1

= −Biτ (3.8)

∂τ

∂R

∣∣∣∣R=0

= 0 (3.9)

And,

τ(R, 0) = 1 (3.10)

3.2 Solution by Laplace Transform

Taking Laplace transform of Eqs. 3.7 to 3.10

L[∂2τ(R, θ)

∂2R+

2

R

∂τ(R, θ)

∂R

]= L

[∂τ(R, θ)

∂θ

](3.11)

And the boundary and initial conditions become

L[∂τ

∂R

]= −L [Biτ ] ⇒ ∂τ

∂R= Biτ (3.12)


L[∂τ

∂R

∣∣∣∣R=0

]= 0⇒ dτ(R, s)

dR

∣∣∣∣R=0

= 0 (3.13)

and

L [τ(R, 0)] = 1⇒ τ(R, 0) = 1 (3.14)

From Eq. 3.11

d2τ

dR2+

2

R

dτ

dR= sτ(R, s)− τ(R, 0) (3.15)

This is the Bessel’s differential equation and its solution after simplification is [24, 25]

τ(R, s) =C2 sinh(

√sR)

R+

1

s(3.16)

Applying boundary condition of Eq. 3.12 and simplifying

C2 =−Bi

s(√s cosh

√s+ (Bi− 1) sinh

√s)

(3.17)

The complete transformed solution is

τ(R, s) =−Bi sinh(

√sR)

s(√s cosh


√s)

+1

s(3.18)

The final solution as a function of R and θ can be found by taking Inverse Laplace

Transform of Eq. 3.18.

τ(R, θ) = L−1 [τ(R, s)] = L−1

[−Bi sinh(

√sR)

s(√s cosh


√s)

]+ L−1

[1

s

](3.19)

The inverse of the above equation can be found by Bromwich’s Contour Integral method

[26]. The method can be given as

if,

∫ ∞0

f(t) = F (s) (3.20)


then,

f(t) =1

2πi

∮F (z)etz︸︷︷︸f(z)

dz = Sum of Residues (3.21)

And residues are given by,

R[f(z); zj] =1

(p− 1)!limz→zj

dp−1

dzp−1[(z − zj)f(z)] (3.22)

Using the Bromwich’s Contour Integral method and simplifying, the inverse of the Eq.

3.19 will give the final solution as below,

τ(R, θ) =T (r, t)− T∞Ti − T∞

=∞∑n=1

4(sinλm − λm cosλm)

Rλm(2λm − sin 2λm)sin(λmR)e−λ

2mθ (3.23)

where λm is the root of the eigen condition λ cotλ = 1−Bi

3.3 Temperature Obtained from Numerical Solution

u urr rr = 0 r = roi− 1 i i+ 1

r rr

Initial condition

Convective

Boundary

Condition

n = 0

n = n+ 1 h

h hh

n = n+ 2 r r

Figure 3.1: Discretization regime

A numerical code using finite volume procedure is developed to obtain temperature

profile. Eq. 3.5 is discretized as

T ni+1 − 2T ni + T ni−1

∆r2+

2

ri

T ni+1 − T ni∆r

=1

α

T n+1i − T ni

∆t(3.24)


Then, temperature at n+ 1 time level at ith node is

T n+1i =

α∆t

∆r2T ni−1 +

(1− 2α∆t

∆r2+

2α∆t

ri∆r

)T ni +

(α∆t

∆r2+

2α∆t

ri∆r

)T ni+1 (3.25)

Temperature at boundary points is given by

Tn+1i = Tni +

Al∆tα

∆v∆rTni−1 −

Al∆tα

∆v∆r+

h∆tArρ∆vCp︸︷︷︸convective

+∆tσbεbArT

ni

3

ρ∆vCp︸︷︷︸radiative

Tni +h∆tArT∞ρ∆vCp︸︷︷︸convective

+∆tσbεbArT

4∞

ρ∆vCp︸︷︷︸radiative

(3.26)

where Ar and Al are right and left surface area respectively and ∆v is difference between

right and left volume. Eq. 3.26 gives solution for combined convective and radiative

boundary condition which is studied later in Chapter 4. Convective effect alone can

be obtained after removing radiative terms from Eq. 3.26. Appendix B has the de-

tailed numerical code. Fig. 3.2 below gives comparison between obtained analytical and

numerical solution for simple case to validate the numerical solution.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1300

350

400

450

500

550

600

650

Dimensionless Radius

Te

mpe

ratu

re in

K

M=3,Numerical Without Radiation,time=1.38 ms

M=3,Numerical With Radiation,time=1.38 ms

M=3,Analytical Without Radiation,time=1.38 ms

(a) Comparison between solutions

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4


Err

or

in P

erc

en

tag

e

Deviation of Temperature Obtained by Numerical Solution

(b) Percentage error in numerical solution

Figure 3.2: Comparison between temperatures obtained from analytical and numerical

solutions for size=50 µm

It is interesting to study how much numerical solution deviates from analytical solution.

Fig. 3.2(b) gives deviation of numerical solution from analytical solution.


3.4 Calculation of Thermal Stresses Developed

It is necessary to calculate thermal stresses to understand fracture pattern of coal particle.

3.4.1 Radial Stresses

Radial stress [27] is given by

σr =2βE

(1− ν)

[1

r3o

∫ ro

0

Tr2dr − 1

r3

∫ r

0

Tr2dr

](3.27)


σr =4βE(Ti − T∞)

(1− ν)

∞∑m=1

4(sinλm − λm cosλm)e−λ2mθ

λm(2λm − sin 2λm)[sinλmλ2m

− cosλmλm

− 1

R3

(−R cosλmR

λm+

sinλmR

λ2m

)](3.28)

Comparison of radial stress is given in Fig. 3.3.

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25


σr i

n M

Pa

M=3,Numerical Without Radiation,t=1.38 ms

M=3,Numerical With Radiation,t=1.38 ms

M=3,Analytical Without Radiation,t=1.38 ms

Figure 3.3: Comparison between σr obtained from analytical and numerical solutions

at M = 3, β = 40× 10−6, size=50 µm


3.4.2 Tangential Stresses

Tangential stress [27] is given by

σt =βE

(1− ν)

[2

r3o

∫ ro

0

Tr2dr +1

r3

∫ r

0

Tr2dr − T]

(3.29)


σt =2βE(Ti − T∞)

(1− ν)

∞∑m=1

4(sinλm − λm cosλm)e−λ2mθ

λm(2λm − sin 2λm)[2 sinλmλ2m

− 2 cosλmλm

− sin(λmR)

R+

1

R3

(−R cos(λmR)

λm+

sin(λmR)

λ2m

)](3.30)

Comparison of tangential stress is given in Fig. 3.4.

0 0.2 0.4 0.6 0.8 1−25

−20

−15

−10

−5

0

5

10

15

20

25


σt i

n M

Pa

M=3,Numerical Without Radiation,t=1.38 ms

M=3,Numerical With Radiation,t=1.38 ms

M=3,Analytical Without Radiation,t=1.38 ms

Figure 3.4: Comparison between σt obtained from analytical and numerical solutions

at M = 3, β = 40× 10−6, size=50 µm


3.5 Summary

Radiation does not have significant effect on temperature for the detonation wave consid-

ered in this chapter. Numerical solution is validated with analytical solution for simple

cases. Numerical method is needed for solution of the more complex problem.

Chapter 4

General Model - Numerical Studies

Previous chapters studied analytical solutions which can be useful in predicting particle

failure in simpler way. Radiation and volatilization has to be incorporated in calculations

to make fragmentation prediction more accurate. Inclusion of radiation and volatilization

will require numerical techniques to obtain solution.

Earlier study of thermal shock by radiation heating was done by Hasselman [15].

It has been shown that the maximum thermal stresses which arise can be calculated,

to a good approximation, by considering the body to be heated by a constant heat

flux. Manson [29] reported that if a body originally at one uniform temperature is

suddenly immersed in a medium of different temperature, a condition of thermal shock

is introduced. It was suggested that in the case of rapid heating the surface stress is

compressive and surface failure may occur as a result of spalling, or as a result of shear

stress induced by the compression. It was reported that failure may first occur at center

and not at the surface.

Earlier study on coal particle fragmentation due to heating were carried out by Chirone

et al. [9]. It was concluded that the material divides into pieces along planes parallel to its

bedding planes. Senneca et al. [14] studied volatilization effect on fragmentation. Semi-

detailed fragmentation model of coal particle under different heating rates was given.

Study of effusion, diffusion and viscous flow of gases was done by Mason et al. [19]. Coal

33

Chapter 4. General Model - Numerical Studies 34

devolatilization and hydrogasification of the volatile matter present in coal was studied

by Anthony et al. [30]. Coal pyrolisis was studied by Gavalas et al. [31]. Stanmore et

al. [32] suggested that thermal stress appears to be the major effect in breakup of small

sized coal particles compared to Volatile Matter (VM) effect.

Few literature are available which uses Weibull’s statistical theory to predict material

fracture. Theory of thermal shock resistance of brittle materials based on Weibull’s

statistical theory was studied by Manson et al. [17]. Weibull parameters of fractured

ceramics hip joint were calculated numerically using Finite Element Method (FEM)

by Fuis et al. [33]. Maximum principal stress is considered in calculation of failure

probability. The analysis of the calculated material parameters show that 2-parameter

Weibull statistical theory can be used as a substitute to 3 parameters theory.

Various literature are available which reported theoretical and experimental investi-

gation of ignition of coal dust behind shock wave. Sichel et al. [34] presented detailed

analytical model, along with experimental results, of the fragmentation of substances

like Coal, Graphite, Diamond, Oats and Research Department eXplosive (RDX) behind

shock wave. Elkotb et al. [35] experimentally studied ignition of organic dust of the size

40 to 80 µm behind shock wave and calculated ignition delay time. Asymptotic analysis

of the shock wave ignition of dust particles were studied by Baek et al. [36]. Ignition

and detonation of coal-particle gas mixtures were theoretically investigated by Fedorov

et al. [37].

This chapter is concerned with deriving various numerical modules required to study

thermal and volatilization induced primary fragmentation of single coal particle. Radi-

ation is included as boundary condition and its effect on temperature and stresses for

lower Mach number is already presented in previous chapter. Volatilization is included

to make prediction accurate. Irregularities and randomness inside coal particle which

were not accounted in the theories presented in previous chapters have been included

through Weibull parameters of WLT presented in this chapter.


4.1 Governing Equations and Numerical Modules

Thermal transport to the coal particle consists of heat transfer from the surrounding

environment to the surface and transport inside the solid media. Heat transferred to the

surface is typically modeled as a convection and since high temperature plasma is used

radiation also becomes important. The transport inside is usually considered as Fourier

conduction.

4.1.1 Heat Transfer Model

Governing heat transfer equation is

1

r2

∂

∂r

(kr2∂T (r, t)

∂r

)= ρCp

∂T (r, t)

∂t(4.1)

and boundary conditions are:

−k∂T∂r

∣∣∣∣r=ro

= h(T − T∞) + σbεb(T4 − T 4

∞) (4.2)

∂T

∂r

∣∣∣∣r=0

= 0 (4.3)


T (r, 0) = Ti (4.4)

Energy conservation equation for shock is given by [38]

hd − hu = −q + Cp(Td − Tu) (4.5)

After simplification it comes

pdpu

=

2qpuvu

+ γ+1γ−1− vd

vu(γ+1γ−1

)vdvu− 1

(4.6)


For non reacting gas dynamic q = 0 is used which is equation for ordinary shock.

Detonation is the case when q 6= 0 where q can be any kind of energy namely combustion

reaction, plasma energy, nuclear etc. In plasma initiated detonation energy q is supplied

through plasma. This is the difference between plasma initiated detonation and other

detonation. For simplicity the effect of energy supplied is included in to Mach number

and detonation is simply represented by a fixed ambient temperature. Rankine Hugoniot

relations are used to calculate conditions behind shock wave

T∞Ti

=[2γM2 − (γ − 1)][(γ − 1)M2 + 2]

(γ + 1)2M2(4.7)

p∞pi

=2γM2 − (γ − 1)

(γ + 1)(4.8)

4.1.2 Volatilization Model

Equations governing volatilization are [14, 19,31]

∂V

∂t= k0 exp

(−EaRuT

)(V ∗ − V )n (4.9)

∂

∂r(Nmolr

2) = r2 ρcMvol

∂V

∂t(4.10)

Where

Nmol = −r2porepε

8µτtRuT

∂p

∂r(4.11)

Boundary conditions for volatilization are:

p|r=r0 = p∞ (4.12)


and

∂p

∂r

∣∣∣∣r=0

= 0 (4.13)

And initial condition for volatilization is:

p|t=0 = p∞ = pi (4.14)

Mezhericher et al. [39] reported that if the length of capillary pore is very short and

the diffusion movement of gas molecules is quick, the spatial change in the temperature

withing pore is neglected. This means the temperature of flowing substance in pore

is assumed to be close to the temperature of solid-substance interface. Particle sizes

considered in this work are of the order of µm and hence length of pore is very short.

Because of high temperature, movement of volatile molecules will be quick. Considering

this fact, the temperature of volatile in the pore is assumed to be close to the temperature

of coal particle.

4.1.3 Solid Mechanics Model

Radial and Tangential stresses are given by [14,27]

σr =2βE

(1− ν)

[1

r3o

∫ ro

0

(T (r, t)− Ti)r2dr − 1

r3

∫ r

0

(T (r, t)− Ti)r2dr

](4.15)

σt =βE

(1− ν)

[2

r3o

∫ ro

0

(T (r, t)− Ti)r2dr +1

r3

∫ r

0

(T (r, t)− Ti)r2dr − (T (r, t)− Ti)]

(4.16)

Total radial stress is given by

σr,tot = σr + p− p∞ (4.17)


4.1.4 Developed Stress Model

The principal stresses over spherical particle in terms of total radial stress (σr,tot) and

tangential stress (σt) is obtained by solving for eigen values of the stress tensor matrix

(σ) given as [40]

σ =

σrr σrθ σrφ

σθr σθθ σθφ

σφr σφθ σφφ

(4.18)

After simplifying, principal stresses are given as below

σ1 = σ0 + 2√|J2/3| cos(θ) (4.19)

σ2 = σ0 − 2√|J2/3| cos

(θ +

π

3

)(4.20)

σ3 = σ0 − 2√|J2/3| cos

(θ − π

3

)(4.21)

where θ is Lode angle and is defined as

θ =1

3cos−1

(− J3

2(|J2/3|)3/2

)(4.22)

where J2 and J3 are the invariants of stress deviator

J2 = SrSθ + SθSφ + SφSr − τ 2rθ − τ 2

θφ − τ 2φr (4.23)

J3 = −(SrSθSφ − Srτ 2θφ − Sθτ 2

φr − Sφτ 2rθ) (4.24)

in which

Sr = σr − σ0, Sθ = σθ − σ0, Sφ = σφ − σ0, (4.25)

σ0 =(σr + σθ + σφ)

3(4.26)


4.1.5 Weibull’s Weak Link Theory and Fracture Criteria Model

Several theories have been applied to understand the fracture of brittle material in the

literature [10,15]. Coal is considered as highly brittle and hence would exhibit a scatter

in its fracture strength which is the result of intrinsic distribution of microscopic flaws.

The brittle characteristic combined with large scatter in fracture strength makes a prob-

abilistic approach relatively more appropriate. Weibull fracture model, also called WLT

which has been widely used for brittle materials [16], has been proven to be more valid

over the critical stress theory [18] for coal and is being used here.

According to WLT sequence of events or objects depend on the support of the whole.

The whole is only as reliable as the weakest member or link. The basic assumption for

the model is that all materials contain inhomogeneities which are distributed at random.

Examples of such inhomogeneities are flaws, cracks etc. When the defects become the

fracture origin, it is found that failure is triggered by the largest defect present or, in

other words, weakest element present. It is assumed that size of flaws is small compared

to distance between them. Failure of entire body is defined as first failure of any element.

It is not possible to indicate an exact value of the breaking load, but it is possible to

indicate a definite probability of the rupture occurring at a given stress or at given time.

As per the WLT the survival probability of unit volume for the case of magnitude

varying but multiaxial stress field can be given as

Ps =3∏j=1

e

[−∫V

(σj−σuσo

)mdVVo

], if σj ≥ σu

0, otherwise

(4.27)

Where σj refers to principal stresses in jth direction, σu is the threshold stress below

which no failure occurs, σo is the characteristic or unit volume fracture strength, which

is the stress level at which 63% of the specimens fail, m is the Weibull modulus or shape

parameter, V is the volume of the specimen (here coal particle) and Vo is the reference


volume. Weibull parameters σu, σo and m are usually to be obtained from experimental

data. For computational purpose Eq. 4.27 is used. Principal stresses are of three

types. Maximum principal stress σ1 is tensile in nature, minimum principal stress σ3 is

compressive in nature and intermediate stress σ2 is in between the other two (see Fig.

4.11). Literature suggests that compressive strength of coal is twenty five times [3] that

of tensile strength. This means coal is weaker in tension. Intermediate stress fluctuates

from compressive to tensile (see Fig. 4.11(b)). Minimum principal stress remains always

negative (see Fig. 4.11(c)) and can be used when coal is under compressive loading.

In this work simpler model [33] with maximum principal stress given by Eq. 4.28 is

considered for calculation after knowing the facts that, coal is weaker in tensile and basis

of the Weibull’s weakest link theory is in the identification of the weakest element in the

chain.

Ps =

e[−

∫V (σ1−σuσo

)m dVVo

], if σ1 ≥ σu

0, otherwise

(4.28)

For two parameter Weibull failure probability σu = 0. The objective here is to include

a Weibull Probability of Failure (POF) estimation in the finite volume solution process.

In the conventional discretized space formulation, recovery of stresses is done at the

points where heat flow variables are computed. POF is computed at the same points

of stress recovery as such positions contain all the necessary information. Eq. 4.28 for

probability of survival of individual volumes Vi for a unit reference volume (Vo = 1)

becomes

Ps,i =

e

[−∫Vi

(σ1,i−σu

σo

)mdV], if σ1,i ≥ σu

0, otherwise

(4.29)


and the overall probability of survival becomes

Ps = Ps,1.Ps,2...Ps,n (4.30)

Ps = e−∑ni (

σ1−σuσo

)m

∆Vi , σi ≥ 0 (4.31)

Failure probability Pf (= 1− Ps)is then calculated as

Pf = 1− e−∑ni (

σ1−σuσo

)m

∆Vi , σi ≥ 0 (4.32)

where n is the number of cells into which the analysed particle is divided for applying

the numerical method, with Vi being the volume of the ith cell. The location of failure is

spotted by tracing the weakest link (as per Weibull’s theory) in terms of local survival (or

failure) probability as given in Eq. 4.29. The location of largest jump in local probability

(∆Pf,i = P nf,i−P n−1

f,i , subscript n is the same time level) causing a local failure and hence

the overall failure of the particle is taken as failure location Rf . If overall probability

reaches the failure probability (Pf,b = 0.999), the particle is considered to have failed.

Rf = Ri for ∆Pf,i = maxi∆Pf,i and P nf , P

nf,i ≥ Pf,b (4.33)

4.2 Fragmentation due to Thermal Effect

The governing equation (Eq. 4.1) is numerically solved for temperature using a finite vol-

ume time stepping method. The material properties for the analyses are those for either

anthracite, bituminous or lignite. Values for the thermal diffusivity, thermal conductiv-

ity, and specific heat of the coal are taken from reference [21] and given in Appendix

A.


4.2.1 Temperature

Temperature on a 50 µm coal particle exposed to M = 7 wave is presented in Fig.

4.1. Temperature rise in time is very gradual except for sudden surge on the surface

temperature at the moment of exposure to hot media. It is generally observed that

temperature gradient on the particle and thermal steady state time depends on the

particle size than the hot side temperature. Time taken for core heating is shorter for

smaller particles and at higher Mach numbers.

0

1000

2000

3000

0 0.2

0.4 0.6

0.8 1

0

1000

2000

3000

T

t

r/r0

T

Figure 4.1: Evolution of temperature in time (in µs) for 50 µm size coal particle at

M = 7

4.2.2 Stresses

Radial stress is unidirectional and all along tensile with the maxima at the center of

particle, while reducing towards the surface. The radial stress gradient is low near the

center and reaches a higher value close to the surface. Radial stresses dominate in the

region where tangential stress crossover occurs. Tangential stress is negative (compres-

sive) close to the surface with its maximum magnitude on the surface. It undergoes sign

change with positive maxima at the core making a tensile interior. With longer particle


exposure to heat, the compressive zone which is limited to surface region penetrates in-

side. If the fissure mechanism is stress reversal, the location of sign change in tangential

stress could give a clue to the fracture location. Stress magnitude of radial and tangen-

tial components is almost the same from the core up to the location of sign reversal in

tangential stress. Beyond this point radial stress reduces to zero while tangential stress

changes direction and increases to large negative value on the surface. The maximum

principal stress follows the trend of tangential stress magnitude, with a dip near location

of sign reversal.

0

1000

2000

3000

0 0.2

0.4 0.6

0.8 1

0

50

100

150

σr

tr/r0

σr

(a) σr (in MPa)

0

1000

2000

3000

0 0.2

0.4 0.6

0.8 1

-200

0

200

σt

t

r/r0

σt

(b) σt (in MPa)

0

1000

2000

3000

0 0.2

0.4 0.6

0.8 1

0

100

200

300

σ1

tr/r0

σ1

(c) σ1 (in MPa)

Figure 4.2: Evolution of stresses in time (in µs) for 50 µm size coal particle at

M = 7, β = 40× 10−6


4.2.3 Probability

Coal particles on exposure to high energy detonation wave heat up and the induced

thermal stresses eventually cause failure of particles. The failure of coal particles under

a given set of physical conditions are expressed in terms of their survival probability as

described by Weibull. Weibull characteristic strength is varied in a range less than 14

MPa which is a typical ultimate tensile strength of anthracite [41] while the Weibull

threshold stress is varied in a finite but lower range. Weibull modulus is set between 3 to

6 as modelled by Chirone et al. [9], except when its variation effect is to be specifically

studied.

0

10

20

30

40

50 0 0.2

0.4 0.6

0.8 1

200

400

600

800

T

t

r/r0

T

(a) T (in K)

0

10

20

30

40

50

0 0.2

0.4 0.6

0.8 1

0

2

4

6

σr

t

r/r0

σr

(b) σr (in MPa)

0

10

20

30

40

50

0 0.2

0.4 0.6

0.8 1

-60

-40

-20

0

20

σt

t

r/r0

σt

(c) σt (in MPa)

0

10

20

30

40

50 0 0.2

0.4 0.6

0.8 1

0

20

40

60

σ1

t

r/r0

σ1

(d) σ1 (in MPa)

0

10

20

30

40

50 0 0.2

0.4 0.6

0.8 1

0

0.2

0.4

0.6

0.8

1

Pf

t

r/r0

Pf

(e) Failure probability

Figure 4.3: Evolution of variables in time (in µs) for 50 µm size coal particle till

fracture at 42 µs for M = 5, β = 40× 10−6, σo = 12, σu = 10, m = 6

Fig. 4.3 presents a typical case of particle being heated till fracture when survival

probability reaches zero. It can be seen that for a typical M = 5 detonation wave flow

across 50 µm particle, fracture is predicted in 42 µs with the set Weibull parameters. In

this time, despite the temperature rise near the surface only, the temperature gradients

induced a sufficient stress for a failure at location about R = 0.9, indicating the role of


high strength wave in causing surface fragmentation (exfoliation) in few microseconds.

The fragmented particle whose dimension vary typically from one-tenth to one-fifth of

the initial particle, (say, 5 µm to 20 µm) will undergo fragmentation like a preheated

particle. Fracture behaviour of such small particle (unheated) is presented in Fig. 4.4

which takes about 10 µs for fragmentation. Unlike the larger particles which exhibited

exfoliation tendency, the smaller particles fragment in the interior. If such tiny particles

are formed due to fragmentation of bigger particles, preheating will result in much faster

fragmentation.

0 5

10 15

20 25

30 35

40 45

0 0.2

0.4 0.6

0.8 1

200

400

600

T

t

r/r0

T

(a) T (in K)

0 5 10

15 20

25 30

35 40

45

0 0.2

0.4 0.6

0.8 1

0 2 4 6 8

10 12 14 16

σr

t

r/r0

σr

(b) σr (in MPa)

0 5 10

15 20

25 30

35 40

45

0 0.2

0.4 0.6

0.8 1

-40

-20

0

20

σt

t

r/r0

σt

(c) σt (in MPa)

0 5

10 15

20 25

30 35

40 45

0 0.2

0.4 0.6

0.8 1

0

20

40

60

σ1

t

r/r0

σ1

(d) σ1 (in MPa)

0 5

10 15

20 25

30 35

40 45

0 0.2

0.4 0.6

0.8 1

0

0.2

0.4

0.6

0.8

1

Pf

t

r/r0

Pf

(e) Failure probability

Figure 4.4: Evolution of variables in time (in µs) for 10 µm size coal particle till

fracture at 10 µs for M = 4, β = 40× 10−6, σo = 12, σu = 4, m = 4

Particle failure being predicted by Weibull theory, time and location of failure are

reflected by the Weibull parameters apart from dependence on operating and material

variables. A study on the effect of such variables/parameters on the spontaneous frag-

mentation of coal due to thermal stresses is presented here in Fig. 4.5, in order to

investigate the detonatability of coal.


4.2.4 Fragmentation Analysis

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70

Failu

re P

robabili

ty

Time µsecs

d=50 ,M=4,σu=3d=50 ,M=6,σu=3d=50 ,M=6,σu=6

(**)d=50 ,M=6,σu=6d=150,M=4,σu=3d=150,M=6,σu=3d=250,M=4,σu=3d=250,M=6,σu=3

(a) Failure probability

0

10

20

30

40

50

60

70

80

90

3.5 4 4.5 5 5.5 6 6.5 7

Fra

ctu

re T

ime(µ

secs)

Mach Number

d=250,σu=3d=250,σu=6d=150,σu=3d=150,σu=6d=50 ,σu=3d=50 ,σu=6

(b) Fracture time

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

3.5 4 4.5 5 5.5 6 6.5 7

Fra

ctu

re location(R

f)

Mach Number


(c) Fracture location

Figure 4.5: Failure characteristics of coal particles of different sizes at different Mach

number when β = 40× 10−6, σo = 9, σu = 6, m = 6

Failure behaviour of coal particle of different size and elastic properties subjected to

various strength of detonation waves upto 100 µs time are presented in Fig. 4.5 and

4.19. Time evolution of failure probability of coal particles on exposure to high energy

detonation wave are presented in Fig. 4.5(a) and 4.19(a). The probability evolution may

be divided into three phases. As observed in Eq. 4.32, probability is directly related to

the induced stresses. The three phases should then correspond to induced stress, atleast

in some particle zones, (a) breaking over threshold stress σu, (b) increasing upto a failure

stress σo and beyond (c) causing expansion of stressed front till a failure happened. The

trend is identical for any coal particle exposed to thermal wave of any strength although

the stretch of the three phases vary in their degree. Fracture time plots Fig. 4.5(b) and

4.19(b) indicate the dependence of spontaneity of heat induced coal fragmentation (in

few µs) on the interplay of hot side temperature (detonation wave Mach number) and

particle size, apart from material properties factored as Weibull parameters. Incomplete

results near low Mach number indicate particles remaining unfragmented till 100 µs.

Location of fracture for particles exposed to waves of different strengths and under mul-

tiple conditions are plotted in Fig. 4.5(c) and 4.19(c). In general, impact of moderate

strength waves (M < 5) on small sized particle (typically < 25µm) caused an interior


fragmentation between R = 0.4 to 0.7 while exfoliation happened on relatively larger

particles and for high strength waves (M > 5), in either case the time for failure being

less than 20 to 30 µs. Detonation waves of lower strength (M < 5) failed to fragment

coarse particles (size 50 µm) even in 100 µs, while smaller particles still fragmented in

less than 30 µs at interior locations. For M < 4, 5 µm and 25 µm particles remain

unfragmented till 100 µs while in the same period 15 µm fractured in less than 10 µs.

300

400

500

600

700

800

900

1000

1100

1200

0 0.2 0.4 0.6 0.8 1

Te

mp

era

ture

(K

)

Radius (r/r0)

Time

(a) 5 µm, not fractured upto 100 µs

300

350

400

450

500

550

0 0.2 0.4 0.6 0.8 1

Radius (r/r0)

Time

(b) 10 µm, fractured at 44.9 µs and R = 0.59

300

350

400

450

500

550

600

0 0.2 0.4 0.6 0.8 1

Radius (r/r0)

Time

(c) 25 µm, fractured at 81.5 µs and R = 0.67

300

350

400

450

500

550

600

0 0.2 0.4 0.6 0.8 1

Radius (r/r0)

Time

(d) 50 µm, not fractured upto 100 µs

Figure 4.6: Variation of temperature (in K) on different size coal particles when

M = 4, β = 40× 10−6, σo = 12, σu = 4, m = 4

Temperature evolution plots given in Fig. 4.6 should be observed to understand

this behaviour. In case of smaller unfragmented particle (size=5 µm), the temperature

gradients are very low due to faster heat penetration while in larger particle (size=50


µm) gradient is restricted to the surface. The well spread gradient aided the failure of

sizes 10, 15 µm particles indicating the role of thermal gradient pattern in causing the

failure. However, if the volatility effects are considered both the temperature as well as

its gradient might determine the local stress and hence the particle fragmentation.

4.3 Fragmentation due to Volatilization

The governing equation (Eq. 4.1) is numerically solved for temperature using a finite

volume time stepping method. Stress field Eq. 4.15 and 4.16 are solved using Simpson’s

formula and principal stresses were obtained through eigenvalues of stress tensor obtained

from Eq. 4.18. Eq. 4.9 is solved using forward time difference formulation and Eq.

4.10 using forward difference formulation. Eq. 4.11 is solved using Euler’s method and

pressure obtained from it is added in radial stress (Eq. 4.15) to obtain total radial stress

given in Eq. 4.17. Detailed numerical code used to solve these equations is given in

Appendix B. The material properties for the analysis are those for either anthracite,

bituminous or lignite. Values of various parameters used are given in Appendix A.

4.3.1 Temperature

Temperature profile of 50 µm size coal particle when subjected to detonation wave of

strength M = 6 is shown in Fig. 4.7.


0

1

2

x 10−3

0

0.5

10

500

1000

1500

2000

t in sR

T in K

400 600 800 1000 1200 1400 1600

(a) T Profile at 1.38 ms

0

0.01

0.02

0

0.5

10

1000

2000

3000

t in sR

T in K

500 1000 1500 2000

(b) T Profile at 20 ms

Figure 4.7: Temperature plots for M = 6, size = 50µm, β = 40× 10−6, V ∗ = 0.3,

σo = 12, σu = 6, m = 6

4.3.2 Volatilization

Volatile matter generation, volatile matter flow and pressure generated due to volatiliza-

tion are shown in figures from 4.8(a) to 4.8(c) respectively. It is observed from Fig.

4.8(a) and 4.8(b) that volatile matter generation and flow of volatilie are higher on sur-

face region. This is because volatile matter generation is governed by first order reaction

and directly depends on temperature. Temperature is comparatively higher on surface

region because heat starts transferring from surroundings to particle via convection and

radiation from outer region. It is observed from Figs. 4.8(c) and 4.9(c) that pressure

generated due to volatilization is higher in inner region.


00.5

1

x 10−3

0

0.5

10

2

4

6

8

x 10−3

t in sR

V in k

g/k

g o

f coal

0 2 4 6 8

x 10−3

(a) V

0

0.5

1

x 10−3

0

0.5

10

1

2

3

x 10−3

t in sR

Nm

ol i

n k

mol/m

2s

0 0.5 1 1.5 2 2.5 3

x 10−3

(b) Nmol

0

0.5

1

x 10−3

0

0.5

1

2.457

2.458

2.459

x 107

t in sR

p Insid

e C

oal P

art

icle

in P

a

2.4565 2.457 2.4575 2.458 2.4585 2.459 2.4595

x 107

(c) p inside coal particle

Figure 4.8: Volatile matter variables at time 1.38 ms, M = 6, size = 50µm,

β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6

Various variables related to volatilization for longer time scales are shown in Fig. 4.9.


0

0.01

0.02

0

0.5

10

0.1

0.2

t in sR

V in

kg

/kg

of

co

al

0 0.05 0.1 0.15 0.2 0.25

(a) V

0

0.01

0.02

0

0.5

10

0.005

0.01

t in sR

Nm

ol i

n k

mo

l/m

2s

2 4 6 8 10

x 10−3

(b) Nmol

00.005

0.010.015

0.02

0

0.5

1

2.46

2.48

2.5

2.52

x 107

t in sR

p Insid

e C

oal P

art

icle

in P

a

2.46 2.47 2.48 2.49 2.5 2.51 2.52

x 107

(c) p inside coal particle

Figure 4.9: Volatile matter variables at longer time scale for time 20 ms, M = 6,

size = 50µm, β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6

4.3.3 Stresses

Total radial stress, tangential stress profiles are shown in Fig. 4.10. There will not be

difference in tangential stress because the volatile pressure is added in to radial stress

only as given by Eq. 4.17.


0

1

2

x 10−3

0

0.5

10

50

100

t in sR

σr,

tot i

n M

Pa

0 20 40 60 80

(a) Radial stress σr,tot

0

1

2

x 10−3

0

0.5

1−200

−100

0

100

t in sR

σt i

n M

Pa

−100 −50 0 50

(b) Tangential stress σt

Figure 4.10: Stresses at time 1.38 ms, M = 6, size = 50µm, β = 40× 10−6,

V ∗ = 0.3, σo = 12, σu = 6, m = 6

0

1

2

x 10−3

0

0.5

10

100

200

300

t in sR

σ1 in M

Pa

0 50 100 150 200 250

(a) Principal stress σ1

0

1

2

x 10−3

0

0.5

1−100

−50

0

50

100

t in sR

σ2 in M

Pa

−60 −40 −20 0 20 40 60 80

(b) Principal stress σ2

0

1

2

x 10−3

0

0.5

1−300

−200

−100

0

t in sR

σ3 in M

Pa

−250 −200 −150 −100 −50 0

(c) Principal stress σ3

Figure 4.11: Principal stresses at time 1.38 ms, M = 6, size = 50µm, β = 40× 10−6,

V ∗ = 0.3, σo = 12, σu = 6, m = 6


Profile of the principal stresses is shown in Fig. 4.11. Principal stresses will differ

from the principal stress obtained from thermal effect alone because total radial stress

is used in stress tensor matrix to calculate principal stresses as obtained from Eq. 4.18.

However overall characteristics of the plots remained same. A fissure similar to σ1 profile

due to thermal effect only (as presented in Fig. 4.2(c)) is observed in Fig. 4.11(a). σ3

remained entirely in negative region i.e. compressive in nature.

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

Dimesionless Radius

Radia

l S

tress in M

Pa

Without Volatilization

V*=0.3

(a) With and without volatilization

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

Dimesionless Radius

σr,

tot in

MP

a

V*=0.3

V*=0.6

(b) With different volatile content

(c) Effect of volatilization on average stress at 2 ms

00.005

0.010.015

0

0.5

10

100

200

t in sR

Perc

enta

ge E

ffect of V

ola

tiliz

ation

0 20 40 60 80 100

(d) Effect of volatilization on average stress at 15 ms

Figure 4.12: Comparison of σr,tot with relation to volatile matter content at 634.8 µs,

M = 6, size 50 µm

Effect of different degree of volatile matter content on radial stress, for the time which

is well below the conventional volatilization time of the order of ms [42], is given in Fig.

4.12. A plot highlighting pressure component of radial stress presented in Figs. 4.12(c)

and 4.12(d) depicted development of volatiles pressure only beyond t ≥ 5µs. Pressure

stress is seen at best only to play a supplementary role and hardly dominated to cause a


fracture, in the time scale of interest (about 3-4 ms, time for shock to travel end-to-end

of the detonator tube). It is observed from Figs. 4.12(c) and 4.12(d) that volatilization

stress is higher in core region and its effect is significant for longer time scale rather than

smaller time scale. This is because pressure generated due to volatilization is higher in

inner region (refer Figs. 4.8(c) and 4.9(c)) and it is added in to radial stress calculation

as given in Eq. 4.17. Generated volatile does not have passage to escape in inner region

hence pressure builds there. From Fig. 4.12(b) it can be seen that there is not much

difference in radial stress with different initial volatile matter content.

4.3.4 Probability

Local failure probability profile is given in Fig. 4.13 for 50 µm size particle which

fragmented at R = 0.99 and time 18.38 µs. It is seen from this figure that in early phase

of heating failure probability is lesser in inner region compared to outer region.

0

0.5

1

1.5

x 10−3

0

0.5

10

0.5

1

t in sR

Local P

f

0 0.2 0.4 0.6 0.8 1

Figure 4.13: Local failure probability for M = 6, size = 50µm, β = 40× 10−6,

V ∗ = 0.3, σo = 12, σu = 6, m = 6


It is interesting to study the effect of volatilization on failure probability at inner region

because σr,tot is higher there (refer Fig. 4.12(a)). Fig. 4.14 shows comparison between

local failure probability with relation to volatile matter content at R = 0.09. Comparison

of failure probability with and without volatilization is presented in Fig. 4.14(a). There

is no change in failure probability in the core region (R = 0.09) where stress due to

volatilization is higher. Failure probability remains same for different volatile matter

content as shown in Fig. 4.14(b).

0 1 2 3 4

x 10−4

0

0.2

0.4

0.6

0.8

1

Time in s

Pf,l

Local Failure Probability

Without Volatilization

V*=0.3

(a) Pf,l Without Volatilization

0 1 2 3 4

x 10−4

0

0.2

0.4

0.6

0.8

1

Time in s

Pf,l

Local Failure Probability

V*=0.3

V*=0.6

(b) Pf,l With Different Volatilization

Figure 4.14: Comparison of failure probability for different initial volatile matter

content at M = 6, size = 50µm, β = 40× 10−6, σo = 12, σu = 6, m = 6, R = 0.09

4.3.5 Fragmentation Analysis

0

0.5

1

1.5

x 10−3

00.2

0.40.6

0.81

−5

0

5

10

15

x 107

Rt in s

First D

erivative o

f T

em

pera

ture

in K

/m

0 2 4 6 8 10

x 107

(a) dTdr

0

0.5

1

1.5

x 10−3

00.2

0.40.6

0.81

−2

−1

0

1

2

3

x 107

Rt in s

First D

erivative o

f σ

1 in M

Pa/m

−1 −0.5 0 0.5 1 1.5 2

x 107

(b) dσ1dr

Figure 4.15: Temperature and stress gradients at time 1.38 ms, M = 6, size = 50µm,

β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6


Fig. 4.15 presents temperature and stress gradient plot for 50 µm size coal particle which

fragmented at 18.38 µs and R = 0.99. It is observed from Fig. 4.15(a) that temperature

gradient is higher at surface region in the initial phase of heating. Similarly, it is observed

from Fig. 4.15(b) that stress gradient is also higher at surface region in initial stage of

heating. It can be concluded that combination of higher temperature and stress gradient

along with higher failure probability (refer Fig. 4.13) in early phase of heating may have

caused earlier fracture at surface region.

For demonstration purpose fracture results for 50 µm size coal particle including

volatilization is given in Table 4.1

Table 4.1: Fracture results for 50 µm size coal particle at different Mach numbers at

β = 40× 10−6, V ∗ = 0.3, σo = 12, σu = 6, m = 6

M Detonation

Temperature in

K

tf in s Tf in K Rf

3 0827.07 8.050828e-04 320.03 0.33

4 1256.88 9.078560e-05 496.37 0.99

5 1807.77 3.513020e-05 483.54 0.99

6 2480.42 1.838160e-05 480.29 0.99

7 3275.07 1.097560e-05 480.17 0.99

8 4191.81 6.964400e-06 481.71 0.99


4.4 Weibull Parameters

0

10

20

30

40

50

60

70

80

90

2 3 4 5 6 7 8 9 10

Fra

ctu

re T

ime

µse

cs

Weibull Modulus

σu=3,σ0= 9σu=6,σ0= 9σu=4,σ0=12σu=8,σ0=12σu=6,σ0=12

(a) M = 5, size=50 µm, Ti = 300 K

10

20

30

40

50

60

70

80

90

2 3 4 5 6 7 8 9 10

Fra

ctu

re T

ime

µse

cs

Weibull Modulus

σu=3,σ0= 9σu=6,σ0= 9σu=4,σ0=12σu=8,σ0=12σu=6,σ0=12

(b) M = 4, size=150 µm, Ti = 500 K

Figure 4.16: Effect of Weibull parameters on coal fracture time at different m

The primary fragmentation of coal due to thermal stress when subjected to high strength

detonation wave is studied in terms of the three Weibull parameters which modelled the

fragmentation.

1. Weibull characteristic strength σo is the stress required to break unit volume of the

material and also is an estimation of available stress for a failure above the threshold

stress. It is the stress bearing capability of material and for a tough material it may

be as high as the ultimate tensile strength. The Weibull characteristic strength σo

of a good quality coal should ideally be the ultimate tensile strength (14.0 MPa is

typical for anthracite [41]). However, the wide gradations in coal and inhomogenities

present will render σo to take values quite below the tensile strength.

2. Weibull threshold stress σu being the stress below which fracture does not happen,

will assume value below the characteristic strength (σo) and naturally far lower to

the ultimate tensile strength.

3. Weibull modulus (m) is an estimation of the homogeneity in the behaviour of the

material. With higher m the homogeneity decreases and the failure more closely


correspond to σu. For σu < σo increasing m predicts an earlier particle failure with

pronounced effects for m < 6 as seen in Fig. 4.16.

4. Failure time is directly related to σo. A low σo implies lesser strength material and

hence smaller failure time. Higher values of σo is that of a tougher material requiring

more stress and time to break the particle. The increase of σu at constant σo (lesser

(σo − σu)) implies lesser initial defects or crevices in material, thus requiring more

stress for breakage and hence particle failure is delayed (see Fig. 4.16).

5. In short, the fracture time increases with the larger size of particles, lower Mach

numbers, higher values of weibull stresses namely, the failure stress (σo), threshold

stress (σu) and lower Weibull modulus(m).

6. It can be seen in Fig. 4.16(a) that with 4 ≤ m ≤ 9, 50 µm particles with char-

acteristic strength (σo) upto 9 or 10 MPa is likely to fragment within 30 µs when

subjected to detonation wave at M = 5, irrespective of Weibull threshold stress σu.

Hence, a typical sample which exhibits m (= 4 to 6) and σo (< 10MPa) with least

value of σu (< σo) is the most sought choice for detonatable coal. However, particles

larger than 50 µm may take a longer time. In order to operate at low Mach number

less than 5 but still have failure time less than 40 µs for even coarser particles 100

or 150 µm, particles preheating above 500 K will be useful.

7. In order to realize spontaneous fragmentation which can potentially cause detona-

tion combustion the following criteria may be conducive

• for small sized particles (< 50 µm) any type of coal will restrict fragmentation

time < 30 µs even at M = 3.

• for large sized particle 100 to 250 µm a preheated particle upto 500 K is likely

to retain a low fragmentation time.


However, the characteristic strength preferred will be < 10 MPa, which can typi-

cally range from lignite to bituminous and low grade anthracite.

8. Since detonation combustion could be a reality only if fracture time is upto a few

tens of micro seconds (say 30 µs), to meet the requirements of achieving this at

low or moderate Mach numbers (say upto 4) and also to meet the need of breaking

from low to moderate size particle (upto 200 µm), the simulation results indicate

that among the coals, the one which will typically exhibit characteristic strength

σo < 10 MPa is a choice, with coal preheating before exposure to detonation wave

a catalytic choice to treat even large and tough particles at even M = 4.

9. Among the different types of coal, anthracite is richest in carbon and known to be

toughest and hence σo should be the maximum. Bituminous coal, richest in volatile

matter has a moderate tensile strength while lignite is the softest and least tensile.

Most varieties of bituminous and lignite which may exhibit characteristic strength

σo < 10 MPa may be a preferred choice for realizing spontaneous fragmentation

leading to detonation combustion of coal. However, preheating may enable any coal

ranging from lignite to a low grade anthracite except perhaps high strength an-

thracite, to become a potential candidate for detonation combustion of coal. While

effectiveness of primary fragmentation is a criteria for coal’s choice, high calorific

value, low ash are conditions always sought after, which then will make the bitu-

minuous coal as a preferred choice for detonation combustion of coal.


4.5 Sensitivity Analysis

4.5.1 Coal Properties

0

20

40

60

80

100

120

140

2 2.5 3 3.5 4 4.5 5 5.5 6

Fra

ctu

re T

ime

(µ

se

cs)

Co-efficient of Thermal Expansion(βx105)

E=2GPa,d=050µmE=2GPa,d=100µmE=4GPa,d=050µmE=4GPa,d=100µmE=6GPa,d=050µmE=6GPa,d=100µm

Figure 4.17: Effect of CTE and Young’s Modulus at M = 4

Physical properties such as thermal conductivity, density, and specific heat capacity are

known to influence the heat transfer and hence thermal stresses on the particle. For most

of our studies we have taken a constant value of thermal conductivity and a constant

density. Since thermal conductivity and density are known to increase with temperature

[21], a constant value at a relatively low temperature can only be conservative about

observations on the heat transfer and fracture characteristics. Coefficient of Thermal

Expansion (CTE) of bituminous coal is known to slightly decrease upto 1500 K and

then surge upward to higher values, while Young’s modulus follows a reverse trend of

increase upto 1500 K and decreasing drastically with higher temperatures [21]. Effect

of variation in Young’s modulus and CTE are studied here and the plot in Fig. 4.17

suggests earlier fracture with increasing Young’s modulus or CTE.


4.5.2 Preheating

0

50

100

150

200

250

300

2 3 4 5 6 7 8 9 10

Fra

ctu

re T

ime (

µ s

ecs)

Mach Number

T=300T=500T=700

Figure 4.18: Effect of preheating at different Mach numbers

Operating combustion in a detonation mode is associated with spontaneous thermal

heating and fracture of particles, for which high strength waves are required. Simulation

results shown in Fig. 4.18 reveal that for relatively low to moderate Mach numbers

(4 < M < 5), for coarse particles (size > 25 µm) the fracture time exceeds 50 µs.

Such conditions can perhaps be switched to a faster fragmentation mode if the coal

is preheated. The fracture time for preheating to 500 K or 700 K show that several

situations of operating/material parameters, have moved to a desirable situation on

preheating.

4.5.3 Detonation Wave Strength

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70

Failu

re P

robabili

ty

Time µsecs

d=5,M=4,σu=3d=5,M=6,σu=3

d=15,M=4,σu=3d=15,M=6,σu=3d=25,M=4,σu=3d=25,M=6,σu=3

(a) Failure probability

0

10

20

30

40

50

60

70

80

90

3 3.5 4 4.5 5 5.5 6 6.5 7

Fra

ctu

re T

ime(µ

secs)

Mach Number


(b) Fracture time

0

0.2

0.4

0.6

0.8

1

1.2

3 3.5 4 4.5 5 5.5 6 6.5 7

Fra

ctu

re location(R

f)

Mach Number


(c) Fracture location

Figure 4.19: Failure characteristics of coal particles of different sizes at different

Mach number and m = 6, σo = 9


Detonation wave strength, in terms of Mach number, influenced coal fragmentation

through the hotside temperature and the relevant prediction for size = 50 µm particle

is presented in Fig. 4.18. For most range of operating conditions, material and fracture

variables at M = 2 and for a wide variable range at M = 3, coal particles hardly frac-

tured. Otherwise, particle fracture time decreased with increasing wave strength till upto

M = 6, beyond which the influence diminished, likely due to heat transfer and gradient

limitations. Moreover, any gain in fracture time at higher Mach number is offset by

higher energy needed for a stronger detonation wave. Hence, a tradeoff exists between

operating at high mach number and having shorter fracture times. In Fig. 4.18, wave

strengths between Mach numbers 4 to 5 shows a transition region, where the energy re-

quirements and gain in fracture time could be balanced, highlighting the desirable range

of operation. Smaller particles (d ≤ 25 µm) experienced interior fragmentation within

20 µs in the same range of Mach numbers 4 to 5 (see Fig. 4.19).

4.6 Summary

Coal particles exposed to a detonation wave experience intense thermal stress. Volatiliza-

tion has lesser effect for particle sizes considered here for the smaller time scales. However,

volatilization effect is considerably high for the longer time scale of up to ms. There are

three major reasons which can delay or limit the contribution of volatiles pressure stress

(i) longer time scales of volatilization compared to heating(or) thermal stress build up

scales (ii) high pressure external hot media and (iii) escape of the volatiles at the sur-

face. Subsequently, the following cases where volatilization may play important role in

fragmentation viz.(i) large sized particles (ii) external pressure is normal i.e. does not

correspond to a detonation shock and (iii) presence of closed pores. Stress pattern de-

veloped on coal depend on particle size, its temperature and hotside temperature (i.e.,

Mach number of detonation wave). Fracture simulation indicated that particles from


size range 5 to 250 µm are likely to fracture in times ranging from 5 to 100 µs depending

on the particle size, its temperature and detonation wave strength. It is observed that

in small particles (d < 25 µm) radial stresses dominate and fragment the core region. In

relatively coarser particles (d ≥ 50 µm), tangential stresses effect early surface attrition

causing many fine particles. The medium sized particles may experience the combination

of both. Small coal particles (d < 25 µm) and preheated coarser particle are likely to

fragment early (< 40 µs) on exposure to detonation waves at Mach number 4 to 5, the

range preferred for optimal fracture time and energy requirements. The fracture times

noted are far lesser than few ms needed for a typical conventional combustion despite

ignoring volatiles pressure stresses. Plasma assisted combustion enhances volatilization

because devolatilization time in plasma assisted combustion is of the order of µs which

is far less than conventional combustion which is of the order of ms [42]. With thermal

stress alone, particle fracture in detonation wave aided coal combustion is qualitatively

different from primary fragmentation in conventional coal combustion. With the ob-

jective to have spontaneous fragmentation (< 30 µs), Weibull parameter investigation

suggests use of any coal with characteristic strength σo < 10 MPa, which may typically

range from lignite to bituminous and a low grade anthracite. However, with fuel value,

ash content etc., also being the criteria, bituminous coal should be the preferred choice

for detonation combustion of coal.

Chapter 5

Fragmentation Statistics

Fragmentation of single coal particle was studied in previous chapters. This chapter is

concerned with the effect of detonation wave on mixture of different size coal particles. In

practical situations ground coal is not of uniform size but it is normally distributed with

particle size ranging from several µm to few mm. For experimental purpose detonation

tube was constructed whose length is 4 m. Time required for single detonation wave

to travel from one end to other end at M = 3 is 3813 µs and for M = 8 travel time

is 1429.9 µs. It is desirable that substantial fraction of coal particle mixture should

fragment before the travel time of the detonation wave. Mass average fracture time is

calculated to find statistical parameters.

Time scale of shock passage across particle being far lower than time scale of thermal

diffusion, any transient behaviour is not of interest. An experimental value of heat

transfer co-efficient(htc) reported by Nettleton et al. [43] for coal heated through shock

for 70 µm particle has been adopted in this thesis. Theoretical variation of htc as

Nu = 2+0.6Re1/2Pr1/3 [44] indicated upto 10% deviation from mean in the temperature

range of 1000-2200 K and particles in the range 40µm to 200µm. Longer travel time(lower

speed) of detonation will ensure longer exposure of particle to detonation wave, but it

also means particle exposure to lower temperature. A 4 m long steel detonation tube is

64

Chapter 5. Fragmentation Statistics 65

constructed and travel time calculated as given below

Travel Time =length

M ×√γRT

=4

3√

1.42× 287× 300= 0.003 813 seconds (5.1)

Details about tsd, tsp, and tst are given in Table 5.1.

Table 5.1: Different time scales for all sizes of particles at different Mach numbers

M Size

(µm)

tsd (s) tsp(s) tst (s) M Size

(µm)

tsd (s) tsp (s) tst (s)

3 5 1.66e-05 9.80e-09 3.81e-03 4 5 1.66e-05 7.35e-09 2.86e-03

10 6.65e-05 1.96e-08 3.81e-03 10 6.65e-05 1.47e-08 2.86e-03

25 4.16e-04 4.90e-08 3.81e-03 25 4.16e-04 3.68e-08 2.86e-03

50 1.66e-03 9.80e-08 3.81e-03 50 1.66e-03 7.35e-08 2.86e-03

75 3.74e-03 1.47e-07 3.81e-03 75 3.74e-03 1.10e-07 2.86e-03

100 6.65e-03 1.96e-07 3.81e-03 100 6.65e-03 1.47e-07 2.86e-03

150 1.50e-02 2.94e-07 3.81e-03 150 1.50e-02 2.21e-07 2.86e-03

200 2.66e-02 3.92e-07 3.81e-03 200 2.66e-02 2.94e-07 2.86e-03

250 4.16e-02 4.90e-07 3.81e-03 250 4.16e-02 3.68e-07 2.86e-03

500 1.66e-01 9.80e-07 3.81e-03 500 1.66e-01 7.35e-07 2.86e-03

750 3.74e-01 1.47e-06 3.81e-03 750 3.74e-01 1.10e-06 2.86e-03

1000 6.65e-01 1.96e-06 3.81e-03 1000 6.65e-01 1.47e-06 2.86e-03

5 5 1.66e-05 5.88e-09 2.29e-03 6 5 1.66e-05 4.90e-09 1.91e-03

10 6.65e-05 1.18e-08 2.29e-03 10 6.65e-05 9.80e-09 1.91e-03

25 4.16e-04 2.94e-08 2.29e-03 25 4.16e-04 2.45e-08 1.91e-03

50 1.66e-03 5.88e-08 2.29e-03 50 1.66e-03 4.90e-08 1.91e-03

75 3.74e-03 8.82e-08 2.29e-03 75 3.74e-03 7.35e-08 1.91e-03

100 6.65e-03 1.18e-07 2.29e-03 100 6.65e-03 9.80e-08 1.91e-03

150 1.50e-02 1.76e-07 2.29e-03 150 1.50e-02 1.47e-07 1.91e-03

200 2.66e-02 2.35e-07 2.29e-03 200 2.66e-02 1.96e-07 1.91e-03

250 4.16e-02 2.94e-07 2.29e-03 250 4.16e-02 2.45e-07 1.91e-03

500 1.66e-01 5.88e-07 2.29e-03 500 1.66e-01 4.90e-07 1.91e-03

750 3.74e-01 8.82e-07 2.29e-03 750 3.74e-01 7.35e-07 1.91e-03

1000 6.65e-01 1.18e-06 2.29e-03 1000 6.65e-01 9.80e-07 1.91e-03

7 5 1.66e-05 4.20e-09 1.63e-03 8 5 1.66e-05 3.68e-09 1.43e-03

10 6.65e-05 8.40e-09 1.63e-03 10 6.65e-05 7.35e-09 1.43e-03

25 4.16e-04 2.10e-08 1.63e-03 25 4.16e-04 1.84e-08 1.43e-03

50 1.66e-03 4.20e-08 1.63e-03 50 1.66e-03 3.68e-08 1.43e-03


75 3.74e-03 6.30e-08 1.63e-03 75 3.74e-03 5.51e-08 1.43e-03

100 6.65e-03 8.40e-08 1.63e-03 100 6.65e-03 7.35e-08 1.43e-03

150 1.50e-02 1.26e-07 1.63e-03 150 1.50e-02 1.10e-07 1.43e-03

200 2.66e-02 1.68e-07 1.63e-03 200 2.66e-02 1.47e-07 1.43e-03

250 4.16e-02 2.10e-07 1.63e-03 250 4.16e-02 1.84e-07 1.43e-03

500 1.66e-01 4.20e-07 1.63e-03 500 1.66e-01 3.68e-07 1.43e-03

750 3.74e-01 6.30e-07 1.63e-03 750 3.74e-01 5.51e-07 1.43e-03

1000 6.65e-01 8.40e-07 1.63e-03 1000 6.65e-01 7.35e-07 1.43e-03

A numerical simulation is run to obtain statistical parameters like mean, SD, CV,

PDF, CDF. Normal Distribution is considered as variable scatter.

5.1 Coal Particle Mixture and Statistics

A typical particle size distribution used in present work is given in Table 5.2 [45].

Table 5.2: Typical particle size distribution

Particle size in µm Percentage (by mass)

0 to 50 7.5 %

50 to 90 18.6 %

90 to 200 31.4 %

200 to 500 25.1 %

> 500 17.3 %

A numerical simulation is run for the time up to 1.38 ms for coal air mixture which

contains 5, 10, 25, 50, 75, 100, 150, 200, 250, 500, 750, and 1000 µm size coal particles.

Coal particles are assumed uniformly distributed over the cross section of detonation

tube. Applied detonation wave Mach numbers ranges from 3 to 8. For demonstration

purposes selected values of parameters are used though numerical code (Appendix B)

is capable of running the simulation for whole range of values for coal and Weibull

parameters given in Appendix A. These parameter values are m = 6, β = 60 × 10−6,


σu = 8, σo = 11, V ∗ = 0.3. Results are given in Table 5.3.

5.2 Statistical Parameters

This chapter is concerned with questions such as: how much fraction of particles in

mixture may fragment within specified time when subjected to detonation wave? Or,

What is the probability that a randomly selected coal particle mixture fractures between

some specified time interval when subjected to detonation wave? With help of statistical

studies there will be more clarity on the answers to these questions after proceeding

further in this chapter. Understanding of some statistical parameters is required which

are useful in studying fragmentation statistics before proceeding further.

PDF

It is a statistical measure that defines a probability distribution for a random variable.

In probability theory, PDF is a function that describes the relative likelihood for this

random variable to take on a given value. If the probability density around a point (some

interval) t is large, that means the random variable X is likely to be close to t. PDF for

Normal Distribution can be given by equation below

f(t) =1

σd√

2πe− (t−µa)2

2σ2d (5.2)

Probability that random variable X lies between a and b is given by

P (a ≤ X ≤ b) =

∫ b

a

f(t) dt (5.3)


Table 5.3: Fracture results for different size coal particles at constant values ofβ = 60× 10−6, V ∗ = 0.3, σo = 11, σu = 8, m = 6

(a) Sizes 5, 10, 25

M Size(µm)

tf (s) Tf (K) Rf

3 5 N.F.4 N.F.5 1.078700e-05 340.10 0.286 5.069200e-06 308.93 0.367 1.794000e-06 448.38 0.998 7.912000e-07 431.49 0.993 10 N.F.4 4.115333e-05 320.40 0.305 1.332000e-05 459.92 0.996 4.853000e-06 434.44 0.997 2.383333e-06 425.20 0.998 1.253333e-06 420.76 0.993 25 2.062134e-04 312.87 0.334 3.420560e-05 435.82 0.995 1.292140e-05 423.70 0.996 6.564200e-06 419.98 0.997 3.799600e-06 418.92 0.998 2.336800e-06 419.28 0.99

(b) Sizes 50, 75, 100

M Size(µm)

tf (s) Tf (K) Rf

3 50 1.479314e-04 437.74 0.994 3.868140e-05 421.74 0.995 1.798600e-05 419.04 0.996 1.040060e-05 419.07 0.997 6.674600e-06 420.14 0.998 4.489600e-06 421.71 0.993 75 1.475588e-04 426.05 0.994 4.717300e-05 419.30 0.995 2.399820e-05 418.98 0.996 1.477060e-05 420.11 0.997 9.959000e-06 421.75 0.998 6.992000e-06 423.67 0.993 100 1.599144e-04 422.06 0.994 5.681000e-05 418.82 0.995 3.056700e-05 419.61 0.996 1.956380e-05 421.24 0.997 1.359760e-05 423.10 0.998 9.788800e-06 425.06 0.99

(c) Sizes 150, 200, 250

M Size(µm)

tf (s) Tf (K) Rf


(d) Sizes 500, 750, 1000

M Size(µm)

tf (s) Tf (K) Rf


Note: N.F. = Not Fragmented.


CDF

In probability theory and statistics, the CDF describes the probability that a real-valued

random variable X with a given probability distribution will be found to have a value

less than or equal to x.

φ(x) = P (X ≤ x) =1

σd√

2π

∫ x

−∞e− (t−µa)2

2σ2d dt (5.4)

Standardized Normal Distribution

A normal distribution with mean µa = 0 and variance σ2d = 1 is known as Standardised

Normal Distribution (SND) and the corresponding random variable is Standardised Ran-

dom Variable (SRV). PDF and CDF of the SRV, which are abbreviated here as SPDF

and SCDF respectively, are given by following equations

F (y) =1√2πe−y

2/2 (5.5)

P (a ≤ Z ≤ b) =

∫ b

a

F (y) dy (5.6)

Φ(z) =1√2π

∫ z

−∞e−y

2/2dy (5.7)

Another statistical parameter which is important to study dependency of two different

variables is CV. It is given by following equation

CV =σdµa

(5.8)

5.3 Time Statistics

Details of results obtained from simulation run on the mixture of coal particle for dif-

ferent Mach numbers is given in Table 5.4. Values of PDF, CDF, SPDF and SCDF are

calculated using particle trend distribution data obtained from simulation. Simulation

was run for 1380 µs.


Table 5.4: Results obtained w.r.t fracture time for coal particle mixture subjected to

detonation waves of different Mach numbers at β = 60× 10−6, V ∗ = 0.3, σo = 11,

σu = 8, m = 6

M Size (µm) tf (s) Avg. Time (s) Time SD (s) PDF CDF SRV SPDF SCDF

5 5 1.0787e-05 1.1049e-04 1.4378e-04 2.1817e+03 0.2440 -0.6934 0.3137 0.2440

10 1.3320e-05 1.1049e-04 1.4378e-04 2.2082e+03 0.2496 -0.6758 0.3175 0.2496

25 1.2921e-05 1.1049e-04 1.4378e-04 2.2040e+03 0.2487 -0.6786 0.3169 0.2487

50 1.7986e-05 1.1049e-04 1.4378e-04 2.2560e+03 0.2600 -0.6433 0.3244 0.2600

75 2.3998e-05 1.1049e-04 1.4378e-04 2.3155e+03 0.2737 -0.6015 0.3329 0.2737

100 3.0567e-05 1.1049e-04 1.4378e-04 2.3775e+03 0.2892 -0.5558 0.3418 0.2891

150 4.5048e-05 1.1049e-04 1.4378e-04 2.5017e+03 0.3245 -0.4551 0.3597 0.3245

200 6.1079e-05 1.1049e-04 1.4378e-04 2.6156e+03 0.3656 -0.3436 0.3761 0.3655

250 7.8485e-05 1.1049e-04 1.4378e-04 2.7068e+03 0.4119 -0.2226 0.3892 0.4119

500 1.8294e-04 1.1049e-04 1.4378e-04 2.4438e+03 0.6928 0.5039 0.3514 0.6928

750 3.1159e-04 1.1049e-04 1.4378e-04 1.0432e+03 0.9190 1.3987 0.1500 0.9190

1000 4.6069e-04 1.1049e-04 1.4378e-04 1.4290e+02 0.9926 2.4356 0.0205 0.9925

6 5 5.0700e-06 8.0375e-05 1.0993e-04 2.8700e+03 0.2467 -0.6850 0.3155 0.2467

10 4.9600e-06 8.0375e-05 1.0993e-04 2.8681e+03 0.2464 -0.6860 0.3153 0.2463

25 6.5600e-06 8.0375e-05 1.0993e-04 2.8966e+03 0.2510 -0.6715 0.3184 0.2510

50 1.0400e-05 8.0375e-05 1.0993e-04 2.9635e+03 0.2622 -0.6365 0.3258 0.2622

75 1.4800e-05 8.0375e-05 1.0993e-04 3.0375e+03 0.2754 -0.5965 0.3339 0.2754

100 1.9600e-05 8.0375e-05 1.0993e-04 3.1147e+03 0.2902 -0.5528 0.3424 0.2902

150 3.0200e-05 8.0375e-05 1.0993e-04 3.2700e+03 0.3240 -0.4564 0.3595 0.3240

200 4.2200e-05 8.0375e-05 1.0993e-04 3.4166e+03 0.3642 -0.3473 0.3756 0.3642

250 5.5300e-05 8.0375e-05 1.0993e-04 3.5358e+03 0.4098 -0.2281 0.3887 0.4098

500 1.3400e-04 8.0375e-05 1.0993e-04 3.2219e+03 0.6872 0.4878 0.3542 0.6871

750 2.3400e-04 8.0375e-05 1.0993e-04 1.3669e+03 0.9189 1.3974 0.1503 0.9188

1000 3.4900e-04 8.0375e-05 1.0993e-04 1.8332e+02 0.9927 2.4435 0.0202 0.9927

7 5 1.7940e-06 6.1544e-05 8.6845e-05 3.6256e+03 0.2457 -0.6880 0.3149 0.2457

10 2.3833e-06 6.1544e-05 8.6845e-05 3.6425e+03 0.2479 -0.6812 0.3163 0.2479

25 3.7996e-06 6.1544e-05 8.6845e-05 3.6827e+03 0.2531 -0.6649 0.3198 0.2530

50 6.6746e-06 6.1544e-05 8.6845e-05 3.7626e+03 0.2638 -0.6318 0.3268 0.2637

75 9.9590e-06 6.1544e-05 8.6845e-05 3.8508e+03 0.2763 -0.5940 0.3344 0.2762

100 1.3598e-05 6.1544e-05 8.6845e-05 3.9444e+03 0.2904 -0.5521 0.3426 0.2904

150 2.1786e-05 6.1544e-05 8.6845e-05 4.1367e+03 0.3235 -0.4578 0.3593 0.3235

200 3.1041e-05 6.1544e-05 8.6845e-05 4.3189e+03 0.3627 -0.3512 0.3751 0.3627

250 4.1230e-05 6.1544e-05 8.6845e-05 4.4698e+03 0.4075 -0.2339 0.3882 0.4075


500 1.0385e-04 6.1544e-05 8.6845e-05 4.0796e+03 0.6869 0.4872 0.3543 0.6869

750 1.8243e-04 6.1544e-05 8.6845e-05 1.7435e+03 0.9180 1.3920 0.1514 0.9180

1000 2.7429e-04 6.1544e-05 8.6845e-05 2.2859e+02 0.9929 2.4497 0.0199 0.9928

8 5 7.9120e-07 4.8191e-05 6.9109e-05 4.5627e+03 0.2464 -0.6859 0.3153 0.2464

10 1.2533e-06 4.8191e-05 6.9109e-05 4.5836e+03 0.2485 -0.6792 0.3168 0.2485

25 2.3368e-06 4.8191e-05 6.9109e-05 4.6321e+03 0.2535 -0.6635 0.3201 0.2535

50 4.4896e-06 4.8191e-05 6.9109e-05 4.7265e+03 0.2636 -0.6323 0.3266 0.2636

75 6.9920e-06 4.8191e-05 6.9109e-05 4.8328e+03 0.2755 -0.5961 0.3340 0.2755

100 9.7888e-06 4.8191e-05 6.9109e-05 4.9468e+03 0.2892 -0.5557 0.3419 0.2892

150 1.6151e-05 4.8191e-05 6.9109e-05 5.1844e+03 0.3215 -0.4636 0.3583 0.3214

200 2.3396e-05 4.8191e-05 6.9109e-05 5.4128e+03 0.3599 -0.3588 0.3741 0.3599

250 3.4440e-05 4.8191e-05 6.9109e-05 5.6595e+03 0.4211 -0.1990 0.3911 0.4211

500 8.1199e-05 4.8191e-05 6.9109e-05 5.1503e+03 0.6835 0.4776 0.3559 0.6835

750 1.4409e-04 4.8191e-05 6.9109e-05 2.2041e+03 0.9174 1.3876 0.1523 0.9173

1000 2.1786e-04 4.8191e-05 6.9109e-05 2.8353e+02 0.9930 2.4550 0.0196 0.9929

For understanding purpose PDF and CDF plots for fracture time for Mach number 6

(refer Table 5.4) is given in Fig. 5.1. It can be seen from Fig. 5.1(b) that almost 91% of

coal particles in the mixture fragment within 234 µs. CV for fracture time is 1.368.

0 1 2 3 4

x 10−4

0

500

1000

1500

2000

2500

3000

3500

4000

Fracture Time in seconds

Pro

ba

bili

ty D

en

sity F

un

ctio

n

PDF plot for fracture time

(a) PDF

0 1 2 3 4

x 10−4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fracture Time in seconds

Cum

ula

tive D

istr

ibution F

unction

CDF plot for fracture time

(b) CDF

Figure 5.1: PDF and CDF plots of fracture time at M = 6, β = 60× 10−6, V ∗ = 0.3,

σo = 11, σu = 8, m = 6


5.4 Volatilization Statistics

PDF and CDF results for volatile matter generated at fracture and flow of volatile at

fracture are given in Table 5.5 and 5.6 respectively while its PDF and CDF plots are

given in Fig. 5.2 and 5.3 respectively. CV for volatile matter generation at fracture is

1.395 and for flow of volatile at fracture it is 1.294.

Table 5.5: Volatile matter generation statistics at M = 6, β = 60× 10−6, V ∗ = 0.3,

σo = 11, σu = 8, m = 6

size

(µm)

Vf

(kg/kg of coal)

Avg. Volatile

(kg/kg of coal)

Volatile SD

(kg/kg of coal)

PDF CDF SRV SPDF SCDF

5 3.8060e-15 3.3350e-11 4.6522e-11 6.6326e+09 0.2368 -0.7168 0.3086 0.2367

10 5.0426e-12 3.3350e-11 4.6522e-11 7.1262e+09 0.2714 -0.6085 0.3315 0.2714

25 3.2238e-12 3.3350e-11 4.6522e-11 6.9533e+09 0.2586 -0.6476 0.3235 0.2586

50 4.3072e-12 3.3350e-11 4.6522e-11 7.0571e+09 0.2662 -0.6243 0.3283 0.2662

75 5.8847e-12 3.3350e-11 4.6522e-11 7.2039e+09 0.2775 -0.5904 0.3351 0.2775

100 7.7060e-12 3.3350e-11 4.6522e-11 7.3667e+09 0.2907 -0.5512 0.3427 0.2907

150 1.1891e-11 3.3350e-11 4.6522e-11 7.7099e+09 0.3223 -0.4613 0.3587 0.3223

200 1.6662e-11 3.3350e-11 4.6522e-11 8.0410e+09 0.3599 -0.3587 0.3741 0.3599

250 2.1945e-11 3.3350e-11 4.6522e-11 8.3215e+09 0.4032 -0.2452 0.3871 0.4031

500 5.4980e-11 3.3350e-11 4.6522e-11 7.6968e+09 0.6790 0.4649 0.3581 0.6790

750 9.7561e-11 3.3350e-11 4.6522e-11 3.3081e+09 0.9162 1.3802 0.1539 0.9162

1000 1.4855e-10 3.3350e-11 4.6522e-11 3.9972e+08 0.9934 2.4762 0.0186 0.9933


0 0.5 1 1.5

x 10−10

0

2

4

6

8

10x 10

9

Volatile Matter Generatedat Fracture in kg/kg of coal

Pro

ba

bili

ty D

en

sity F

un

ctio

n

PDF Plot for Volatile MatterGeneration at Fracture

(a) PDF

0 0.5 1 1.5

x 10−10

0.2

0.4

0.6

0.8

1

Volatile Matter Generatedat Fracture in kg/kg of coal

Cu

mu

lative

Dis

trib

utio

n F

un

ctio

n

CDF Plot for Volatile MatterGeneration at Fracture

(b) CDF

Figure 5.2: PDF and CDF plots of volatile matter generated at fracture at M = 6,

β = 60× 10−6, V ∗ = 0.3, σo = 11, σu = 8, m = 6

Table 5.6: Volatile matter flow statistics at M = 6, β = 60× 10−6, V ∗ = 0.3, σo = 11,

σu = 8, m = 6

size

(µm)

Flow of

Volatile at

Fracture

(kmol/m2.s)

Avg. Flow

(kmol/m2.s)

Flow SD

(kmol/m2.s)

PDF CDF SRV SPDF SCDF

5 2.6891e-14 4.0309e-10 5.2143e-10 5.6750e+08 0.2198 -0.7730 0.2959 0.2197

10 4.3076e-11 4.0309e-10 5.2143e-10 6.0284e+08 0.2450 -0.6904 0.3143 0.2449

25 3.2683e-11 4.0309e-10 5.2143e-10 5.9448e+08 0.2387 -0.7104 0.3100 0.2387

50 5.1285e-11 4.0309e-10 5.2143e-10 6.0935e+08 0.2499 -0.6747 0.3177 0.2499

75 7.6226e-11 4.0309e-10 5.2143e-10 6.2862e+08 0.2654 -0.6269 0.3278 0.2654

100 1.0468e-10 4.0309e-10 5.2143e-10 6.4953e+08 0.2836 -0.5723 0.3387 0.2835

150 1.6853e-10 4.0309e-10 5.2143e-10 6.9147e+08 0.3264 -0.4499 0.3606 0.3264

200 2.3837e-10 4.0309e-10 5.2143e-10 7.2786e+08 0.3760 -0.3159 0.3795 0.3760

250 3.1224e-10 4.0309e-10 5.2143e-10 7.5357e+08 0.4308 -0.1742 0.3929 0.4308

500 7.1736e-10 4.0309e-10 5.2143e-10 6.3802e+08 0.7267 0.6027 0.3327 0.7266

750 1.1579e-09 4.0309e-10 5.2143e-10 2.6836e+08 0.9261 1.4475 0.1399 0.9260

1000 1.6208e-09 4.0309e-10 5.2143e-10 5.0044e+07 0.9902 2.3354 0.0261 0.9902


0 0.5 1 1.5 2

x 10−9

0

2

4

6

8x 10

8

Flow of Volatile Matter

at Fracture in kmol/m2s

Pro

babili

ty D

ensity F

unction

PDF Plot for VolatileMatter Flow at Fracture

(a) PDF

0 0.5 1 1.5 2

x 10−9

0.2

0.4

0.6

0.8

1

Flow of Volatile Matter

at Fracture in kmol/m2s

Cum

ula

tive D

istr

ibution F

unction

CDF plot for VolatileMatter Flow at Fracture

(b) CDF

Figure 5.3: PDF and CDF plots of volatile matter flow at fracture at M = 6,

β = 60× 10−6, V ∗ = 0.3, σo = 11, σu = 8, m = 6

5.5 Significance of Statistical Studies

As mentioned in the beginning of Section 5.2, this chapter is concerned with some im-

portant questions. Considering results presented in Table 5.4 for M = 6, these questions

can be better understood as explained below.

Q. 1: How much coal particle mixture may fragment between 6 to 134 µs?

With help of statistical analysis this question can be answered as below

P (6 ≤ Z ≤ 134) =

∫ 134

6

f(t) dt

= φ(134)− φ(6)

Standardising

= Φ

(134− µa

σd

)− Φ

(6− µaσd

)= Φ

(134− 80.38

109.93

)− Φ

(6− 80.38

109.93

)= Φ(0.49)− Φ(−0.68)


SCDF values for SRVs 0.49 and -0.68 from distribution table are [46]

= 0.6873− 0.2483

= 0.4396

This means 43.96% of the coal particle mixture may fragment within asked time

period. It can be seen from Fig. 5.1(a) that area under the curve between 6 to 134

µs is approximately 45%. Both analytical and graphical values of PDF matches

and either of the method can be used to calculate PDF.

Q. 2: How much time it will take to fragment 90% of the mixture?

If required time is t seconds then

P (Z ≤ t) = P (−∞ ≤ Z ≤ t) = 0.9

⇒ φ(t)− φ(−∞) = 0.9

Standardising

= Φ

(t− µaσd

)− Φ

(−∞− µa

σd

)= 0.9

⇒ Φ

(t− 80.38

109.93

)− Φ

(−∞− 80.38

109.93

)= 0.9

SRV value from SCDF table at 0.9 is 1.29 [46]

⇒ t− 80.38

109.93− 0 = 1.29

⇒ t = 222.2 µs

It may take 222.2 µs to fragment 90% of coal particle mixture. It can be seen from

Fig. 5.1(b) that 90% of the coal mixture fragment at approximately 225 µs. Both

analytical and graphical values of CDF matches and either of the method can be

used to calculate CDF.


5.6 Results and Discussions

5 5.5 6 6.5 7 7.5 840

50

60

70

80

90

100

110

120

Mach number

Avera

ge

Fra

ctu

re t

ime

in

µs

(a) tavg

5 5.5 6 6.5 7 7.5 81.3

1.32

1.34

1.36

1.38

1.4

1.42

1.44

Mach number

Fra

ctu

re T

ime

CV

(b) CV

Figure 5.4: Comparison of average fracture time and coefficient of variation with

different Mach numbers

1. Smaller particles ≤ 25 µm can fracture at about 1/3 of their radius at lower Mach

number. Time taken is longer.

2. Larger particles ≥ 50 µm always fracture at the surface and in much smaller time.

3. Interior fracture occurs at lower temperature when compared to surface fracture.

4. The temperature due to detonation wave must be low and particle size small for

interior fracture to occur. Otherwise fracture always starts at the surface.

5. Table 5.4 and Fig. 5.4(a) suggests that average fracture time tavg and SD reduce as

Mach number increases.

6. It is observed from Table 5.4 and Fig. 5.4(a) that average fracture time of the coal

particle mixture at different Mach numbers are 110.49 µs for M = 5, 80.38 µs for

M = 6, 61.54 µs for M = 7 and 48.19 µs for M = 8. These are far less than largest

detonation travel time of 3813 µs for M = 3.


7. PDF plots for fracture time, volatile matter generation at fracture and flow of

volatile at fracture given in Fig. 5.1(a), 5.2(a) and 5.3(a) respectively gives typical

”bell shaped” curve characteristic of normal distribution. This means our assump-

tion of fracture time and volatile matter having relation with fracture is justifiable.

8. It is evident from Fig. 5.1(b) that almost 91% of the coal particles in mixture

fragment withing 234 µs. This time is far less than the largest detonation travel

time of 3813 µs for M = 3.

9. Most of the particles have fractured before the average time irrespective of oper-

ational Mach number. Particles of the size ≥ 500 µm only take more than the

average time to fracture. This is because of lower surface area to volume ratio for

larger coal particles. It takes more time for propagation of heat required to cause

fracture.

10. CV for fracture time is 1.368, for volatile matter generated at fracture it is 1.395

and for flow of volatile at fracture it is 1.294.

11. Lower the CV smaller the residual of expected/predicted value. This means good

model fit and prediction is more reliable.

12. It is observed from Fig. 5.4 that average fracture time is reducing and CV is

increasing with increasing Mach number. Since CV is indication of good model

fit and reliability, increased CV with Mach number means at higher Mach number

results may not be reliable. Hence, numerical simulation is a good model fit at lower

Mach number, preferably between 5 to 6.

5.7 Summary

Average fracture time are far less than the detonation travel time. Fracture time and

volatilization statistics represent good model fit and reliable predictions because CV is


lower for both cases. Significant fraction of coal particle mixture fragment at far lesser

time than the travel time of detonation. Relation of CV with Mach number suggest

that results may not be reliable at higher Mach numbers. Plasma initiated detonation

combustion leads to faster fragmentation rates compared to conventional combustion.

Furthermore, the process mimics constant volume combustion and therefore will likely

result in higher system and combustion efficiencies. Bituminous coal is the preferred

choice for detonation combustion. Anthracite can be used with preheating.

Chapter 6

Conclusions

The focus of present research is numerical study of primary fragmentation and statistics

of coal particles when subjected to plasma initiated detonation wave. Research started

with analytical solution of simplified assumptions and simple failure theories and evolved

to numerical solution of more realistic conditions with use of more accurate failure theory.

The numerical solution is validated with analytical solution for simple cases. Various

numerical models outlined to effectively study the problem of primary fragmentation

of single coal particle. Effect of thermal stresses and volatilization on fragmentation

studied. Numerical results suggest that volatilization does not have significant effect on

fragmentation for the particle sizes considered in this work. Industrial combustors use

mixture of different size and property coal particles. Hence, it necessitates study of coal

particle mixture statistics. Following important questions arise regarding objective of

statistical studies for application of detonation initiated combustion technology.

1. How much fraction of mixture of coal particles will fragment when subjected to det-

onation wave? Or, in other words, how efficient is detonation initiated combustion

technology in fragmenting coal mixture?

2. How much amount of coal mixture will fragment between specific time period?

3. How fragmentation time of mixture can be reduced? Or, in other words, how

79

Chapter 6. Conclusions 80

fragmentation process can be made faster?

An attempt is made to answer these questions in present work. Simulation results

suggest that 90% of coal particle mixture fragment well within travel time of detonation

wave for the tube considered in this work. Answer to question no. 2 can be obtained

from PDF plots. There are multiple ways by which fragmentation time can be reduced.

1. Numerical results suggest that it is recommended to use bituminous coal for det-

onation combustion. Lignite has lesser σo and σu values which is crucial factor in

faster fragmentation. Fragmentation time can be reduced if lignite is used instead

of bituminous.

2. Numerical results suggest that preheated coal particles fragment in lesser time.

Hence, preheating the coal particle mixture will reduce fragmentation time.

3. Numerical results suggest that fragmentation time of coal particle reduces as Mach

number of detonation wave is increased. Hence, increase in Mach number may

reduce fragmentation time. This solution should be implemented after proper check

of energy balance. The estimation need to be done between energy supplied to

generate detonation and energy obtained from combustion of coal mixture. Also,

statistical studies on coal particle mixture suggest that CV increases with increasing

Mach number making prediction unreliable for higher Mach numbers.

Scope of Present Work

A complete combustion process can be divided into in primary fragmentation, pyrolysis

and combustion reaction. Combustion completion time depends on time taken in each of

these steps viz. time of primary fragmentation, time of pyrolysis, and time of combustion

reactions. Among these different time scales, primary fragmentation is slowest and it

is studied in this work, along with time of pyrolysis. A complete understanding of the

Chapter 6. Conclusions 81

subject comes when effect of detonation on reaction times is studied. Though time of

combustion reactions is least among time scales of all the three processes, it needs to be

studied to completely understand detonation initiated combustion technology. There is

also scope of validating proposed theory with experiments.

Appendix A

Values of Parameters Used

Values of various parameters used [7, 9, 12,14,21,41–43,47]

Parameter Value

β 20, 40, 60 (×10−6) 1/K [42]

α 1.67 × 10−7 m2/s [12]

Ti 300 K

pi 101 325 Pa

ν 0.37 [42]

k 0.254 W/(m.K) [12]

ρ 1230 kg/m3 [12]

Cp 1220 J/(kg.K) [12]

E 3000 MPa [42]

N 1

σuu 14 MPa

γ 1.42

h 15240 W/(m2.K) [43, 47]

σb 5.67× 10−8 W/(m2.K4)

εb 0.85 [14]

82

Appendix A. Values of Parameters Used 83

Ru 8314.5 J/(kmol.K)

m 3, 4, 5, 6 [7, 9]

σo 6, 7, 8, 9, 10, 11, 12 MPa

σu 6, 7, 8, 9, 10, 11, 12 MPa [41]

ε 0.003 [14]

τt 3

Ea 72 × 106 J/kmol [14]

ko 7050 1/s [14]

Mvol 40 kg/kmol [14]

n 1 [14]

rpore 5 × 10−7 m [14]

V ∗ 0.3, 0.45, 0.6 [14]

Pf,b 0.999

Appendix B

Numerical Code in C Language

1 # include <s t d i o . h>2 # include <math . h>3 # include <s t d l i b . h>4 # define PI 3.1415926535897932384626435 # include <time . h>67 double temp ( int k ) ;8 double P r i nS t r e s s ( ) ;9 double Fai lProb ( ) ;

10 double vo l ( ) ;11 double sigRR ( int k ) ;12 double VolPr inSt re s s ( ) ;13 double convFV ( ) ;14 double ModStressSimp ( ) ;15 double d e r i v a t i v e ( ) ;16 double InitCond ( ) ;17 double ValAtCent ( int j , double s s [ ] , double Tt [ ] ) ;18 double maximum(double aa , double bb) ;19 double FractCr i t2 ( ) ;20 double Thermal FailProb ( ) ;21 double Thermal FractCrit2 ( ) ;22 double Thermal der ivat ive ( ) ;23 double F i l e P r i n t ( ) ;24 double StdNormalDistr (double x [ ] ) ;25 double Thermal Fi l ePr int ( ) ;26 double Thermal StdNormalDistr (double x [ ] ) ;27 double fn (double xxx ) ;28 double trapezium (double a , double b , int step , double (∗ f ) (double x ) ) ;2930 int i =0, imax=100 , n=0, nmax=300000 , j , m=0, mmax, mmin=0, M=6, wbmod=6,

sigU =6, sigO =12, v s ta r =30, b e t a i n t =40;31 int i count ;3233 /∗ v a r i a b l e s r e l a t e d to s i z e loop ∗/

84

Appendix B. Numerical Code in C Language 85

34 int s i z e 1 [12 ]={5 , 10 , 25 , 50 , 75 , 100 , 150 , 200 , 250 , 500 , 750 , 1000} ,i i , s i z e =50, j j ;

3536 /∗ v a r i a b l e s used in temperature c a l c u l a t i o n f unc t i on s ∗/37 double T[ 1 0 0 0 0 0 ] , Tn[ 1 0 0 0 0 0 ] , r [ 1 5 0 ] ;38 double alpha =0.000000167 , beta =0.000040 , E=3000.0 , mu=0.37;39 double de l t , de l r , r0 =0.0 , rmax , tmax=0.00138 , rmin =0.0 , tmin =0.0 , mult ,

mult1 , mult2 ;40 double Tinf =800.0 , T in i t =300.00 , t [ 1 0 1 0 00 0 ] , t0 =0.0 ;41 double hsigma r , hs igma l , hc =15240.0 , kc =0.254 , Tnn , s i g b =0.0000000567 ,

epsc =0.85 , to l , rho =1230.0 , cp =1220.0;42 double multT le f t , multT center , multT right , multT const ;43 double a r , q r , a l , q l , v o l d i f f ;4445 /∗ v a r i a b l e s used in r a d i a l and t a n g en t i a l s t r e s s c a l c u l a t i o n ∗/46 double f u l l i n t , f u l l i n t 4 , f u l l i n t 5 , f u l l i n t 5 1 , r r r ;47 double f u l l i n t 2 , f u l l i n t 3 , y , z , rr , TT[ 1 0 0 0 0 0 ] , pa r t i n t , pa r t in t11 ,

pa r t in t1 , d e l r 2 ;48 double s ig R [ 1 0 0 0 0 0 ] , s i g T [ 1 0 0 0 0 0 ] ;49 double f u l l i n t 4 1 , f u l l i n t 5 1 , f u l l i n t 5 1 1 , par t in t , part int11 , pa r t i n t 1 ;50 double sumComp=0.0 , sumPart =0.0 , rrc , TTc , tempxrrc , rrp , TTp, tempxrrp ,

sumComp1 , sumPart1 ;5152 /∗ v a r i a b l e s used in maximum p r i n c i p a l s t r e s s c a l c u l a t i o n ∗/53 double r r , vo l r , r l , v o l l , s i g 0 [ 1 0 0 0 0 0 ] , s i g 1 [ 1 0 0 0 0 0 ] ,

s i g 2 [ 1 0 0 0 0 0 ] , s i g 3 [ 1 0 0 0 0 0 ] ;54 double tau xy [ 1 0 0 0 0 0 ] , tau yz [ 1 0 0 0 0 0 ] , tau zx [ 1 0 0 0 0 0 ] , sx [ 1 0 0 0 0 0 ] ,

sy [ 1 0 0 0 0 0 ] , sz [ 1 0 0 0 0 0 ] ;55 double J3 , J2 , J22 , th ;56 double t h s i g 0 [ 1 0 0 0 0 0 ] , t h s i g 1 [ 1 0 0 0 0 0 ] , t h s i g 2 [ 1 0 0 0 0 0 ] ,

t h s i g 3 [ 1 0 0 0 0 0 ] ;57 double th tau xy [ 1 0 0 0 0 0 ] , th tau yz [ 1 0 0 0 0 0 ] , th tau zx [ 1 0 0 0 0 0 ] ,

th sx [ 1 0 0 0 0 0 ] , th sy [ 1 0 0 0 0 0 ] , t h s z [ 1 0 0 0 0 0 ] ;58 double th J3 , th J2 , th J22 , th th ;5960 /∗ f a i l u r e p r o b a b i l i t y c a l c u l a t i o n s ∗/61 double pf1 [ 1 0 0 0 0 0 ] , pf10 =0.0 , pf2 [ 1 0 0 0 0 0 ] , p f t [ 1 0 0 0 0 0 ] , p f [ 5 0 0 1 0 0 ] [ 1 1 0 ] ,

ps [ 1 0 0 0 0 0 ] , pf21 , eta ;62 double th p f1 [ 1 0 0 0 0 0 ] , th p f10 =0.0 , th p f2 [ 1 0 0 0 0 0 ] , t h p f t [ 1 0 0 0 0 0 ] ,

t h p f [ 5 0 0 1 0 0 ] [ 1 1 0 ] , th ps [ 1 0 0 0 0 0 ] , th pf21 , eta ;6364 /∗ v a r i a b l e s used in v o l a t i l i z a t i o n ∗/65 double vn [ 1 0 0 0 0 0 ] , v [ 1 0 0 0 0 0 ] , k0 =7050.0 , Ea=72000000.0 , g s cns t =8314.5 ,

vmax , N[ 1 0 0 0 0 0 ] , rhoc =1600.0 , Mvol =40.0;66 double p [ 1 0 0 0 0 0 ] , v i s c =0.0001 , tau =3.0 , rpor =0.0000005 , e p s i l o n =0.003 ,

pb , p4ac , patm=101325.0 , pp [ 1 0 0 0 0 0 ] , sig RR [ 1 0 0 0 0 0 ] ;6768 /∗ v a r i a b l e used in d e r i v a t i v e c a l c u l a t i o n func to in ∗/69 double dtbydr [ 1 0 0 0 0 0 ] , d2tbydr2 [ 1 0 0 0 0 0 ] , ds igbydr [ 1 0 0 0 0 0 ] , DELpfn ,

DELpfi , f p f =0.999 , f r a c l o c , dpbydr [ 1 0 0 0 0 0 ] ;70 double th dtbydr [ 1 0 0 0 0 0 ] , th d2tbydr2 [ 1 0 0 0 0 0 ] , th ds igbydr [ 1 0 0 0 0 0 ] ,

th DELpfn , th DELpfi , t h f r a c l o c ;71


72 /∗ v a r i a b l e s used in f r a c t u r e c r i t e r i a l c a l c u l a t i o n ∗/73 double f r a c l o c , maxR, maxj , f t ime , f temp ;74 double t h f r a c l o c , th maxR , th maxj , th f t ime , th f temp ;7576 /∗ v a r i a b l e used in cond i t i on behind shock wave∗/77 double p num , p den , p i n f ;7879 /∗ f i l e p o in t e r s ∗/80 FILE ∗ fp , ∗ f t , ∗op , ∗od , ∗ top , ∗ tod , ∗ fd , ∗ fsnd , ∗ t fp , ∗ t f t , ∗ t f snd ;81 /∗ v a r i a l b l e s used to c a l c u l a t e normal d i s t r i b u t i o n data ∗/8283 double f t i m e a r r a y [ 2 0 ] , f v o l , f v o l a r r a y [ 2 0 ] , f v o l f l o w ,

f v o l f l o w a r r a y [ 2 0 ] , f v o l p r e s , f v o l p r e s a r r a y [ 2 0 ] , f v o l s i g 1 ,f v o l s i g 1 a r r a y [ 2 0 ] , f t emp array [ 2 0 ] ;

84 double t h f t i m e a r r a y [ 2 0 ] , t h f s i g 1 , t h f s i g 1 a r r a y [ 2 0 ] ,th f t emp ar ray [ 2 0 ] ;

8586 /∗ v a r i a l b l e s r e qu i r ed to auto name f i l e s generated us ing ” s n p r i n t f ”∗/87 char Mfi l e [ ]= ”M” ;88 char s i g u f i l e [ ]= ” sigU ” ;89 char s i g O f i l e [ ]= ” sigO ” ;90 char m f i l e [ ]= ”m” ;91 char vmaxf i l e [ ]= ”vmax” ;92 char BetaFi l e [ ]= ” beta ” ;93 char S i z e F i l e [ ]= ” s i z e ” ;9495 /∗some other v a r i a b l e s ∗/96 double num, den , gama=1.42;97 double c1 , c2 ; int i 1 ;9899

100 int main ( )101 {102 c l o c k t s t a r t = c lock ( ) ; // t h i s w i l l p r i n t program running time103 //rmax=s i z e /1000000.0 ;104 d e l t =(tmax−tmin ) /nmax ;105 // d e l r=(rmax−rmin )/imax ;106 InitCond ( ) ;107 // f o r (M=3;M<=8;M++) //108 //{109 num=(2.0∗gama∗pow(M, 2 . 0 )−(gama−1.0) ) ∗ ( ( gama−1.0)∗pow(M, 2 . 0 ) +2.0) ;110 den=pow(M∗(gama+1.0) , 2 . 0 ) ;111 Tinf=Tin i t ∗num/den ;112 p num=2.0∗gama∗pow(M, 2 . 0 )−(gama−1.0) ;113 p den=(gama−1.0) ;114 // f o r ( b e t a i n t =20; b e t a i n t <=60; b e t a i n t+=20)115 //{116 beta=b e t a i n t /1000000 .0 ;117 /∗ f o r ( sigO=11; sigO<=12;sigO++)118 {119 f o r ( sigU=8; sigU<=10;sigU+=1)120 {121 f o r (wbmod=3;wbmod<=6;wbmod++)


122 {123 f o r ( v s t a r =30; vs tar <=60; v s t a r+=15)124 { // ∗/125 vmax=vs ta r / 1 0 0 . 0 ;126 // icount =0; // t h i s i count shou ld be wr i t t en in f o r loop o f

s i z e127 j j =0;128 // f o r ( i i =0; i i <=11; i i++)129 //{130 // s i z e=s i z e 1 [ i i ] ;131 rmax=s i z e /1000000 .0 ;132 d e l r =(rmax−rmin ) /imax ;133 // p r i n t f (”% l f \n” , rmax) ; /∗ t h i s works f i n e up to t h i s po in t ∗/134 i count =0;135 for (n=0;n<=nmax ; n++){136 InitCond ( ) ;137 t [ n]= t0+n∗ d e l t ;138 p i n f=patm∗p num/p den ;139 //@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@140 // convec t i v e BC: thermal+v o l a t i l i z a t o i n141 //@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@142 convFV ( ) ;143 ModStressSimp ( ) ;144 Pr i nS t r e s s ( ) ;145 vo l ( ) ;146 Vo lPr inSt re s s ( ) ;147 d e r i v a t i v e ( ) ;148 Fai lProb ( ) ;149 F i l e P r i n t ( ) ;150 i f ( p f t [ n]> f p f ) {151 FractCr i t2 ( ) ;152 // f t ime a r r a y [ j j ]= f t ime ;153 }154 }//n loop155 // p r i n t f (”\n”) ;156 // } // s i z e loop157 StdNormalDistr ( f t i m e a r r a y ) ;158 StdNormalDistr ( f t emp array ) ;159 StdNormalDistr ( f v o l a r r a y ) ;160 StdNormalDistr ( f v o l f l o w a r r a y ) ;161 //Thermal StdNormalDistr ( t h f t ime a r r a y ) ;162 // }// v s t a r loop163 // }//wbmod loop164 // }// sigU loop165 // }// sigO loop166 // }// b e t a i n t loop167 //} //M loop168 c l o c k t stop = c lock ( ) ;169 double e lapsed = (double ) ( stop − s t a r t ) ∗ 1000 .0 / CLOCKS PER SEC;170 p r i n t f ( ”Time e lapsed in ms : %f \n” , e l apsed ) ;171 return 0 ;172 }173 /∗##########################################################


174 ############ ###########175 ############ FUNCTION SECTION ###########176 ############ ###########177##########################################################∗/178 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗179 i n i t i a l cond i t i on180 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/181 double InitCond ( )182 {183 i f (n==0){184 for ( i =0; i<=imax ; i++){185 T[ i ]= Tin i t ;186 v [ i ] = 0 . 0 ;187 vn [ i ] = 0 . 0 ;188 N[ i ] = 0 . 0 ;189 p [ i ]=patm ;190 p i n f=patm ;191 pf [ n ] [ i ] = 0 . 0 ; p f t [ n ] = 0 . 0 ;192 t h p f [ n ] [ i ] = 0 . 0 ;193 t h p f t [ n ] = 0 . 0 ;194 t [ n]= t0+n∗ d e l t ;195 r [ i ]= r0+i ∗ d e l r ;196 Tn [ i ]= Tin i t ;197 dtbydr [ i ] = 0 . 0 ;198 d2tbydr2 [ i ] = 0 . 0 ;199 ds igbydr [ i ] = 0 . 0 ;200 th dtbydr [ i ] = 0 . 0 ;201 th d2tbydr2 [ i ] = 0 . 0 ;202 th ds igbydr [ i ] = 0 . 0 ;203 s ig R [ i ] = 0 . 0 ;204 s ig T [ i ] = 0 . 0 ;205 s i g 1 [ i ] = 0 . 0 ;206 s i g 2 [ i ] = 0 . 0 ;207 s i g 3 [ i ] = 0 . 0 ;208 t h s i g 1 [ i ] = 0 . 0 ;209 t h s i g 2 [ i ] = 0 . 0 ;210 t h s i g 3 [ i ] = 0 . 0 ;211 // p r i n t f (”%7.6 f %8.7 f %8.4 f %8.4 f %8.4 f %8.4 f %11.8 l f %7.5 l f \n” ,

t [ n ] , r [ i ] , T[ i −1] , T[ i ] , T[ i +1] , Tn[ i ] , v [ i ] , p [ i ] ) ;212 }213 }214 // p r i n t f (”\n”) ;215 return 0 . 0 ;216 }217 /∗@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@218 @@@@@@ FUNCTIONS RELATED TO TEMPERATURE @@@@@219 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@∗/220 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗221 f o u r i e r HT: convec t i v e boundary heat t r a n s f e r222 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/223224 double convFV ( )225 {


226 d e l r =(rmax−rmin ) /imax ;227 d e l t =(tmax−tmin ) /nmax ;228229 for ( i =0; i<=imax ; i++){230 r [ i ]= r0+i ∗ d e l r ;231232 i f ( i ==0){233 r [ i ]=0.1∗ rmax/imax ;234 ValAtCent ( i ,T, r ) ;235 T[ i−1]=T[ i +1] ;236 }237 mult=alpha ∗ d e l t / d e l r ;238 Tn [ i ]=mult∗T[ i −1]/ d e l r + (1.0−2.0∗mult/ de l r −2.0∗mult/ r [ i ] ) ∗T[ i ] +

( mult/ d e l r +2.0∗mult/ r [ i ] ) ∗T[ i +1] ;239240 i f ( i==imax ) {241 r l=r [ i −1] ;242 a l =4.0∗PI∗ r l ∗ r l ;243 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;244 r r=r [ i ] ;245 a r =4.0∗PI∗ r r ∗ r r ;246 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;247 v o l d i f f=vo l r−v o l l ;248249 mul tT l e f t=a l ∗ d e l t ∗ alpha /( v o l d i f f ∗ d e l r ) ;250 multT center=a l ∗ d e l t ∗ alpha /( v o l d i f f ∗ d e l r ) +

d e l t ∗hc∗ a r /( rho∗ v o l d i f f ∗cp ) +d e l t ∗ s i g b ∗ epsc ∗ a r ∗pow(T[ i ] , 3 . 0 ) /( rho∗ v o l d i f f ∗cp ) ;

251 multT const=hc∗ d e l t ∗ a r ∗Tinf /( rho∗ v o l d i f f ∗cp ) +d e l t ∗ s i g b ∗ epsc ∗ a r ∗pow( Tinf , 4 . 0 ) / ( rho∗ v o l d i f f ∗cp ) ;

252253 Tn [ i ]=T[ i ] + mul tT l e f t ∗T[ i −1] − multT center ∗T[ i ] + multT const ;254 }255 i f (n==nmax) {256 // p r i n t f (”%3.1 l f %7.2 l f %7.6 f %8.7 f %8.2 l f %8.2 l f %8.2 l f %8.2 l f %l f

%l f %l f \n” , M, Tinf , t [ n ] , r [ i ] , T[ i −1] , T[ i ] , T[ i +1] , Tn[ i ] ,mu l tT l e f t , multT center , multT const ) ;

257 // p r i n t f (”%3.1 l f %7.2 l f %7.6 f %8.7 f %8.2 l f %8.2 l f %8.2 l f %8.2 l f \n” ,M, Tinf , t [ n ] , r [ i ] , T[ i −1] , T[ i ] , T[ i +1] , Tn[ i ] ) ;

258 }259 T[ i ]=Tn [ i ] ;260 }261 return ( 0 . 0 ) ;262 }263 /∗@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@264 @@@@@@ FUNCTIONS RELATED TO STRESS @@@@@265 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@∗/266 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗267 s t r e s s us ing simpson 1/3 ru l e modi f ied (1 4 1) /6268 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/269 double ModStressSimp ( )270 {271 d e l r =(rmax−rmin ) /imax ;


272 sumComp=0.0;273 /∗0 to rmax i n t e g r a t i o n beg in s here ∗/274 for ( i =0; i<=imax ; i++){275 i f ( i ==0){ // t h i s has to be invoked when f o r ( i =0; i<=imax ; i++) loop

used276 sumComp=0.0;277 }278 r r c =0.5∗( r [ i ]+ r [ i −1]) ;279 TTc=0.5∗(T[ i ]+T[ i −1]) ;280 tempxrrc=(TTc−Tin i t ) ∗ r r c ∗ r r c ;281282 sumComp=sumComp+(temp ( i −1) + 4.0∗ tempxrrc + temp ( i ) ) ;283 // i f (n==nmax){ p r i n t f (”%3d %g %g %g %g\n” , i , temp ( i ) , T[ i ] , r [ i ] ,

sumComp) ;}284 }285286 sumComp1=d e l r ∗sumComp/(6 . 0∗pow(rmax , 3 . 0 ) ) ;287 /∗0 to r i n t e g r a t i o n beg in s here ∗/288 for ( i =0; i<=imax ; i++){289 i f ( i ==0){290 sumPart =0.0 ;291 }292 mmin=0;293 mmax=i ;294 sumPart =0.0 ;295296 for (m=0;m<=mmax;m++){297 i f (m==0){298 sumPart =0.0 ;299 }300 else {301 rrp =0.5∗( r [m]+ r [m−1]) ;302 TTp=0.5∗(T[m]+T[m−1]) ;303 tempxrrp=(TTp−Tin i t ) ∗ rrp ∗ rrp ;304305 sumPart=sumPart+(temp (m−1)+4.0∗ tempxrrp+temp (m) ) ;306 // i f (n==nmax){ p r i n t f (”%3d %3d %g %g %g %g %g\n” , i , m, temp (m) ,

T[m] , r [m] , r [mmax] , sumPart ) ;}307 }308 }309 sumPart1=d e l r ∗sumPart / (6 . 0∗pow( r [mmax] , 3 . 0 ) ) ;310311 mult2=beta ∗E/(1−mu) ;312 s ig R [ i ]=2.0∗mult2 ∗(sumComp1−sumPart1 ) ;313314 s ig T [ i ]=mult2 ∗ (2 . 0∗ sumComp1+sumPart1−(T[ i ]−Tin i t ) ) ;315 i f ( i ==1){316 ValAtCent ( i , s ig R , r ) ;317 ValAtCent ( i , s ig T , r ) ;318 }319 i f ( i ==0){320 ValAtCent ( i , s ig R , r ) ;321 ValAtCent ( i , s ig T , r ) ;


322 }323 i f ( i==imax ) {324 s ig R [ i ] = 0 . 0 ;325 }326 i f (n==nmax) {327 // p r i n t f (”%3.1 l f %7.2 l f %7.6 f %8.7 f %8.2 l f %l f %l f %8.2 l f

%8.2 l f \n” , M, Tinf , t [ n ] , r [ i ] , Tn[ i ] , sumComp1 , sumPart1 ,s ig R [ i ] , s i g T [ i ] ) ;

328 }329 }330 return 0 . 0 ;331 }332 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗333 temperature func to in used in i n t e g r a t i o n334 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/335 double temp ( int k )336 {337 i f ( k==0){338 r [ k ]=0.1∗ rmax/imax ;339 }340 return ( (T[ k]−Tin i t ) ∗ r [ k ]∗ r [ k ] ) ;341 }342 /∗@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@343 @@@@@@ FUNCTIONS RELATED TO THERMAL ANALYS @@@@@344 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@∗/345 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗346 p r i n c i p a l s t r e s s347 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/348 double P r i nS t r e s s ( )349 {350 for ( i =0; i<=imax ; i++){351 r [ i ]= r0+i ∗ d e l r ;352 i f ( i ==0){353 r [ i−1]=r [ i +1] ;354 r [ i ]=0.1∗ rmax/imax ;355 r l=r [ i ] ;356 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;357 r r =(r [ i +1]+r [ i ] ) / 2 . 0 ;358 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;359 th tau xy [ i ]=( v o l l ∗ s ig T [ i−1]+ v o l r ∗ s ig T [ i ] ) /( v o l l+v o l r ) ;360 }361362 t h s i g 0 [ i ] = ( s ig R [ i ]+ s ig T [ i ]+ s ig T [ i ] ) / 3 . 0 ;363 th sx [ i ] = s ig R [ i ] − t h s i g 0 [ i ] ;364 th sy [ i ] = s ig T [ i ] − t h s i g 0 [ i ] ;365 t h s z [ i ] = s ig T [ i ] − t h s i g 0 [ i ] ;366367 r l =(r [ i−1]+r [ i ] ) / 2 . 0 ;368 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;369 r r =(r [ i ]+ r [ i +1]) / 2 . 0 ;370 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;371372 th tau xy [ i ]=( v o l l ∗ s ig T [ i−1]+ v o l r ∗ s ig T [ i ] ) /( v o l l+v o l r ) ;


373374 i f ( i==imax ) {375 r [ i ]=rmax ;376 r [ i +1]=r [ i ] ;377 r r =(r [ i +1]+r [ i ] ) / 2 . 0 ;378 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;379 r l=r [ i −1] ;380 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;381 th tau xy [ i ]=( v o l l ∗ s ig T [ i−1]+ v o l r ∗ s ig T [ i ] ) /( v o l l+v o l r ) ;382 }383384 th tau yz [ i ]= s ig T [ i ] ;385 th tau zx [ i ]= s ig T [ i ] ;386387 th J3=−th sx [ i ]∗ th sy [ i ]∗ t h s z [ i ] + th sx [ i ]∗pow( th tau yz [ i ] , 2 . 0 ) +

th sy [ i ]∗pow( th tau zx [ i ] , 2 . 0 ) + t h s z [ i ]∗pow( th tau xy [ i ] , 2 . 0 ) ;388 th J2=th sx [ i ]∗ th sy [ i ] + th sy [ i ]∗ t h s z [ i ] + t h s z [ i ]∗ th sx [ i ] −

pow( th tau xy [ i ] , 2 . 0 ) − pow( th tau yz [ i ] , 2 . 0 ) −pow( th tau zx [ i ] , 2 . 0 ) ;

389390 th J22=fabs ( th J2 / 3 . 0 ) ;391392 th th =(1 .0/3 .0 ) ∗ acos(− th J3 /(2 . 0∗pow( th J22 , 1 . 5 ) ) ) ;393394 t h s i g 1 [ i ]= t h s i g 0 [ i ] + 2 .0∗ s q r t ( th J22 ) ∗ cos ( th th ) ;395 t h s i g 2 [ i ]= t h s i g 0 [ i ] − 2 .0∗ s q r t ( th J22 ) ∗ cos ( th th+PI / 3 . 0 ) ;396 t h s i g 3 [ i ]= t h s i g 0 [ i ] − 2 .0∗ s q r t ( th J22 ) ∗ cos ( th th−PI / 3 . 0 ) ;397398 i f ( i ==0){399 ValAtCent ( i , t h s i g 1 , r ) ;400 ValAtCent ( i , t h s i g 2 , r ) ;401 ValAtCent ( i , t h s i g 3 , r ) ;402 }403 i f (n==0){404 t h s i g 1 [ i ] = 0 . 0 ;405 t h s i g 2 [ i ] = 0 . 0 ;406 t h s i g 3 [ i ] = 0 . 0 ;407 }408 // p r i n t f (”%7.6 f %8.7 f %l f %l f %l f %l f \n” , t [ n ] , r [ i ] , th J2 , th J22 ,

th J3 , t h t h ) ;409410 // i f (n==nmax)411 i f ( ( n%300)==0){412 // p r i n t f (”%10.4 l e %4.2 l f %11.4 l e %11.4 l e %11.4 l e %8.2 l f , %8.2 l f

%8.2 l f %8.2 l f %8.2 l f %8.2 l f \n” ,413 // t [ n ] , ( r [ i ] / rmax) , th J3 , th J22 , t h t h , Tn[ i ] , s i g R [ i ] ,

s i g T [ i ] , t h s i g 1 [ i ] , t h s i g 2 [ i ] , t h s i g 3 [ i ] ) ;414 }415 }416 return 0 . 0 ;417 }418 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗419 thermal f a i l u r e p r o b a b i l i t y


420 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/421 double Thermal FailProb ( )422 {423 char ThermalOverallProb [ BUFSIZ ] ;424 /∗ wr i t e s o v e r a l l f a i l u r e p r o b a b i l i t y datas ∗/425 s n p r i n t f ( ThermalOverallProb , s izeof ( ThermalOverallProb ) ,

” th %s %d %s %d %s %d %s %d %s %d %s %d Overal lProb . txt ” ,S i z e F i l e , s i z e , Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU ,s i g O f i l e , sigO , mf i l e ,wbmod) ;

426 top=fopen ( ThermalOverallProb , ”a” ) ;427428 th p f10 =0.0 ;429 for ( i =0; i<=imax ; i++){430 eta =4.0∗PI∗pow(rmax , 3 . 0 ) / 3 . 0 ;431 i f ( i ==0){432 r [ i−1]=r [ i +1] ;433 r [ i ]=0.1∗ rmax/imax ;434 r r =(r [ i +1]+r [ i ] ) / 2 . 0 ;435 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;436 th p f1 [ i ]=( t h s i g 1 [ i ]− sigU ) / sigO ;437 th p f2 [ i ]= v o l r ∗pow( th p f1 [ i ] ,wbmod) ;438 th ps [ i ]=exp(− th p f2 [ i ] / eta ) ;439 t h p f [ n ] [ i ]=1.0− th ps [ i ] ;440 }441442 v o l r =4.0∗PI∗pow( r [ i ] , 3 . 0 ) / 3 . 0 ;443 v o l l =4.0∗PI∗pow( r [ i −1 ] , 3 . 0 ) / 3 . 0 ;444445 i f ( t h s i g 1 [ i ]>(1.0∗ sigU ) ) {446 th p f1 [ i ]=( t h s i g 1 [ i ]− sigU ) / sigO ;447 }448 else {449 th p f1 [ i ] = 0 . 0 ;450 }451 th p f2 [ i ]=( vo l r−v o l l ) ∗pow( th p f1 [ i ] ,wbmod) ;452 th ps [ i ]=exp(− th p f2 [ i ] / eta ) ;453 t h p f [ n ] [ i ]=1.0− th ps [ i ] ;454455 i f ( i==imax ) {456 v o l r =4.0∗PI∗pow( r [ i ] , 3 . 0 ) / 3 . 0 ;457 v o l l =4.0∗PI∗pow( r [ i −1 ] , 3 . 0 ) / 3 . 0 ;458 th p f1 [ i ]=( t h s i g 1 [ i ]− sigU ) / sigO ;459 th p f2 [ i ]=( vo l r−v o l l ) ∗pow( th p f1 [ i ] ,wbmod) ;460 th ps [ i ]=exp(− th p f2 [ i ] / eta ) ;461 t h p f [ n ] [ i ]=1.0− th ps [ i ] ;462 }463 i f ( i ==0){464 th p f1 [ i ] = 0 . 0 ;465 th p f2 [ i ] = 0 . 0 ;466 th ps [ i ] = 0 . 0 ;467 t h p f [ n ] [ i ] = 0 . 0 ;468 // t h p f [ n ] [ i ]= t h p f [ n ] [ i +1]+(r [ i ]−r [ i +1]) ∗( t h p f [ n ] [ i+1]− t h p f [ n ] [ i +2])

/ ( r [ i+1]−r [ i +2]) ;


469 }470 i f (n==0){471 t h p f [ n ] [ i ] = 0 . 0 ;472 }473 // i f (n==nmax)474 i f ( ( n%300==0))475 // i f ( ( n%100==0) && i==100)476 {477 // p r i n t f (”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %3.1 l f %10.4 l e

%10.4 l e %11.1 l f %6.2 l f %6.2 l f %7.2 l f %d %d %d %6.2 l f %6.2 l f%7.2 l f %11.4 l e %11.4 l e %11.4 l e %7.4 l f \n” ,

478 //M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn[ i ] ,vmax , vn [ i ] ,N[ i ] , p [ i ] , s i g R [ i ] ,sigRR( i ) , t h s i g T [ i ] , wbmod , sigU , sigO , t h s i g 1 [ i ] , t h s i g 2 [ i ] ,t h s i g 3 [ i ] , t h d t b y d r [ i ] ,t h d2 t byd r2 [ i ] , t h d s i g b y d r [ i ] , t h p f [ n ] [ i ] ) ;

479 }480 th p f2 [ i ]= th p f10+th p f2 [ i ] ;481 th p f10=th p f2 [ i ] ;482 // t h p f 21=t h p f 2 [ i ] ;483 }484 // t h p f t [ n]=1−exp(− t h p f 2 1 / e ta ) ;485 t h p f t [ n]=1−exp(− th p f10 / eta ) ;486 i f ( ( n%3000)==0){487 f p r i n t f ( top , ”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %4.2 l f %d %2d %2d

%7.4 l f %7.4 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) , T[ i −1] ,vmax ,wbmod, sigU , sigO , t h p f [ n ] [ i ] , t h p f t [ n ] ) ;

488 // p r i n t f (”%d %7.2 l f %10.4 l e %10.4 l e %8.2 l f %8.2 l f %8.2 l f %8.2 l f %6.4 l f%9.2 l e %9.2 l e \n” , M, Tinf , t [ n ] , r [ i ] ,T[ i −1] , T[ i ] ,T[ i +1] ,Tn[ i ] ,t h p f t [ n ] , t h p f10 , e ta ) ;

489 }490 f c l o s e ( top ) ;491 return 0 . 0 ;492 }493 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗494 thermal : var ious d e r i v a t i v e s495 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/496 double Thermal der ivat ive ( )497 {498 for ( i =0; i<=imax ; i++){499 th dtbydr [ i ]=(T[ i ]−T[ i −1]) /( r [ i ]− r [ i −1]) ;500 th d2tbydr2 [ i ]=(T[ i −1]−2.0∗T[ i ] + T[ i +1]) / pow ( ( r [ i ]− r [ i −1]) , 2 . 0 ) ;501 th ds igbydr [ i ]=( t h s i g 1 [ i ]− t h s i g 1 [ i −1]) / ( r [ i ]− r [ i −1]) ;502503 i f ( i ==0){504 ValAtCent ( i , th dtbydr , r ) ;505 ValAtCent ( i , th d2tbydr2 , r ) ;506 ValAtCent ( i , th ds igbydr , r ) ;507 }508 i f (n==0){509 th dtbydr [ i ] = 0 . 0 ;510 th d2tbydr2 [ i ] = 0 . 0 ;511 th ds igbydr [ i ] = 0 . 0 ;


512 }513 i f (n==nmax) {514 // p r i n t f (”%3.1 l f %7.2 l f %g %g %4.2 l f %8.2 l f %11.2 l f %16.2 l f

%12.2 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn[ i ] , t h d t b y d r [ i ] , t h d2 t byd r2 [ i ] , t h d s i g b y d r [ i ] ) ;

515 }516 }517 return 0 . 0 ;518 }519 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗520 thermal f r a c t u r e c r i t e r i a521 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/522 double Thermal FractCrit2 ( )523 {524 tod=fopen ( ” Thermal Overa l l data . txt ” , ”a” ) ;525 th maxj =−10.0;526527 for ( i =1; i<=imax ; i++){528 th DELpfn = t h p f [ n ] [ i ] − t h p f [ n−1] [ i ] ;529 i f ( ( th DELpfn>th maxj ) && t h p f [ n ] [ i ]>0.99∗ f p f ) {530 th maxj=maximum( th maxj , th DELpfn ) ;531 th maxR=r [ i ] / rmax ;532 t h f t i m e=t [ n ] ;533 th f temp=Tn[ i ] ;534 t h f s i g 1 = t h s i g 1 [ i ] ;535 break ;536 }537 }538 i f ( i count==0 && th maxj>0.0 && t h p f t [ n]> f p f ) {539 t h f r a c l o c=th maxR ;540 // p r i n t f (”%d %7.2 l f %10.8 l f %10.8 l f %4.2 l f %8.2 l f %d %d %d %3.1 l f

%4.2 l f %g %6.4 l f %6.4 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn[ i ] , wbmod , sigU , sigO ,vmax , t h f r a c l o c , th maxj , t h p f [ n ] [ i ] , t h p f t [ n ] ) ;

541 // p r i n t f (”FRACTURED: %d %7.2 l f %10.4 l e %8.2 l e %8.2 l f %d %2d %2d%3.1 l f %4.2 l f %10.4 l e \n” , M, Tinf , t h f t ime , rmax , th f t emp ,wbmod , sigU , sigO , vmax , t h f r a c l o c , th maxj ) ;

542 f p r i n t f ( tod , ”FRACTURED: %d %7.2 l f %12.6 l e %8.2 l e %8.2 l f %2d %d %2d%2d %4.2 l f %6.2 l f %7.2 l f %6.2 l f %7.2 l f %7.2 l f %7.4 l f %7.4 l f \n” ,M, Tinf , th f t ime , rmax , th f temp ,be ta in t ,wbmod, sigU , sigO , t h f r a c l o c ,s ig R [ i ] , s i g T [ i ] , t h s i g 1 [ i ] , t h s i g 2 [ i ] ,t h s i g 3 [ i ] , t h p f [ n ] [ i ] , t h p f t [ n ] ) ;

543544 t h f t i m e a r r a y [ j j ]= t h f t i m e ;545 th f t emp ar ray [ j j ]= th f temp ;546 t h f s i g 1 a r r a y [ j j ]= t h f s i g 1 ;547548 p r i n t f ( ”%2d %12.6 l e %7.2 l f %6.2 l f \n” , j j , t h f t i m e a r r a y [ j j ] ,

th f t emp ar ray [ j j ] , t h f s i g 1 a r r a y [ j j ] ) ;549 j j ++;550 i count =1;551 }


552 f c l o s e ( tod ) ;553 return 0 . 0 ;554 }555 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗556 th sep func t i on to p r i n t f i l e s557 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/558 double Thermal Fi l ePr int ( )559 {560 char t h f i l e n a m e [ BUFSIZ ] ;561 char th FileName [ BUFSIZ ] ;562 /∗wr i t e temp v o l a t i l i z a t i o n matter f l ow sigR and sigT on f i l e ∗/563 s n p r i n t f ( th f i l e name , s izeof ( t h f i l e n a m e ) ,

” th %s %d %s %d %s %d %s %d %s %d %s %d . txt ” , S i z e F i l e , s i z e ,Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU , s i g O f i l e , sigO ,mf i l e ,wbmod) ;

564 t fp=fopen ( th f i l e name , ”a” ) ;565566 for ( i =0; i<=imax ; i++){567 i f ( ( n%3000==0)) {568 f p r i n t f ( t fp , ”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %6.2 l f

%7.2 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn [ i ] , s i g R [ i ] ,s i g T [ i ] ) ;

569 }570 }571 f c l o s e ( t f p ) ;572 /∗wr i t e p r o b a b i l i t y s t a t s on f i l e ∗/573 s n p r i n t f ( th FileName , s izeof ( th FileName ) ,

” th %s %d %s %d %s %d %s %d %s %d %s %d prob . txt ” , S i z e F i l e , s i z e ,Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU , s i g O f i l e , sigO ,mf i l e ,wbmod) ;

574 t f t=fopen ( th FileName , ”a” ) ;575576 for ( i =0; i<=imax ; i++){577 i f ( ( n%3000==0)) {578 f p r i n t f ( t f t , ”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %4.2 l f %d %2d

%2d %6.2 l f %7.2 l f %7.2 l f %11.4 l e %11.4 l e %11.4 l e %7.4 l f%7.4 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,T[ i ] ,vmax ,wbmod, sigU , sigO , t h s i g 1 [ i ] , t h s i g 2 [ i ] ,t h s i g 3 [ i ] , th dtbydr [ i ] , th d2tbydr2 [ i ] , th ds igbydr [ i ] , t h p f [ n ] [ i ] ,t h p f t [ n ] ) ;

579 }580 }581 f c l o s e ( t f t ) ;582 return 0 . 0 ;583 }584 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗585 f unc t i on to c a l c u l a t e normal d i s t r i b u t i o n data586 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/587 double Thermal StdNormalDistr (double x [ ] )588 {589 char th std norm data [ BUFSIZ ] ;


590 s n p r i n t f ( th std norm data , s izeof ( th std norm data ) ,” th Normal Dist Data %s %d %s %d %s %d %s %d %s %d . txt ” , Mfi le ,M,BetaFi le , be ta in t , s i g u f i l e , sigU , s i g O f i l e , sigO , mf i l e ,wbmod) ;

591 t f snd=fopen ( th std norm data , ”a” ) ;592593 int i ;594 double d i f f x [ 1 2 ] , z [ 1 2 ] , sumx=0.0 , avgx ;595 double s u m s q r d i f f x , s q r d i f f x 0 =0.0 , s q r d i f f x [ 1 2 ] , var iance ,

ph i z [ 1 2 ] , ph i x [ 1 2 ] ;596597 for ( i =0; i <=11; i++){598 i f ( i<=2){599 sumx=sumx+0.075∗x [ i ] / 3 . 0 ;600 sumx=sumx ;601 }602 else i f ( i>=3 && i<=4){603 sumx=sumx+0.186∗x [ i ] / 2 . 0 ;604 sumx=sumx ;605 }606 else i f ( i>=5 && i<=6){607 sumx=sumx+0.314∗x [ i ] / 2 . 0 ;608 sumx=sumx ;609 }610 else i f ( i>=7 && i<=9){611 sumx=sumx+0.251∗x [ i ] / 3 . 0 ;612 sumx=sumx ;613 }614 else i f ( i>=10){615 sumx=sumx+0.173∗x [ i ] / 2 . 0 ;616 sumx=sumx ;617 }618 //sumx=sumx ;619 }620 avgx=sumx ;621 // p r i n t f (”% l f \n” , avgx ) ;622 for ( i =0; i <=11; i++){623 d i f f x [ i ]=x [ i ]−avgx ;624 s q r d i f f x [ i ]=pow( d i f f x [ i ] , 2 . 0 ) ;625 s u m s q r d i f f x=s q r d i f f x 0+s q r d i f f x [ i ] ;626 // p r i n t f (”%d %l f %l f %l f \n” , i , d i f f x [ i ] , s q r d i f f x [ i ] ,

s um s q r d i f f x ) ;627 s q r d i f f x 0=s u m s q r d i f f x ;628 }629 for ( i =0; i <=11; i++){630 var iance=s q r t ( s u m s q r d i f f x /(12−1) ) ;631 z [ i ]= d i f f x [ i ] / var iance ;632 // p r i n t f (”%d %l f %l f \n” , i , variance , z [ i ] ) ;633 phi x [ i ]=0.5∗(1 .0+ e r f ( ( x [ i ]−avgx ) / ( var iance ∗ s q r t ( 2 . 0 ) ) ) ) ;634 ph i z [ i ]= trapezium (−4.0 , z [ i ] , 100 , fn ) / s q r t ( 2 . 0∗PI ) ;635636 f p r i n t f ( t f snd , ”%4d %11.4 l e %11.4 l e %11.4 l e %11.4 l e %6.4 l f %8.4 l f

%6.4 l f %6.4 l f %7.4 l f \n” ,


637 s i z e 1 [ i ] , x [ i ] , avgx , var iance ,exp(−pow ( ( x [ i ]−avgx ) / var iance , 2 . 0 ) / 2 . 0 ) /( var iance ∗ s q r t (2∗PI ) ) ,ph i x [ i ] , z [ i ] , ( exp(−pow( z [ i ] , 2 . 0 ) / 2 . 0 ) / s q r t ( 2 . 0∗PI ) ) ,ph i z [ i ] , ( va r i ance /avgx ) ) ;

638 }639 f c l o s e ( t f snd ) ;640 return 0 . 0 ;641 }642 /∗@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@643 @@@@@@ FUNCTIONS RELATED TO VOLATILE STRESS @@@@@644 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@∗/645 /∗ ∗∗∗∗∗∗∗∗∗∗∗646 v o l a t i l e647 ∗∗∗∗∗∗∗∗∗∗∗ ∗/648 double vo l ( )649 {650 c1=(rpor ∗ rpor ∗ e p s i l o n ) / (8 . 0∗ v i s c ∗ tau∗ gscns t ) ;651 c2=rhoc /Mvol ;652 for ( i =0; i<=imax ; i++){653 i f ( i ==0){654 N[ i ] = 0 . 0 ;655 }656 vn [ i ] = v [ i ] + d e l t ∗k0∗exp(−Ea/( g scns t ∗T[ i ] ) ) ∗(vmax−v [ i ] ) ;657 N[ i +1]=(N[ i ]∗pow( r [ i ] , 2 . 0 ) +

0 .5∗ c2 ∗ ( ( vn [ i ]−v [ i ]+vn [ i +1]−v [ i +1]) / d e l t ) ∗pow ( ( r [ i ]+0.5∗ d e l r ) , 2 . 0 ) ∗ d e l r ) /pow( r [ i +1 ] , 2 . 0 ) ;

658 i f ( i ==0){659 r [ i ]=0.1∗ rmax/imax ;660 ValAtCent ( i , vn , r ) ;661 N[ i ] = 0 . 0 ;662 }663 i f ( i==imax ) {664 p [ i ]= p i n f ;665 }666 i f (n==nmax) {667 // p r i n t f (”%3.1 l f %7.2 l f %7.6 f %8.7 f %8.2 l f %12.8 l f %16.8 l f

%12.1 l f \n” , M, Tinf , t [ n ] , r [ i ] , Tn[ i ] , vn [ i ] , N[ i ] , p [ i ] ) ;668 }669 v [ i ]=vn [ i ] ;670 }671 for ( i 1 =0; i1<=imax ; i 1++){672 i=imax−i 1 ;673674 i f ( i ==0){675 r [ i ]=0.1∗ rmax/imax ;676 p [ i−1]=p [ i +1] ;677 }678 /∗Euler ’ s method∗/679 p [ i−1]=p [ i ]−( r [ i−1]−r [ i ] ) ∗N[ i ]∗T[ i ] / ( p [ i ]∗ c1 ) ;680681 /∗//RK 4 method682 k1= −d e l r ∗N[ i ]∗T[ i ] / ( p [ i ]∗ c1 ) ;683 k2= −d e l r ∗N[ i ]∗T[ i ] / ( ( p [ i ]+0.5∗ k1 )∗c1 ) ;


684 k3= −d e l r ∗N[ i ]∗T[ i ] / ( ( p [ i ]+0.5∗ k2 )∗c1 ) ;685 k4= −d e l r ∗N[ i ]∗T[ i ] / ( ( p [ i ]+k2 )∗c1 ) ;686 p [ i−1]=p [ i ]−( k1+2.0∗ k2+2.0∗ k3+k4 ) /6 . 0 ; ∗/687688 i f ( i==imax ) {689 p [ i ]= p i n f ;690 }691 }692 return 0 . 0 ;693 }694 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗695 s ig R v o l a t i l e s t r e s s696 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/697 double sigRR ( int k )698 {699 pp [ k ] = (p [ k]− p i n f ) /1000000 .0 ;700 sig RR [ k ] = s ig R [ k ] + pp [ k ] ;701 return ( sig RR [ k ] ) ;702 }703 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗704 v o l a t i l i z a t i o n p r i n c i p a l s t r e s s Popov pp . 491705 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/706 double VolPr inSt re s s ( )707 {708 for ( i =0; i<=imax ; i++){709 r [ i ]= r0+i ∗ d e l r ;710 i f ( i ==0){711 r [ i−1]=r [ i +1] ;712 r [ i ]=0.1∗ rmax/imax ;713 r l=r [ i ] ;714 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;715 r r =(r [ i +1]+r [ i ] ) / 2 . 0 ;716 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;717 tau xy [ i ]=( v o l l ∗ s ig T [ i−1]+ v o l r ∗ s ig T [ i ] ) / ( v o l l+v o l r ) ;718 }719720 s i g 0 [ i ]=( sigRR ( i )+s ig T [ i ]+ s ig T [ i ] ) / 3 . 0 ;721 sx [ i ] = sigRR ( i ) − s i g 0 [ i ] ;722 sy [ i ] = s ig T [ i ] − s i g 0 [ i ] ;723 sz [ i ] = s ig T [ i ] − s i g 0 [ i ] ;724725 r l =(r [ i−1]+r [ i ] ) / 2 . 0 ;726 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;727 r r =(r [ i ]+ r [ i +1]) / 2 . 0 ;728 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;729730 tau xy [ i ]=( v o l l ∗ s ig T [ i−1]+ v o l r ∗ s ig T [ i ] ) / ( v o l l+v o l r ) ;731732 i f ( i==imax ) {733 r [ i ]=rmax ;734 r [ i +1]=r [ i ] ;735 r r =(r [ i +1]+r [ i ] ) / 2 . 0 ;736 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;


737 r l=r [ i −1] ;738 v o l l =4.0∗PI∗pow( r l , 3 . 0 ) / 3 . 0 ;739 tau xy [ i ]=( v o l l ∗ s ig T [ i−1]+ v o l r ∗ s ig T [ i ] ) / ( v o l l+v o l r ) ;740 }741742 tau yz [ i ]= s ig T [ i ] ;743 tau zx [ i ]= s ig T [ i ] ;744745 J3=−sx [ i ]∗ sy [ i ]∗ sz [ i ] + sx [ i ]∗pow( tau yz [ i ] , 2 . 0 ) +

sy [ i ]∗pow( tau zx [ i ] , 2 . 0 ) + sz [ i ]∗pow( tau xy [ i ] , 2 . 0 ) ;746 J2=sx [ i ]∗ sy [ i ] + sy [ i ]∗ sz [ i ] + sz [ i ]∗ sx [ i ] − pow( tau xy [ i ] , 2 . 0 ) −

pow( tau yz [ i ] , 2 . 0 ) − pow( tau zx [ i ] , 2 . 0 ) ;747748 J22=fabs ( J2 / 3 . 0 ) ;749750 th =(1 .0/3 .0 ) ∗ acos(−J3 / (2 . 0∗pow( J22 , 1 . 5 ) ) ) ;751752 s i g 1 [ i ] = s i g 0 [ i ] + 2 .0∗ s q r t ( J22 ) ∗ cos ( th ) ;753 s i g 2 [ i ] = s i g 0 [ i ] − 2 .0∗ s q r t ( J22 ) ∗ cos ( th+PI / 3 . 0 ) ;754 s i g 3 [ i ] = s i g 0 [ i ] − 2 .0∗ s q r t ( J22 ) ∗ cos ( th−PI / 3 . 0 ) ;755756 i f ( i ==0){757 ValAtCent ( i , s i g 1 , r ) ;758 ValAtCent ( i , s i g 2 , r ) ;759 ValAtCent ( i , s i g 3 , r ) ;760 }761 // p r i n t f (”%7.6 f %8.7 f %l f %l f %l f %l f \n” , t [ n ] , r [ i ] , J2 , J22 , J3 , th ) ;762 i f ( ( n%300)==0)763 // i f (n==nmax)764 {765 // p r i n t f (”%10.4 l e %4.2 l f %11.4 l e %11.4 l e %11.4 l e %8.2 l f , %8.2 l f

%8.2 l f %8.2 l f %8.2 l f %8.2 l f %8.2 l f \n” ,766 // t [ n ] , ( r [ i ] / rmax) , J3 , J22 , th , Tn[ i ] , s i g R [ i ] , sigRR( i ) ,

s i g T [ i ] , s i g 1 [ i ] , s i g 2 [ i ] , s i g 3 [ i ] ) ;767 // p r i n t f (”%3.1 l f %7.2 l f %7.6 f %8.7 f %8.2 l f , %12.8 l f %14.6 l f %11.1 l f

%8.2 l f %8.2 l f %8.2 l f %8.2 l f %8.2 l f %8.2 l f \n” ,M, Tinf , t [ n ] , r [ i ] ,Tn[ i ] , vn [ i ] ,N[ i ] , p [ i ] , s i g R [ i ] , sigRR( i ) ,s i g T [ i ] , s i g 1 [ i ] , s i g 2 [ i ] , s i g 3 [ i ] ) ;

768 }769 }770 return 0 . 0 ;771 }772 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗773 var ious f i r s t order d e r i v a t i v e s774 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/775 double d e r i v a t i v e ( )776 {777 for ( i =0; i<=imax ; i++){778 dtbydr [ i ]=(T[ i ]−T[ i −1]) /( r [ i ]− r [ i −1]) ;779 d2tbydr2 [ i ]=(T[ i −1]−2.0∗T[ i ] + T[ i +1]) / pow ( ( r [ i ]− r [ i −1]) , 2 . 0 ) ;780 ds igbydr [ i ]=( s i g 1 [ i ]− s i g 1 [ i −1]) / ( r [ i ]− r [ i −1]) ;781 dpbydr [ i ]=(p [ i ]−p [ i −1]) /( r [ i ]− r [ i −1]) ;782


783 i f ( i ==0){784 ValAtCent ( i , dtbydr , r ) ;785 ValAtCent ( i , d2tbydr2 , r ) ;786 ValAtCent ( i , ds igbydr , r ) ;787 ValAtCent ( i , dpbydr , r ) ;788 }789 i f (n==0){790 dtbydr [ i ] = 0 . 0 ;791 d2tbydr2 [ i ] = 0 . 0 ;792 ds igbydr [ i ] = 0 . 0 ;793 dpbydr [ i ] = 0 . 0 ;794 }795 i f (n==nmax) {796 // p r i n t f (”%3.1 l f %7.2 l f %g %g %4.2 l f %8.2 l f %11.2 l f %16.2 l f

%12.2 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn[ i ] , d t bydr [ i ] , d2tbydr2 [ i ] , d s i g byd r [ i ] ) ;

797 }798 }799 return 0 . 0 ;800 }801 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗802 f a i l u r e p r o b a b i l i t y803 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/804 double Fai lProb ( )805 {806 char Overal lProb [ BUFSIZ ] ;807 /∗ wr i t e s o v e r a l l f a i l u r e p r o b a b i l i t y data ∗/808 s n p r i n t f ( Overal lProb , s izeof ( Overal lProb ) ,

”%s %d %s %d %s %d %s %d %s %d %s %d %s %d Overal lProb . txt ” ,S i z e F i l e , s i z e , Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU ,s i g O f i l e , sigO , mf i l e ,wbmod, vmaxf i le , v s ta r ) ;

809 op=fopen ( OverallProb , ”a” ) ;810811 pf10 =0.0 ;812 for ( i =0; i<=imax ; i++){813 eta =4.0∗PI∗pow(rmax , 3 . 0 ) / 3 . 0 ;814 i f ( i ==0){815 r [ i−1]=r [ i +1] ;816 r [ i ]=0.1∗ rmax/imax ;817 r r =(r [ i +1]+r [ i ] ) / 2 . 0 ;818 v o l r =4.0∗PI∗pow( r r , 3 . 0 ) / 3 . 0 ;819 pf1 [ i ]=( s i g 1 [ i ]− sigU ) / sigO ;820 pf2 [ i ]= v o l r ∗pow( pf1 [ i ] ,wbmod) ;821 ps [ i ]=exp(−pf2 [ i ] / eta ) ;822 pf [ n ] [ i ]=1.0−ps [ i ] ;823 }824825 v o l r =4.0∗PI∗pow( r [ i ] , 3 . 0 ) / 3 . 0 ;826 v o l l =4.0∗PI∗pow( r [ i −1 ] , 3 . 0 ) / 3 . 0 ;827828 i f ( s i g 1 [ i ]>(1.0∗ sigU ) ) {829 pf1 [ i ]=( s i g 1 [ i ]− sigU ) / sigO ;830 }


831 else {832 pf1 [ i ] = 0 . 0 ;833 }834 pf2 [ i ]=( vo l r−v o l l ) ∗pow( pf1 [ i ] ,wbmod) ;835 ps [ i ]=exp(−pf2 [ i ] / eta ) ;836 pf [ n ] [ i ]=1.0−ps [ i ] ;837838 i f ( i==imax ) {839 v o l r =4.0∗PI∗pow( r [ i ] , 3 . 0 ) / 3 . 0 ;840 v o l l =4.0∗PI∗pow( r [ i −1 ] , 3 . 0 ) / 3 . 0 ;841 pf1 [ i ]=( s i g 1 [ i ]− sigU ) / sigO ;842 pf2 [ i ]=( vo l r−v o l l ) ∗pow( pf1 [ i ] ,wbmod) ;843 ps [ i ]=exp(−pf2 [ i ] / eta ) ;844 pf [ n ] [ i ]=1.0−ps [ i ] ;845 }846 i f ( i ==0){847 pf1 [ i ] = 0 . 0 ;848 pf2 [ i ] = 0 . 0 ;849 ps [ i ] = 0 . 0 ;850 pf [ n ] [ i ] = 0 . 0 ;851 // p f [ n ] [ i ]= p f [ n ] [ i +1]+(r [ i ]−r [ i +1]) ∗( p f [ n ] [ i+1]−p f [ n ] [ i +2]) /

( r [ i+1]−r [ i +2]) ;852 }853 i f (n==0){854 pf [ n ] [ i ] = 0 . 0 ;855 }856 i f ( ( n%300==0)) {857 // p r i n t f (”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %3.1 l f %10.4 l e

%10.4 l e %11.1 l f %6.2 l f %6.2 l f %7.2 l f %d %d %d %6.2 l f %6.2 l f%7.2 l f %11.4 l e %11.4 l e %11.4 l e %7.4 l f \n” ,

858 //M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn[ i ] ,vmax , vn [ i ] ,N[ i ] , p [ i ] , s i g R [ i ] ,sigRR( i ) , s i g T [ i ] , wbmod , sigU , sigO , s i g 1 [ i ] , s i g 2 [ i ] ,s i g 3 [ i ] , d t bydr [ i ] , d2tbydr2 [ i ] , d s i g byd r [ i ] , p f [ n ] [ i ] ) ;

859 }860 pf2 [ i ]= pf10+pf2 [ i ] ;861 pf10=pf2 [ i ] ;862 // pf21=pf2 [ i ] ;863 }864 // p f t [ n]=1−exp(−pf21 / e ta ) ;865 p f t [ n]=1−exp(−pf10 / eta ) ;866 // i f (n==nmax)867 i f ( ( n%3000)==0){868 f p r i n t f ( op , ”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %4.2 l f %d %2d %2d

%7.4 l f %7.4 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) , T[ i −1] ,vmax ,wbmod, sigU , sigO , pf [ n ] [ i ] , p f t [ n ] ) ;

869 // p r i n t f (”%d %7.2 l f %10.4 l e %10.4 l e %8.2 l f %8.2 l f %8.2 l f %8.2 l f %6.4 l f%9.2 l e %9.2 l e \n” , M, Tinf , t [ n ] , r [ i ] ,T[ i −1] , T[ i ] ,T[ i +1] ,Tn[ i ] ,p f t [ n ] , pf10 , e ta ) ;

870 }871 f c l o s e ( op ) ;872 return 0 . 0 ;873 }


874 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗875 v o l a t i l e f r a c t u r e c r i t e r i a876 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/877 double FractCr i t2 ( )878 {879 od=fopen ( ” Overa l l da ta . txt ” , ”a” ) ;880 maxj=−10.0;881 for ( i =1; i<imax ; i++){882 DELpfn=pf [ n ] [ i ]−pf [ n−1] [ i ] ;883 i f ( ( DELpfn>maxj ) && pf [ n ] [ i ]>0.99∗ f p f ) {884 maxj=maximum( maxj , DELpfn) ;885 maxR=r [ i ] / rmax ;886 f t i m e=t [ n ] ;887 f v o l = vn [ i ] ;888 f v o l f l o w = N[ i ] ;889 f v o l p r e s = p [ i ] ;890 f v o l s i g 1 = s i g 1 [ i ] ;891 f temp=Tn[ i ] ;892 break ;893 }894 }895 i f ( i count==0 && maxj>0.0 && pf t [ n]> f p f ) {896 f r a c l o c=maxR;897 // p r i n t f (”%d %7.2 l f %10.8 l f %10.8 l f %4.2 l f %8.2 l f %d %d %d %3.1 l f

%4.2 l f %g %6.4 l f %6.4 l f \n” , M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn[ i ] , wbmod , sigU , sigO , vmax , f r a c l o c , maxj , p f [ n ] [ i ] , p f t [ n ] ) ;

898 // p r i n t f (”FRACTURED: %d %7.2 l f %10.4 l e %8.2 l e %8.2 l f %d %2d %2d%3.1 l f %4.2 l f %10.4 l e \n” , M, Tinf , f t ime , rmax , f temp , wbmod ,sigU , sigO , vmax , f r a c l o c , maxj ) ;

899 f p r i n t f ( od , ”FRACTURED: %d %7.2 l f %12.6 l e %8.2 l e %8.2 l f %2d %d %2d %2d%4.2 l f %4.2 l f %10.4 l e %10.4 l e %11.1 l f %6.2 l f %6.2 l f %7.2 l f %6.2 l f%7.2 l f %7.2 l f %7.4 l f %7.4 l f \n” , M, Tinf , f t ime , rmax , f temp ,be ta in t ,wbmod, sigU , sigO , vmax , f r a c l o c , vn [ i ] ,N[ i ] , p [ i ] ,s i g R [ i ] , sigRR ( i ) , s i g T [ i ] , s i g 1 [ i ] , s i g 2 [ i ] , s i g 3 [ i ] , p f [ n ] [ i ] ,p f t [ n ] ) ;

900901 f t i m e a r r a y [ j j ]= f t i m e ;902 f t emp array [ j j ]= f temp ;903 f v o l a r r a y [ j j ]= f v o l ;904 f v o l f l o w a r r a y [ j j ]= f v o l f l o w ;905 f v o l p r e s a r r a y [ j j ]= f v o l p r e s ;906 f v o l s i g 1 a r r a y [ j j ]= f v o l s i g 1 ;907908 p r i n t f ( ”%2d %12.6 l e %7.2 l f %12.6 l e %12.6 l e %12.6 l e %6.2 l f \n” , j j ,

f t i m e a r r a y [ j j ] , f t emp array [ j j ] , f v o l a r r a y [ j j ] ,f v o l f l o w a r r a y [ j j ] , f v o l p r e s a r r a y [ j j ] , f v o l s i g 1 a r r a y [ j j ] ) ;

909 j j ++;910911 i count =1;912 }913 f c l o s e ( od ) ;914 return 0 . 0 ;915 }


916 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗917 s epara t e func t i on to p r i n t f i l e s918 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/919 double F i l e P r i n t ( )920 {921 char f i l e n a m e [ BUFSIZ ] ;922 char FileName [ BUFSIZ ] ;923 /∗wr i t e temp v o l a t i l i z a t i o n matter f l ow sigR

and sigT on f i l e ∗/924 s n p r i n t f ( f i l e name , s izeof ( f i l e n a m e ) ,

”%s %d %s %d %s %d %s %d %s %d %s %d %s %d . txt ” , S i z e F i l e , s i z e ,Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU , s i g O f i l e , sigO ,mf i l e ,wbmod, vmaxf i le , v s ta r ) ;

925 fp=fopen ( f i l e name , ”a” ) ;926927 for ( i =0; i<=imax ; i++){928 i f ( ( n%3000==0)) {929 f p r i n t f ( fp , ”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %3.1 l f %10.4 l e

%10.4 l e %11.1 l f %6.2 l f %6.2 l f %7.2 l f \n” ,M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,Tn [ i ] ,vmax , vn [ i ] ,N[ i ] , p [ i ] , s i g R [ i ] , sigRR ( i ) , s i g T [ i ] ) ;

930 }931 }932 f c l o s e ( fp ) ;933 /∗wr i t e p r o b a b i l i t y s t a t s on f i l e ∗/934 s n p r i n t f ( FileName , s izeof ( FileName ) ,

”%s %d %s %d %s %d %s %d %s %d %s %d %s %d prob . txt ” , S i z e F i l e , s i z e ,Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU , s i g O f i l e , sigO ,mf i l e ,wbmod, vmaxf i le , v s ta r ) ;

935 f t=fopen ( FileName , ”a” ) ;936937 for ( i =0; i<=imax ; i++){938 i f ( ( n%3000==0)) {939 f p r i n t f ( f t , ”%d %7.2 l f %10.4 l e %8.2 l e %4.2 l f %8.2 l f %4.2 l f %d %2d

%2d %6.2 l f %6.2 l f %7.2 l f %6.2 l f %6.2 l f %7.2 l f %11.4 l e %11.4 l e%11.4 l e %7.4 l f %7.4 l f %11.4 l e \n” ,M, Tinf , t [ n ] , rmax , ( r [ i ] / rmax) ,T[ i ] ,vmax ,wbmod, sigU , sigO , t h s i g 1 [ i ] , t h s i g 2 [ i ] ,t h s i g 3 [ i ] , s i g 1 [ i ] , s i g 2 [ i ] ,s i g 3 [ i ] , dtbydr [ i ] , d2tbydr2 [ i ] , ds igbydr [ i ] , p f [ n ] [ i ] ,p f t [ n ] , dpbydr [ i ] ) ;

940 }941 }942 f c l o s e ( f t ) ;943 return 0 . 0 ;944 }945 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗946 f unc t i on to c a l c u l a t e normal d i s t r i b u t i o n data947 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/948 double StdNormalDistr (double x [ ] )949 {950 char std norm data [ BUFSIZ ] ;


951 s n p r i n t f ( std norm data , s izeof ( std norm data ) ,” Normal Dist Data %s %d %s %d %s %d %s %d %s %d %s %d . txt ” ,Mfi le ,M, BetaFi le , be ta in t , s i g u f i l e , sigU , s i g O f i l e , sigO ,mf i l e ,wbmod, vmaxf i le , v s ta r ) ;

952 fsnd=fopen ( std norm data , ”a” ) ;953954 int i ;955 double d i f f x [ 1 2 ] , z [ 1 2 ] , sumx=0.0 , avgx ;956 double s u m s q r d i f f x , s q r d i f f x 0 =0.0 , s q r d i f f x [ 1 2 ] , var iance ,

ph i z [ 1 2 ] , ph i x [ 1 2 ] ;957958 for ( i =0; i <=11; i++){959 i f ( i<=2){960 sumx=sumx+0.075∗x [ i ] / 3 . 0 ;961 sumx=sumx ;962 }963 else i f ( i>=3 && i<=4){964 sumx=sumx+0.186∗x [ i ] / 2 . 0 ;965 sumx=sumx ;966 }967 else i f ( i>=5 && i<=6){968 sumx=sumx+0.314∗x [ i ] / 2 . 0 ;969 sumx=sumx ;970 }971 else i f ( i>=7 && i<=9){972 sumx=sumx+0.251∗x [ i ] / 3 . 0 ;973 sumx=sumx ;974 }975 else i f ( i>=10){976 sumx=sumx+0.173∗x [ i ] / 2 . 0 ;977 sumx=sumx ;978 }979 //sumx=sumx ;980 }981 avgx=sumx ;982 // p r i n t f (”% l f \n” , avgx ) ;983 for ( i =0; i <=11; i++){984 d i f f x [ i ]=x [ i ]−avgx ;985 s q r d i f f x [ i ]=pow( d i f f x [ i ] , 2 . 0 ) ;986 s u m s q r d i f f x=s q r d i f f x 0+s q r d i f f x [ i ] ;987 // p r i n t f (”%d %l f %l f %l f \n” , i , d i f f x [ i ] , s q r d i f f x [ i ] ,

s um s q r d i f f x ) ;988 s q r d i f f x 0=s u m s q r d i f f x ;989 }990 for ( i =0; i <=11; i++){991 var iance=s q r t ( s u m s q r d i f f x /(12−1) ) ;992 z [ i ]= d i f f x [ i ] / var iance ;993 // p r i n t f (”%d %l f %l f \n” , i , variance , z [ i ] ) ;994 /∗ CDF can a l s o be wr i t t en in the form of error func t i on .995 Here f o r v a r i a b l e x i t i s in form of error func t i on and996 f o r v a r i a b l e z (SNRV) i t i s c a l c u l a t e d from in t e g r a t i o n ∗/997 phi x [ i ]=0.5∗(1 .0+ e r f ( ( x [ i ]−avgx ) / ( var iance ∗ s q r t ( 2 . 0 ) ) ) ) ;998 ph i z [ i ]= trapezium (−4.0 , z [ i ] , 100 , fn ) / s q r t ( 2 . 0∗PI ) ;


9991000 f p r i n t f ( fsnd , ”%4d %11.4 l e %11.4 l e %11.4 l e %11.4 l e %6.4 l f %8.4 l f

%6.4 l f %6.4 l f %7.4 l f \n” ,1001 s i z e 1 [ i ] , x [ i ] , avgx , var iance ,

exp(−pow ( ( x [ i ]−avgx ) / var iance , 2 . 0 ) / 2 . 0 ) /( var iance ∗ s q r t (2∗PI ) ) ,ph i x [ i ] , z [ i ] , ( exp(−pow( z [ i ] , 2 . 0 ) / 2 . 0 ) / s q r t ( 2 . 0∗PI ) ) ,ph i z [ i ] , ( va r i ance /avgx ) ) ;

1002 }1003 f c l o s e ( f snd ) ;1004 return 0 . 0 ;1005 }1006 /∗@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@1007 @@@@@@ COMMON FUNCTIONS @@@@@1008 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@∗/1009 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗1010 f unc t i on to c a l c u l a t e va l u e s at cente1011 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/1012 double ValAtCent ( int j , double s s [ ] , double Tt [ ] )1013 {1014 s s [ j ] = s s [ j +1] + (Tt [ j ]−Tt [ j +1]) ∗ ( s s [ j +1]− s s [ j +2]) /(Tt [ j +1]−Tt [ j +2]) ;1015 return s s [ j ] ;1016 }1017 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗1018 f unc t i on to c a l c u l a t e maximu o f two1019 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/1020 double maximum(double aa , double bb)1021 {1022 double zz ;1023 i f ( ( aa−bb)>0){1024 zz=aa ;1025 }1026 else {1027 zz=bb ;1028 }1029 return zz ;1030 }1031 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗1032 normal d i s t r i b u t i o n func t i on1033 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/1034 double fn (double xxx )1035 {1036 return ( exp(−pow( xxx , 2 . 0 ) / 2 . 0 ) ) ;1037 }1038 /∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗1039 t r a p e z o i d a l i n t e g r a t o i n1040 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗/1041 double trapezium (double a , double b , int step , double (∗ f ) (double x ) )1042 {1043 int i ;1044 double h , J0 , Jnew , J , f ab , x ;1045 h=(b−a ) / s tep ;1046 J0 =0.0 ;1047 i =0;


1048 x=a+i ∗h ;10491050 (∗ f ) ( x ) ;1051 f ab=h∗( f ( a )+f (b) ) / 2 . 0 ;10521053 for ( i =1; i<s tep ; i++){1054 x=a+i ∗h ;1055 J=h∗ f ( x ) ;1056 Jnew=J0+J ;1057 // p r i n t f (”%d %l f %l f %l f %l f \n” , i , x , J , Jnew , J0 ) ;1058 J0=Jnew ;1059 }1060 // p r i n t f (”% l f %l f ” , Jnew , f a b ) ;1061 Jnew=Jnew+f ab ;1062 // p r i n t f (”% l f \n” , Jnew) ;1063 return ( Jnew) ;1064 }

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