Explosion Dynamics E-Book

Ali S. Rangwala

• Herausgeber: Wiley-VCH GmbH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 466

Veröffentlichungsjahr: 2023

Beschreibung

Explosion Dynamics

Structured and comprehensive introductory guide to understanding and applying explosion dynamics concepts

Explosion Dynamics thoroughly explores the physical phenomena of explosions and enables readers to understand controlling variables that govern temperature, pressure, and rate of increase in pressure respectively, while also providing a mathematical framework for characterizing and applying key concepts.

To promote seamless reader comprehension, this comprehensive textbook provides working examples, case studies, and assignments for self-study, as well as additional material such as property data for common gases and dusts, which supports the examples presented throughout the text.

Written by two highly qualified authors, topics covered in Explosion Dynamics include:

• Similitude theory, similarity solutions, nonlinear systems of differential equations, gas dynamics, and chemical kinetics
• How a flammable mixture of gas or vapor or a suspension of powder, dust particles, or droplets forms in the industrial processing of hazardous materials
• Range of temperature, pressure, and concentration in which a flame can ignite and propagate
• How the “rate-of-pressure-rise” affects the overall explosion hazard and the viability of various explosion protection measures

Providing a structured and comprehensive approach to the subject, Explosion Dynamics is an indispensable textbook that allows chemistry and engineering students, along with professional engineers and professionals in the chemical and food industries, to understand the fundamental mathematics and physics involved in explosions and develop appropriate protection and prevention measures.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

1 Introduction

1.1 What Is an Explosion? Types of Explosions Covered in this Book

1.2 Controlling Parameters of a Combustible Gas/Vapor Explosion Hazard

1.3 Flame Propagation

1.4 Mixture Concentration – Definition of Flammability Limits

1.5 Minimum Ignition Energy (MIE) and Auto Ignition Temperature (AIT)

Exercise Problems

Nomenclature

Greek Symbols

Subscripts

Other Notations

References

Notes

2 Synopses of Explosion Incidents

2.1 Hydrogen Cylinder Trailer Module Explosion

2.2 Nylon Flock Dust Explosion

2.3 Flammable Vapor Explosion at Ink and Paint Manufacturing Plant

2.4 Jahn Foundry Dust Explosion

2.5 Upper Big Branch Coal Mine Explosion

Nomenclature

Subscripts

Other Notations

References

3 Explosion in a Closed Vessel

3.1 Introduction

3.2 The Movement of a Flame in a Premixed Gas–Air Mixture

3.3 Explosion Pressure Rise vs. Time in a Confined Vessel (Theory)

3.4 The Closed Vessel as an Experimental Platform for Standard Testing

3.5 Influence of Flame Wrinkling, Turbulence, and Dust on Flame Propagation

Nomenclature

Greek Symbols

Subscripts

Other Notations

References

Notes

4 Explosion in a Vented Vessel

4.1 Introduction: How Does Pressure Develop in a Vented Vessel?

4.2 Explosion in a Vented Vessel Neglecting Transient Effects

4.3 Pressure Generation in a Vented Vessel with Transient Effects

4.4 Flame Instabilities

4.5 Flame Front Turbulence and the Concept of Turbulent Burning Velocity

S

T

4.6 Pressures Generated in Vented Vessels – Experiments

4.7 Modeling Pressure Generated as a Function of Time in a Vented Vessel

4.8 Pressure Developed Outside the Enclosure

4.9 Vent Design in Engineering Codes and Standards

Nomenclature

Greek Symbols

Subscripts

Superscript

Other Notations

References

Notes

5 Accumulation of a Flammable Mixture in an Enclosure

5.1 Introduction

5.2 Gas Filling in an Enclosure with

Forced

Ventilation

5.3 Gas Filling in an Enclosure with

Passive

Ventilation

5.4 Criteria for Mixture Uniformity

5.5 Concentration Buildup in an Enclosure by a Liquid Spill

5.6 Concentration Buildup in an Enclosure due to Dust

Nomenclature

Greek Symbols

Subscripts

Other Notations

References

Notes

6 Dimensionless Analysis

6.1 Introduction

6.2 Dimensional and Nondimensional Quantities

6.3 The Buckingham Pi Theorem

6.4 Procedure for Obtaining Pi Terms

Nomenclature

Greek Symbols

Subscripts

References

Notes

7 Vapor Cloud Explosions

7.1 Introduction

7.2 Shape of Overpressure Curves: Pressure Wave, Shock Wave, and Blast Wave

7.3 The Classical Model of Pressure Developed by a Spherically Expanding Flame

7.4 TNT Equivalent Model

7.5 The Multienergy Models

7.6 Baker–Strehlow–Tang Model

7.7 TNO Model

7.8 The Williams Model

7.9 Computational Fluid Dynamics (CFD) Modeling of VCE

7.10 Summary

Nomenclature

Greek Symbols

Subscripts

Other Notations

References

Notes

8 Dust Flames and Dust Explosions

8.1 Introduction

8.2 Elements of a Dust Explosion

8.3 Flame Structure – What Is a Dust Flame?

8.4 Dust Explosion Test Platforms

8.5 Powder and Dust Processing Equipment

8.6 Dust Hazard Analyses

8.7 Dust Explosion Venting

Exercise Problems

Nomenclature

Greek Symbols

Subscripts

Other Notations

References

Notes

9 Other Explosion Protection Methods

9.1 Introduction

9.2 Gas/Vapor Concentration Dilution

9.3 Inerting

9.4 Explosion Suppression Systems

9.5 Isolation

Nomenclature

Greek Symbols

Subscripts

Other Notations

References

Note

Appendix A: A Review of Chemistry and Thermodynamics

A.1 Mole fraction (

X

i

) and a Mass fraction (

Y

i

)

A.2 Stoichiometry

A.3 Combustion Chemistry

A.4 Pressure

A.5 Energy Terms and Adiabatic Flame Temperature

A.6 Equivalence Ratio

A.7 Heat and Heat Capacity

A.8 Entropy and Isentropic Process

Exercises

References

Notes

Appendix B: Mathematica Code for Solved Examples

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Flammability properties of some common gas air mixtures in air.

Chapter 2

Table 2.1 Calculated heptane vapor concentrations in building.

Table 2.2 Jahn Foundry incident dust explosibility parameters [10, 11].

Table 2.3 UBB mine coal proximate analysis.

Chapter 3

Table 3.1 Laminar burning velocity, peak pressure, and temperature for mixtu...

Table 3.2 Operating conditions for 20, 36, and 1000 l vessels. The reservoir...

Chapter 4

Table 4.1 Vented gas explosions in large enclosures (more than 1 m

3

).

Table 4.2 Burning velocity expressions.

Chapter 5

Table 5.1 Properties of pentane.

Table 5.2 Temperature of surface.

Table 5.3 Time to evaporate pentane layer.

Chapter 6

Table 6.1 Fundamental dimensions.

Table 6.2 Physical parameters controlling the blast wave pressure decay from...

Chapter 7

Table 7.1 Pressure damage thresholds for buildings.

Table 7.2 TNT equivalency for several hydrocarbons in industry.

E

TNT

 = 4187 ...

Table 7.3 Flame speed mach number selection table (

M

w

).

Table 7.4 Selection of blast strength (1–10) when solving VCE problems using...

Table 7.5 Summary of results of side on pressure and duration of pressure pu...

Table 7.6 Summarizes the results from the graphical solution using the Baker...

Chapter 8

Table 8.1 Stoichiometric combustion of some common dust and calculation of t...

Table 8.2 Heat of combustion of substances per mole of O

2

consumed.

Table 8.3 MIE data example.

Table 8.4 Dust ignition temperature tests.

Table 8.5 Comparison of ignition temperature data.

Table 8.6 Parameters involved in a dust explosion hazard analysis.

Chapter 9

Table 9.1 Chao and Dorofeev model empirical parameter values.

List of Illustrations

Chapter 1

Figure 1.1 Classification of different types of explosions based on peak pre...

Figure 1.2 Aerial view of damage and debris caused by a black powder explosi...

Figure 1.3 Comparison of pressure profiles between a detonation and a confin...

Figure 1.4 Conceptual flow of events in a building explosion,

P

,

T

,

V

, and

n

Figure 1.5 Vapor cloud explosion (VCE) at Buncefield, UK, on 11 December 200...

Figure 1.6 The evolution of an 8% ethylene–air flame in a cubic vessel of vo...

Figure 1.7 Flame propagation in: (a) spherical freely propagating flame (lam...

Figure 1.8 Planar, curved, and wrinkled flame. The area of the flame increas...

Figure 1.9 (a) Laminar burning velocity (cm/s) vs. pressure (atm) and (b) La...

Figure 1.10 Snapshots of GASEQ,

Figure 1.11 Snapshot of GASEQ,

Figure 1.12 Flame propagation in a tube with (a) open ends, and (b) nozzle e...

Figure 1.13 Burned and unburned gas velocities with the coordinate system on...

Figure 1.14 Laminar burning velocity of methane–air premixed flame at differ...

Figure 1.15 ASTM standard test method for measuring upper and lower flammabi...

Figure 1.16 ASTM standard test method for measuring minimum ignition energy....

Chapter 2

Figure 2.1 A photograph of the hydrogen cylinder trailer module.

Figure 2.2 Damaged trailer module and cab.

Figure 2.3 Malden Mills Monomac Building first floor layout.

Figure 2.4 Electrostatic flocking process.

Figure 2.5 Monomac Building Fire after the collapse of upper stories.

Figure 2.6 Remains of Monomac Building.

Figure 2.7 Danvers manufacturing facility layout [8].

Figure 2.8 Large mixing tank 3 at the Danvers facility [8].

Figure 2.9 Aerial view of damage to facility and surrounding area.

Figure 2.10 Remains of a portion of production building.

Figure 2.11 Jahn Foundry shell mold building layout.

Figure 2.12 Jahn Foundry shell mold building fume exhaust system.

Figure 2.13 Shell mold station flexible ducts.

Figure 2.14 Shell mold building damage viewed from SE.

Figure 2.15 Resin dust burn residue of fresh air duct and hood tops.

Figure 2.16 Flaking and agglomerated duct burn residue on left and ash resid...

Figure 2.17 Upper big branch coal mine area of explosion initiation.

Figure 2.18 Coal mining longwall shearer.

Figure 2.19 Tail drum of the longwall shearer.

Figure 2.20 Location of shearer and mine floor cracks.

Figure 2.21 MSHA coke content sample map; circle = extra large and large, cr...

Figure 2.22 Extent of flame propagation in UBB mine.

Figure 2.23 Rock dust percentage required to inert low volatility and high v...

Chapter 3

Figure 3.1 Adiabatic constant volume explosion: (a) Temperature vs. vol% for...

Figure 3.2 Snapshot of GASEQ code used to solve Example 3.1.

Figure 3.3 Propagation of a spherical deflagration. Fuel‐rich, H

2

–air mixtur...

Figure 3.4 Propagation of a flame in a confined vessel and corresponding pre...

Figure 3.5 Screenshot of GASEQ showing constant volume explosion pressure fo...

Figure 3.6 Pressure rise vs. time for a confined explosion of tetramethylsil...

Figure 3.7 Pressure rise vs. time for a flame propagating in a cylindrical v...

Figure 3.8 Flame propagating in a tube (a) idealized planar flame, (b) nonpl...

Figure 3.9 20 l explosion sphere as an experimental bench‐scale platform for...

Figure 3.10 One of the authors standing next to a 1 m

3

explosion sphere at F...

Figure 3.11 Influence of turbulence on the pressure rise vs. time in a 20 l ...

Figure 3.12 The maximum pressure

P

max

and

Figure 3.13 Schematic of turbulent flame propagation in a closed vessel. As ...

Figure 3.14 Schematic of three different types of particle‐flow interactions...

Chapter 4

Figure 4.1 Schematic representation of deflagration in a vented vessel. With...

Figure 4.2 Explosion in a cylindrical vessel with a vent. Methane–air mixtur...

Figure 4.3 Pressure variation in a vented enclosure during an explosion.

Figure 4.4 Example of RT instability – rapid flame acceleration in a vented ...

Figure 4.5 Pressure vs. time for rear, central, and front ignition for 5.25 ...

Figure 4.6 Wrinkling of flame because of an obstacle.

Figure 4.7 Influence of ignition location on vented pressure.

Figure 4.8 Influence of vent size on explosion pressure.

Figure 4.9 Vented explosion pressure in dust explosions. Observe the lack of...

Figure 4.10 Turbulent intensity variation vs. time for two vessels because o...

Figure 4.11 Deflagration index,

K

st

(bar m/s) for a large‐scale vented dust ...

Figure 4.12 Schematic representation of a spherical vessel of volume

V

with ...

Figure 4.13 Snapshot of GASEQ for adiabatic constant volume explosion of hyd...

Figure 4.14 Plot of pressure (kPa) vs. time (s).

Figure 4.15 (a) Vented mass,

m

v

(kg), burned mass,

m

b

(kg) vs. time in secon...

Figure 4.16 Pressure development outside because of an external explosion pr...

Figure 4.17 Overpressure because of an external explosion.

Figure 4.18 Low obstacle density configuration in

blast and fire engineering

...

Figure 4.19 High obstacle density configuration in BFETS tests (25.6 m × 8 m...

Figure 4.20 Piping obstacle array used in Baker Risk vented explosion tests....

Figure 4.21 Explosion vent panels on a dust collector.

Figure 4.22 Deployed explosion vent panel on a wall.

Chapter 5

Figure 5.1 Leaking of a gas in an enclosure of volume

V

(m

3

). Fresh air ente...

Figure 5.2 (a) Mole fraction of CO in the enclosure vs. time of CO for ...

Figure 5.3 Flow velocity through a vent with passive ventilation because of ...

Figure 5.4 Enclosure with a hydrogen gas leak with a flow rate of 3/s) a...

Figure 5.5 (a) Mole fraction of H

2

vs. leakage rate. (b) Neutral plane heigh...

Figure 5.6 Sketch of problem.

Figure 5.7 Time for evaporation of pentane layer.

Figure 5.8 Explosive hazard estimation in two compartments of the same volum...

Figure 5.9 Dimensions of the dust pile.

Chapter 6

Figure 6.1 Displacement of a mass

M

o

because of a force

F

(

t

).

Figure 6.2 Trinity atomic explosion (1945).

Chapter 7

Figure 7.1 Deflagration, turbulent deflagration, and detonation in a vapor c...

Figure 7.2 Shapes of shock and pressure waves.

P

s

denotes the side on pressu...

Figure 7.3 Pressure profile generated by a flame propagating through a vapor...

Figure 7.4 Pressure difference

P

o

–

P

B

observed in 1D, 2D, and 3D flame prop...

Figure 7.5 Side on overpressure (atm) vs. distance (m) from an explosion wit...

Figure 7.6 Baker Strehlow Tang (BST) peak side on overpressure vs. nondimens...

Figure 7.7 Baker Strehlow Tang (BST) nondimensional impulse vs. nondimension...

Figure 7.8 TNO curves for nondimensional side on overpressure vs. scaled dis...

Figure 7.9 TNO curves for nondimensional duration of impulse vs. scaled dist...

Figure 7.10 Facility with a vapor cloud.

Figure 7.11 Top and front views of cylinder bank zone 1 is the volume of acc...

Figure 7.12 Top and front views of the spherical storage tanks. Zone 2 is th...

Figure 7.13 TNO curves for nondimensional side on overpressure vs. scaled di...

Figure 7.14 TNO curves for nondimensional duration of impulse vs. scaled dis...

Figure 7.15 Baker Strehlow Tang graphical solution. Nondimensional side on p...

Figure 7.16 Nondimensional impulse vs. scaled distance and .

Figure 7.17 (a) Torn balloon approximately one second before ignition and (b...

Figure 7.18 Williams model for VCE.

Figure 7.19 Vapor cloud explosion in a process unit.

Figure 7.20 Multiple building VCE shielding effects as calculated with a VCE...

Chapter 8

Figure 8.1 Elements of a dust explosion.

Figure 8.2 Three different types of dust encountered in flames: (a) combusti...

Figure 8.3 Dust flames stabilized at the mouth of a tube. The particle sizes...

Figure 8.4 Schematic of a dust flame propagation showing controlling zones o...

Figure 8.5 Schematic illustration of the structure of premixed dust–air flam...

Figure 8.6 Dust preburn and postburn microscopic images. Ranganathan [4] Wor...

Figure 8.7 Experimental setup – Hybrid Flame Analyzer (HFA).

Figure 8.8 Influence of turbulent intensity on the flame shape and burning v...

Figure 8.9 Pilot flame ignition and flame anchoring system in the HFA. A per...

Figure 8.10 Laminar and turbulent iron, coal, and sand dust burner flames. T...

Figure 8.11 Influence of inert sand particles on a laminar premixed flame. F...

Figure 8.12 (a) Radiative fraction of the total heat release rate. Compariso...

Figure 8.13 Comparison of

P

max

and

K

st

values from 1 m

3

and 20‐l spheres....

Figure 8.14 Comparison of 20‐l sphere and 1 m

3

vessel

K

st

values.

Figure 8.15 Max pressure and (

dP

/

dt

)

max

V

1/3

data for two iron dust samples ...

Figure 8.16 Vertical tube MIE test apparatus, also called Hartmann tube.

Figure 8.17 MIKE 3 MIE test apparatus.

Figure 8.18 Distribution of MIE values reported by Bartknecht [29] as compil...

Figure 8.19 Test apparatus for ignition of combustible dust layers: (a) hot ...

Figure 8.20 Flaming ignition in dust layer hot surface ignition test.

Figure 8.21 Godbert–Greenwald vertical furnace for dust cloud ignition tests...

Figure 8.22 BAM horizontal oven for dust cloud ignition tests.

Figure 8.23 Experimental apparatus for dust deflagration analysis by Dobashi...

Figure 8.24 Hammermill schematic diagram.

Figure 8.25 Ball mill schematic diagram.

Figure 8.26 Rotary drum schematic.

Figure 8.27 Spray dryer schematic diagram.

Figure 8.28 Calculated particle concentrations along cyclone wall.

Figure 8.29 Small baghouse dust collector.

Figure 8.30 Small cartridge dust collector.

Figure 8.31 FM Global vent dust explosion venting correlation.

Figure 8.32 Calculated vs. actual experimental vent area for indicated VDI 3...

Figure 8.33 Cylindrical flameless explosion vent construction.

Figure 8.34 Photograph of a flameless explosion vent.

Figure 8.35 Vented dust explosion with and without flameless vent.

Chapter 9

Figure 9.1 Explosion suppression system schematic.

Figure 9.2 Radial positions of the flame front and suppressant agent vs. tim...

Figure 9.3 Experimental setup for a deflagration suppression experiment in a...

Figure 9.4 Pressure development during a suppressed dust collector explosion...

Figure 9.5 Influence of activation pressure on the maximum pressure (

P

red

) i...

Figure 9.6 Schematic of an isolation system.

Figure 9.7 Schematic of a flap valve. (a) Normal operating condition. (b) Cl...

Figure 9.8 Rotary valve.

Figure 9.9 Maximum allowable gap width to prevent deflagration propagation....

Figure 9.10 Diverter valve.

Figure 9.11 Combined explosion suppression and isolation system.

Figure 9.12 Dust collector explosion suppression and isolation system.

Figure 9.13 Calculated flame development for a stoichiometric propane–air ex...

Figure 9.14 Minimum required distance to isolation device vs...

Figure 9.15 EN 15089 testing arrangement for active isolation systems [22]. ...

Figure 9.16 End‐of‐line deflagration flame arrestor installed on the vent li...

Figure 9.17 Polymer foam flame arrestor before (right) and after (left) flam...

Guide

Cover

Title Page

Copyright

Preface

Table of Contents

Begin Reading

Appendix A: A Review of Chemistry and Thermodynamics

Appendix B: Mathematica Code for Solved Examples

Index

End User License Agreement

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Explosion Dynamics

Fundamentals and Practical Applications

Ali S. Rangwala and Robert G. Zalosh

Authors

Prof. Ali S. RangwalaWorcester Polytechnic InstituteDepartment of Fire Protection Engin.Gateway Park II 1210Worcester, MA 01609USA

Prof. Robert G. ZaloshFirexplo20 Rockland St.Wellesley, MA 02481USA

Cover Image: Courtesy of Ali S. Rangwala

All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing‐in‐Publication DataA catalogue record for this book is available from the British Library.

Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.

© 2023 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Print ISBN: 978‐3‐527‐34938‐8ePDF ISBN: 978‐3‐527‐83337‐5ePub ISBN: 978‐3‐527‐83336‐8oBook ISBN: 978‐3‐527‐84462‐3

Typesetting Straive, Chennai, India

Preface

This book focuses primarily on explosions due to the combustion of flammable gas–air mixtures and combustible dust clouds. It evolved from the lecture materials developed for a graduate course offered to fire protection engineering and chemical engineering students at Worcester Polytechnic Institute. These students have an understanding of Thermodynamics, Chemistry, Heat and Mass Transfer, Fluid Flow, and Ordinary Differential Equations commensurate with an undergraduate chemical, mechanical, or fire protection engineering curriculum.

Besides serving as a textbook in an academic program, the book is intended to provide guidance to practicing engineers working on various industrial explosion protection applications. Although there are already several excellent books on dust explosions and others on gas explosions, this book should serve as a resource for safety professionals seeking one good resource for both types of explosions as well as explosion protection and blast wave hazard evaluations.

Some generic questions that are answered in an explosion dynamics context in this book are:

How does a flammable mixture of gas or vapor or a suspension of powder or dust particles or droplets form in the industrial processing of these materials?

What are gas or dust cloud limits of ignitability, or in other words, what is the range of temperature, pressure, and concentration in which a flame can ignite and propagate?

What is the relationship between the flame propagation rate and the associated explosion pressure, and how is it influenced by the combustibility properties of the gas or dust cloud?

How does the “rate‐of‐pressure‐rise” affect the overall explosion hazard and the viability of various explosion protection measures?

How does pressure development within the flammable gas or combustible dust cloud relate to the blast wave pressures propagating away from the cloud and away from the equipment in which the explosion originated?

This book explains the physical and thermochemical phenomena pertinent to these questions and provides a mathematical framework for characterizing and applying the answers. Examples, questions, and solutions are offered in each chapter, and there are also brief summaries of how these phenomena influenced what happened in some industrial explosion incidents. The authors hope that readers find the phenomenological explanations and examples interesting in the way that many people are inherently attracted and energized by a good story about an explosion.

We are grateful to Dr. Sharanya Nair for developing the solved examples and drawing many of the figures. We thank Hsin‐Hsu (Matt) Ho for his help with digitizing the graphs in Chapter 7. We would like to acknowledge the valuable contributions of Dr. Scott Rockwell, whose PhD research at the Combustion Laboratory at Worcester Polytechnic Institute provided the captivating front cover photos. We also extend our gratitude to Brian Elias and Dr. Li Chang for their efforts in creating the content featured in the Annex section.

14 October 2022

Ali S. Rangwala and Robert G. Zalosh

Worcester, MA

1Introduction

1.1 What Is an Explosion? Types of Explosions Covered in this Book

To introduce the concept of explosion protection, one must first understand what is an “explosion?” The dictionary definition of an explosion is “the action of going off with a loud noise or of bursting under the influence of a suddenly developed internal energy.” A more relevant definition related to the scientific study of the problem is the release of energy to generate a pressure wave of finite amplitude traveling away from the source. This energy may have been stored in various forms such as nuclear, chemical, electrical, or pressure energy[1]. The release of energy is not considered explosive unless it is rapid and concentrated enough to produce a pressure wave that one can hear. Even though many explosions damage their surroundings, an explosion doesn't need to create external damage.

Explosions can occur in any media, such as air or condensed phases like liquid or solids. In all cases, the critical aspect is the generation of energy and pressure, which is released in a short time. The magnitude of energy release and its rate of release thus constitute the basis of the classification of different types of explosions. Zalosh [2] describes this using a peak pressure generated vs. a time scale for energy release, as shown in Figure 1.1. The peak pressure is directly related to the total amount of energy1, and the time scale is a result of the spatial scale and reaction rate or the speed with which the energy is released during the explosion. For example, when dynamite is ignited, the chemical reaction front proceeds through the solid at a speed of 4900 m/s. Thus, a 50 cm (0.5 m) stick would release all of its energy in 0.5/4900 = 102 μs. For a gas detonation explosion, typical detonation velocities are in the range of 1500–4000 m/s. For example, stoichiometric acetylene (C2H2)–air mixture's detonation velocity can be calculated from a chemical equilibrium code [3] and equal to 1868 m/s. Thus, in this case, the energy release in a 0.5 m radius would occur in 0.5/1868 = 268 μs. The corresponding energy released would be the heat of combustion of acetylene in air (48.22 kJ/g) times its density (1.2 kg/m3) times the volume () given by 30.3 MJ! The corresponding pressure is equal to 18.2 atm (267.5 psig).

Figure 1.1 Classification of different types of explosions based on peak pressure and time to different kinds of energy release.

Source: Zalosh [2].

Explosions can be either deflagrations or detonations, depending on whether the speed of the chemical reaction front propagating through the combustible mixture is less than or greater than sound speed in the unburned fuel–air mixture. (Sound speed is approximately equal to 347 m/s if the fuel concentration is small compared to the air concentration.2) As shown in Figure 1.1, the peak pressures generated in detonations are at least twice as large as those in deflagrations, and the time scale is often at least an order of magnitude smaller. To begin, let us briefly describe the different types of explosions shown in Figure 1.1 to understand the significance of peak pressure and time for energy release.

1.1.1 Nuclear Explosions

As shown in Figure 1.1, nuclear explosions release the most amount of energy per unit volume. Therefore, they generate the highest pressure on the top right‐hand corner of Figure 1.1. Also, the reaction speed is exceptionally high for nuclear explosions, with a tremendous amount of energy released in a microsecond. Both the exceptionally high magnitude of pressures and the extremely short time scales make nuclear explosions extremely damaging.

1.1.2 Pressure Vessel Bursts

Progressing further in a direction of increasing time scale in Figure 1.1, a pressure vessel burst is the release of energy of compression in high‐pressure vessels. The release of pressure takes place in a time for a crack3 to propagate sufficiently far to allow the vessel shell to split open. This is typically on the order of 10 μs. The peak pressure is approximately equal to the vessel pressure at the time of bursting, Pb. The isentropic expansion energy, Eburst, for an ideal gas released during the vessel burst is [1]:

(1.1)

where Pb = vessel pressure at the time of bursting, Pa = pressure of ambient air (1 atm = 14.7 psia = 101 kPa at sea level), V = vessel volume, and γ = ratio of specific heats for the gas in the vessel (equals 1.4 for air).

1.1.3 Explosives

Explosions caused by explosives, usually condensed phase have time scales of the order of 100 μs. Figure 1.2 shows an aerial view of the aftermath of an explosion incident involving a special black powder composed of ascorbic acid (combustible powder), potassium nitrate (strong oxidizer), and potassium perchlorate (highly reactive oxidizer). The latter two ingredients were in the form of granular solids requiring milling prior to being mixed in two combination milling/blending machines located where indicated by the arrow in Figure 1.2. There were about 34 kg (75 lb) of black powder in each machine, and the first explosion triggered a second explosion, with the combined effects causing two fatalities in addition to the destruction shown in the photograph. As shown, the relatively small amount of explosive created significant damage to property in a radius of 30.5 m (100 ft). This radius is also called as a “blast debris radius,” associated with a blast wave, i.e. a pressure disturbance propagating into the atmosphere away from the source of energy release. We will discuss the damage potential of blast waves based on the initial energy release and distance from the release point in Chapter 7. The knowledge is useful for safe citing of industrial facilities.

Figure 1.2 Aerial view of damage and debris caused by a black powder explosion, with arrow indicating origin.

Source: U.S. Department of Justice.

Energies released by condensed‐phase explosives are often quoted in terms of the trinitrotoluene (TNT) equivalent weight. One kilogram of TNT has an explosive energy of 4.2 × 106 J. Most condensed‐phase high explosives have an explosive energy per unit mass that is similar to that of TNT. For example, the explosive energy of pentolite (50/50) is 5.1 × 106 J/kg, and that of royal demolition explosive (RDX) is 5.4 × 106 J/kg. The corresponding TNT equivalent of pentolite is 5.1/4.2 = 1.2 kg‐pentolite/kg‐TNT, and that of RDX is 5.4/4.2 = 1.3 kg‐RDX/kg‐TNT.

1.1.4 Closed Vessel Detonation

As discussed earlier, a detonation propagates at a speed greater than the speed of sound. A closed vessel detonation is usually the detonation of a flammable gas that is enclosed in a vessel, for example, a pipeline. In this case, ignition leads to a deflagration, which starts slowly, but rapidly accelerates to a detonation after propagating through the pipe for a distance called a run‐up distance. These distances are usually large (60–100 tube diameters) and the transition occurs in piping but is very improbable in vessels and equipment unless there is a combination of a fast‐burning gas mixture and a highly turbulent flame accelerating situation. The transition from deflagration to detonation is also highly complex. A flammable gas can also be made to detonate without a “run up” by providing a sufficiently large ignition energy. For example, Carlson [4] determined the minimum energy for initiation of detonation in stoichiometric gas–oxygen mixtures, using exploding wires to initiate detonation. The ignition energy to cause direct detonation of a stoichiometric propane–oxygen mixture is 2.5 J [4]. On the other hand, the minimum ignition energy (MIE) to ignite (sustain a propagating flame) in the same mixture is four orders of magnitude lower at 0.26 mJ as shown in Table 1.1. Thus, a combustible gas–air mixture likely will form a sustained flame, which may accelerate to a detonation rather than detonate directly since ignition with such a large energy source is usually unlikely.

Table 1.1 Flammability properties of some common gas air mixtures in air.

Fuel

T

ad

(K)

[3]

S

L

(cm/s)

[5]

MIE (mJ)

[6]

d

q

(mm)

[6]

AIT (K)

[6]

LFL %Fuel

[7]

UFL %Fuel

[7]

φ

LFL

[8]

φ

UFL

[8]

LOC N

2

−

air

[9]

LOC CO

2

−

air

[9]

r

st

%Fuel

[6]

Δ

H

C

(kJ/mol)

[6]

Δ

H

C

(kJ/g)

[6]

H

2

2400

312

0.018

0.55

673

 4

 75

0.14

2.54

4.6

4.6

29.5

 241.8

119.96

CO

2370

 46

—

1.73

882

12.5

 74

0.34

6.76

5.1

5.1

29.5

 283

 10.1

CH

4

2226

 40

0.28

2.5

810

 5

 15

0.46

1.64

11.1

13.1

 9.47

 802.3

 50.1

C

2

H

2

2541

166

0.017

0.55

578

 2.5

100

0.19

∞

NA

NA

 7.74

1255.5

 48.22

C

2

H

4

2370

 80

0.09

1.25

763

 2.7

 36

0.41

6.1

8.5

10.2

 6.53

1323.1

 47.16

C

2

H

6

2260

 42.5

[10]

0.25

2

745

 3

 12.4

0.5

2.72

9.5

11.9

 5.65

1428.6

 47.5

C

3

H

8

2257

 46

0.26

2.10

743

 2.1

  9.5

0.51

2.83

10.7

12.8

 4.02

2043.1

 46.3

C

4

H

10

2260

 45

0.26

2.20

638

 1.8

  8.4

—

—

10.6

13.0

 3.12

2656

 45.7

Tad: Adiabatic flame temperature; dq: Quenching distance; UFL: Upper flammability limit; SL: Laminar burning velocity; AIT: Auto‐ignition temperature; ɸ: Equivalence ratio; MIE: Minimum ignition energy; LFL: Lower flammability limit; LOC: Limiting oxygen concentration; rst: Stoichiometric volume concentration of fuel; ΔHC: Heat of combustion.

1.1.5 Steam Explosions

A steam explosion produces peak pressures in the range of 2–70 bar (30–1000 psig), within a millisecond. As shown in Figure 1.1, steam explosions have similar time scales as a closed vessel detonation but lower peak pressures. A steam explosion is not caused by combustion. Instead, a steam explosion is a physical explosion caused by the extremely rapid vaporization of water due to heat transfer from a second liquid that is at a temperature far in excess of the water's boiling point and in direct contact with the water. As the second liquid is usually either molten metal or some other melt, a steam explosion is a violent melt–water interaction. If the water is replaced with some other liquid that has a much lower boiling point than the hot liquid, the more general term is vapor explosion. Vapor explosion examples include Freon‐22 and heated mineral oil, water and liquid nitrogen, and liquid ethane and water [11]. Steam and vapor explosions are a concern in nuclear power plant accidents with water‐cooled reactor core temperatures sufficiently high to produce molten nuclear reactor fuel rods or cladding [11].

Vapor explosions occur only if certain thermodynamic and hydrodynamic conditions are satisfied. The thermodynamic condition is that the liquid–liquid contact surface temperature, Tcontact, must be greater than the spontaneous nucleation temperature, Tsn, for water, that is, the temperature at which vapor bubbles first appear in the absence of any heated surfaces. The equation for Tcontact is:

(1.1)

where TH is the hot liquid temperature, TC is the cold liquid temperature, and kρcp is the product of thermal conductivity, density, and specific heat for either the cold or hot liquid depending on the subscript.

For example, if molten copper at a temperature of 1400 °C is immersed in 20 °C water, the interfacial contact temperature as per Eq. (1.1) is 1341 °C. If molten cuprous oxide at a temperature of 1330 °C is immersed in 20 °C water, the calculated interfacial contact temperature is 954 °C. In both cases, the contact temperature is substantially higher than the spontaneous nucleation temperature for water, which is very sensitive to surface tension changes due to additives or contaminants but can be as high as 270 °C. Thus, molten copper interactions with water can indeed be explosive. Similar results are observed with many other molten metals and with kraft smelt immersions into water. The latter have been associated with black liquor recovery boiler accidents at paper mills. For the vaporization to occur rapidly enough and in sufficient volume to generate potentially damaging pressures, it is necessary to have ample liquid–liquid interfacial contact area.

1.1.6 Closed Vessel Deflagrations

All the explosions discussed earlier, except steam explosions, and pressure vessel bursts were detonations, where the speed of propagation of the reaction front is greater than the speed of the sound. Such explosions are particularly dangerous because of the high detonation pressures and the short time scale of energy release. We will now move on to explosions in which the speed of propagation is slower than the speed of the sound. Such explosions are called deflagrations. In small closed vessel deflagrations, peak pressures can reach up to 8–10 bar with time scales in the range of 1–10 ms. Deflagrations in buildings have longer time scales. Such accidental deflagration causes the enclosure to open or burst (because of either deflagration venting or structural failure), and the released blast wave will exert pressure loads on adjacent structures. The pressure vs. time trace because of confined deflagration is much different from pressures discussed earlier from condensed‐phase explosives and burst pressure vessels. Figure 1.3 shows the pressure wave generated by a blast wave generated by a condensed‐phase explosive (100‐ton TNT) at a location 125 m away by a dashed curve. The experiments were performed by Kingery et al. [12] where pressure rise vs. time at different distances from a 100‐ton TNT explosion were measured. As shown, a peak pressure of 1.4 bar is attained in a short time scale of ∼1 ms. The overpressure then decays over a time scale of ∼100 ms followed by a smaller negative pressure pulse. The solid line in Figure 1.3 is obtained from test data published by the Steel Construction Institute [13]. It shows the pressure felt by the walls of an enclosure (25.6 × 8 × 8 m3) during the deflagration of a 9.4% natural gas–air mixture ignited in the center. The deflagration pressure cannot reach the peak pressure of ∼8 bar because of the venting and occurs at 1.5 bar instead because of three openings in the enclosure walls with a total coverage area of 160 m2. The peak pressure of 1.5 bar occurs much slower than the detonation at 400 ms. In addition, it should be noted that the pressure trace by the TNT explosion in Figure 1.3 is 125 m away from the source. At ∼30 m from the TNT explosion, the peak pressure is 26 bar [12]!

Figure 1.3 Comparison of pressure profiles between a detonation and a confined deflagration. Pressure profile for the 100‐ton TNT detonation (dashed curve) at 125 m (410 ft) from the blast center [12]. The natural gas–air deflagration occurs in a large scale 25.6 m× 8 m× 8 m test rig [13]. Natural gas concentration equals 9.4% (equivalence ratio = 1.05), and ignition is at the center. Total vent area = 160 m2.

One example of deflagration in a closed vessel is the Center Wing Tank explosion that occurred during the TWA 800 flight on 17 July 1996. The flammable vapor in the Center Wing Tank of the Boeing 747 on that flight came from a small quantity of Jet A fuel in the tank. As the fuel was heated from air conditioning equipment under the Center Wing Tank, and the partial pressure of tank air was reduced as the Boeing 747 climbed after takeoff, the fuel–air equivalence ratio increased well into the flammable range. Ignition occurred at an altitude of 4300 m (14 000 ft), at which the ambient pressure is 0.585 bar. The closed vessel deflagration pressure Pm of 6 bar was significantly higher than the strength of the Center Wing Tank structures, leading to a massive breakup of the Boeing 747 [14].

1.1.7 Building Deflagrations

Building deflagrations can be because of gas, droplet, or dust (tiny solid particles suspended in air) explosions. Although small in terms of peak pressures generated, compared to the other types of explosions shown in Figure 1.1, it should be noted that it is of sufficient magnitude to cause building collapse. For example, to cause significant damage to a brick wall takes only approximately 2 psig or 0.14 bar (14 kPa)! For typical hydrocarbon fuels, the maximum explosion pressure is roughly 8–10 bars. Since this pressure is enormous when compared to the strength of most industrial structures, small pockets of flammable gasses in a building as shown in Figure 1.4 are sufficient to cause extensive damage.

Figure 1.4 Conceptual flow of events in a building explosion, P, T, V, and n are pressure, temperature, volume, and number of moles. Subscript 1 represents the initial state of the gas pocket and subscript 2 represents the final state after combustion of the small pocket of gas of initial volume V1.

Figure 1.4 shows a conceptual model for a confined deflagration in a room partially filled with flammable gas. If gas is ignited, a flame will grow spherically outward from the point of ignition. As the flame grows, it consumes fuel and causes a rise in pressure that depends on the volume and concentration of the flammable gas. Building partial volume deflagrations are one of the most common industrial accidents. A case study related to this (Danvers Explosion) is discussed in Chapter 2.

1.1.8 Vapor Cloud Explosions

Vapor cloud explosions (VCE) usually occur because of a large vapor cloud release and consequent mixing with ambient air, combined with highly obstructed or partially confined areas. The obstructions and partial confinements create zones, where the deflagration accelerates because of turbulence and in some cases can lead to detonations. Peak pressures in VCEs are of the same order of magnitude as those in building deflagrations, but the energy release times are usually longer because the flammable clouds are usually much larger than those that form inside buildings. One of the most significant industrial accidents recently in Buncefield, UK, was a vapor cloud explosion [15] as shown in Figure 1.5.

Figure 1.5 Vapor cloud explosion (VCE) at Buncefield, UK, on 11 December 2005. The explosion originated because of a vapor cloud release from a Jet A fuel storage facility. The resulting vapor cloud was estimated to be 120 000 m2 with an average height of 3 m. The black dashed line indicates the area affected by scorching and overpressure damage. The white dashed line shows the areas in the tank farm where sustained bund fires occurred after the explosion. The numbers indicate the location of vehicles, drums, and other enclosures, which exhibited damage consistent with overpressures above 200 kPa.

Source: Taveau [15]/John Wiley & Sons.

Example 1.1

Calculate the burst energy for a 10 m3 vessel that ruptures when it is filled with air at a pressure of 6890 kPa (1000 psi).

Solution

The solution of this problem is based on Eq. (1.1)

At sea level, Pb − Pa = 6890 − 101 = 6789 kPa, γ = 1.4.

Thus,

The burst energy is 169 MJ = 169/4.2 = 40.2 kg of TNT.

1.2 Controlling Parameters of a Combustible Gas/Vapor Explosion Hazard

Given a combustible gas–air or vapor–air or dust–air mixture, the following parameters must be evaluated by an engineer to analyze the magnitude of the explosion hazard:

1. The laminar burning velocity defined, in a spatial frame fixed to the flame front, as the velocity of the unburned mixture approaching the reaction front in the normal direction. The velocity is a function of the concentration or equivalence ratio of the mixture, ambient pressure, and temperature.

2. Volume of mixture and geometry of the space.

3. The rate of pressure rise denoted by .

A flame is a surface in the gas phase where a rapid chemical change occurs in a thin layer accompanied by heat generation. The unburned gas velocity approaching normal to the flame surface in the absence of turbulence is called the laminar burning velocitySL. The laminar burning velocity is a fundamental thermokinetic property of the mixture composition and concentration, indicating the fuel consumption rate at the reaction zone or flame surface. Hence, it does not change with the increase or decrease of the flow speed. The laminar burning velocity of typical hydrocarbon air flames range from 10 to 80 cm/s with a flame thickness of ∼0.1 mm. The highest velocity is that of H2–air mixtures, which can be up to 300 cm/s (6.3 mph), or around an average jogging speed. Further details about the laminar burning velocity, its formulation, and its variation with pressure and temperature will be discussed in Chapter 3.

The speed with which the flame travels through the gas–air mixture, measured with respect to some fixed position, is called the flame speed. Flame speed is not the same as burning velocity. The flame speed and the burning velocity are related by the expression

(1.2)

where E is an expansion ratio caused because of an increase in temperature. Typical flame temperatures of hydrocarbon–air mixtures are around 2500 K, and thus the expansion ratio is ∼2500/300 K ∼ 8. β is a parameter that is related to the fact that the unburned gas may be in motion and the combustion reaction is enhanced because of instabilities and turbulence (β > 1), as is often manifested in a wrinkled flame surface. The flame speed in gas deflagrations can be very high ∼100 m/s because of turbulent flame acceleration. An important aspect of both flame speed and pressure development is the degree of confinement or the geometry of the space in which the combustible mixture is contained. Usually, if the combustible vapor–air mixture is not at least partially confined, then pressure effects are not observed. The deflagration is usually in the form of a flash fire and poses a thermal hazard rather than an explosion hazard. For example, unconfined gunpowder (75% KNO3, 15% charcoal, and 10% sulfur) will burn rapidly if ignited. Still, it will not explode if it is not wrapped tightly in a cartridge to make a firework. Increasing the gunpowder volume will result in more energy being produced, thereby creating more power in the explosion [16].

Under certain conditions, especially with significant vapor releases, pressure effects can occur if the flame or reaction front accelerates as it propagates through the gas–air mixture. This acceleration and corresponding enhanced reaction rate are usually because of either initial turbulence in the gas cloud, turbulence induced by unburned gas flow around obstacles, or instabilities in the flame front that lead to turbulence. Such explosions, discussed earlier, are called VCEs. VCEs are rare compared to the more common confined gas deflagration explosions. This is because it is unlikely that large quantities of vapor (usually in tens to hundreds of tons) are released in the open.

A more likely scenario is the release of smaller quantities of vapor, dust, and mist within some form of confinement, which is provided by the equipment or the industrial process compartment or section of the industrial plant. If a flammable mixture is formed under such conditions and is ignited, then a confined gas explosion will occur. Such equipment and building explosions usually cause damage to the structure in which they originate. In the case of dust explosions, propagation into adjacent compartments is also possible because the pressure wave from the initial explosion causes built‐up dust layers in ceilings and joists in the entire facility to get suspended. The explosion hazard in equipment can be controlled using explosion suppression or deflagration venting systems, and the compartment explosions can be further protected by using deflagration vents, whose design is covered in NFPA 68 [5]. We will also discuss deflagration venting in Chapter 4.

The energy released in explosions can be propagated from the source by three mechanisms:

i. Shock/blast wave

ii. Projectiles often in the form of fragments

iii. Thermal radiation.

Of these, the blast wave is the common form of far‐field damage from explosions where large quantities of explosive mixtures are involved. Damage by flying fragments is important in building explosions and in pressure vessel bursts. Thermal radiation is an important consideration in dust explosions. In addition, during dust explosions, besides fragments, burning particles can also lead to injury to personnel.

1.3 Flame Propagation

In many industrial explosion accidents, the explosion develops because of a chemical combustion reaction. High amounts of energy release occur because of the self‐sustained propagation of a localized combustion zone propagating through the given gas–air or combustible dust cloud mixture. Figure 1.6 shows an example of an explosion in a 0.56 m × 0.56 m × 0.56 m (0.18 m3) cubic vessel with a vent of area 0.063 m2. The vessel is filled with an 8% ethylene–air mixture and ignited in the center as shown in Figure 1.6[17]. The flame propagates spherically as shown in Figure 1.7. Initially, at t = 12 ms it is laminar but at t = 20 ms the flame shows signs of wrinkling, and at t = 26 ms, burned gasses are observed escaping from the vent at the bottom. The venting leads to additional turbulence. Higher burning rates and consequently higher pressures are achieved because of the venting.

Figure 1.6 The evolution of an 8% ethylene–air flame in a cubic vessel of volume 0.18 m3, with a bottom vent area of 0.063 m2. The flame grows spherically till around t = 26 ms when venting of the gasses causes the flame shape to become nonspherical. The surface of the flame also becomes wrinkled at around t = 20 ms. The maximum pressure in the vented compartment equals 0.87 atm or 12.8 psig.

Source: Zalosh [17], American Institute of Chemical Engineers.

Figure 1.7 Flame propagation in: (a) spherical freely propagating flame (laminar), (b) spherical freely propagating flame (turbulent), and (c) spherical flame propagating in a confined vessel with a vent.

The speed at which the combustion wave propagates with respect to a fixed position is called the flame speed. The velocity at which the unburned gas enters the flame front in a direction normal to it is called the laminar burning velocity SL. The laminar burning velocity, SL, is a fundamental thermokinetic property of the gas–air mixture and depends on the composition of the gas–air mixture, gas or dust concentration, temperature, and pressure. The flame speed, Sf, is a function of SL, thermal expansion, change in number of moles, and the initial velocity of the gas. To further demonstrate the differences between SL and Sf, three cases of spherical flame propagation with central ignition are shown in Figure 1.7. All gasses are at rest when ignition takes place. The following questions can now be asked:

1. What is the flame speed or what is the rate at which the flame grows in size?

2. What is the consumption rate of unburned fuel at the flame surface?

3. What happens when the flame becomes stretched or wrinkled because of instabilities and/or turbulence as shown in

Figure 1.7

b?

4. What happens when the flame is confined in a vessel with an opening, as shown in

Figure 1.7

c?

The answers to the first two questions should clarify the difference between the flame speed Sf and the laminar burning velocity SL. The third question will clarify the important role of flame wrinkling and turbulence in flame propagation and bridge our understanding of industrial accidents where large‐scale turbulent premixed flame spread occurs compared to flames studied in the laboratory that cannot be simulated at such large scales. The fourth question is of importance to explosion safety as when the propagating flame is confined, it will cause an increase in pressure within the confinement. Some of this pressure will be relieved by the flow of gasses leaving the enclosure, which is a process called venting. The pressure rise in the enclosure with time and corresponding movement of gasses out of the enclosure through an opening are coupled to the propagating flame's motion that can be turbulent. This complex problem is one of the critical aspects that we will discuss in detail in Chapters 3 and 4.

Let us consider a combustible gas–air mixture that is ignited at the center causing a spherical flame to propagate freely outward without any confinement as shown in Figure 1.7 radially. The flame divides the burned gasses on the inside denoted with a subscript b and the unburned gasses outside denoted by a subscript u. Let us assume the following:

i. The flame front moves at a velocity that is low relative to the velocity of sound. Typical velocities in HC–air mixtures in case 1 are ∼10 m/s, an order of magnitude lower than sound speed (330 m/s).

ii. Ideal gas law applies.

iii. The flame is perfectly spherical, giving: .

Since the gas mixture flame is assumed to be spherical (at least until the vent opens),

(1.3)

where Af, N is the area of the flame. The subscript “N” denotes that the flame area evolves in a direction normal to the direction of the flame front. It denotes the minimum area that the flame can acquire, normal to the direction of propagation. This is an important assumption that is made here. If the flame front area is not spherical, for example, it is wrinkled as shown in Figure 1.7b, the actual area of the flame denoted by Af will be much more. However, since rb has not increased, , will equal the minimum area possible for the flame to assume a perfectly spherical shape.

The rate of production of burned gas is given by

(1.4)

The left‐hand side denotes the production of burned gasses and the term on the right is the product of the mass flux of unburned gas approaching the flame (ρuSL), times its actual surface area, Af. The subscript N is dropped here because the flame surface need not be smooth, as shown in Figure 1.7b. If the flame is wrinkled, there is more surface for the chemical reaction of combustion to take place, causing more production of the burned gas. Writing mb, equal to the product of the density of burned gas times its volume gives,

(1.5)

(1.6)

Note that SL is the laminar burning velocity or the velocity of the unburned gas approaching the flame in a direction perpendicular to the surface of the flame. So imagine, you are able to be on the flame front. You will see unburned gasses moving toward you at a velocity equal to SL. The unburned gasses will combust and exit the flame at a velocity that you will perceive as leaving behind you at a temperature equal to the flame temperature. Also, in Eq. (1.6),  = Af,NSf, where Sf is the velocity with which the spherical flame grows or the flame speed. Equation (1.6) now becomes,

(1.7)

(1.8)

Equation (1.8) gives the relationship between the laminar flame speed and the laminar burning velocity for a generalized case of the spherical flame propagating outward. The second term on the right‐hand side of Eq. (1.8) can be simplified using the ideal gas law , where MWb equals the molecular weight of the burned gas, Pb, is the pressure and Tb is the temperature of the burned gas. R is the universal gas constant (8.314 J/mol‐K).

(1.9)

With an assumption that Tb and MWb will not change with time,4 the differential term in Eq. (1.9) is equal to denoting the rate of pressure rise in the burned gas. For a freely expanding flame Figure 1.7a or b, this quantity is small and can be neglected, causing the second term on the right‐hand side of Eq. (1.8) to be equal to zero. Thus the flame speed is given by

(1.10)

And for a case where the flame is smooth (Figure 1.7a), can be further simplified as Af = Af,N,

(1.11)

For freely propagating premixed hydrocarbon gas flames,  ∼ 1.01 and  ∼ 1, thus making Sf related to SL in the simplest case as

(1.12)

where E is called an expansion factor and is a measure of the increase in volume created by combustion

(1.13)

Assuming no losses, the burned gas temperature is equal to the flame temperature and for most hydrocarbon–air flames. Equation (1.13) shows that, in a simplified case with similar molecular weights of unburned reactants and products, flame speed is equal to the burning velocity multiplied by an expansion ratio that is equal to the flame temperature divided by the unburned gas temperature. However, in practice, as the flame grows its surface can no longer be considered planar and the assumption of Af = Af, N no longer applies. Figure 1.8 shows a sketch of a flame propagating in a tube. Instabilities and turbulence create flame distortions causing the smooth planar flame to transition to a wrinkled flame front, which is usually the shape of the flame in industrial deflagrations. The first sketch shown in Figure 1.8 shows a planar flame. However, such a flame occurs only during the initial stages of the flame propagation when the flow is laminar. Very soon the flame becomes wrinkled as shown in the middle sketch of Figure 1.8