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I PUBLICATIONS

13fl7B0 [I ' NBSIR 80-2018 (DOT)

A Third Appraisal of Methods for

Estimating Self-Reaction Hazards

Kenneth L Churney and David Garvin

Division of Chemical Thermodvnamics

Center for Thermodynamics ano Molecular Sctence

National Measurement Laboratory

National Bureau of Starxfards

U S. Department of Commerce

Washington, D.C. 20234

I March 1980

Issued August 1 980

Interim Report for Period January 1976 • August 1978

Prepared for azardous Materials Operations

transportation Bureau

Transportation

k, D.C. 20590

NBSIR 80-2018 (DOT)

A THIRD APPRAISAL OF METHODS FOR

ESTIMATING SELF-REACTION HAZARDS

Kenneth L Churney and David Garvin

Division of Chemical Thermodynamics

Center for Thermodynamics and Molecular Science

National Measurement Laboratory

National Bureau of Standards

U S. Department of Commerce

Washington, D.C. 20234

March 1980

Issued August 1 980

Interim Report for Period January 1976 - August 1978

Prepared for Office of Hazardous Materials Operations

Materials Transportation Bureau

Department of Transportation

Washington, D.C. 20590

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology, and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Abstract

A study has been made of thermal explosion theory and thermal

hazard tests. From this definitions have been developed, suitable for

use in a transportation context, of thermally unstable materials-package

configurations and thermal instability hazards. A thermally unstable

materials package configuration is one:

(1) composed of substances that can decompose or react with the

evolution of heat,

(2) for which a thermal mechanism of initiation of a temperature

rise is present,

and (3) in which a resulting large increase in the temperature of the

material can occur.

A thermal instability hazard exists if decomposition of the materialpackage configuration results in destruction of the package or neighboring

objects or poses a safety threat. Typical destructions are explosion,

rupture of package, and fire.

The properties of the material and package needed to predict the

hazard are identified. and methods for their measurement, using heat

conduction and adiabatic calorimetry, are analyzed. It is concluded

that, for least ambiguous scale-up of results to bulk shipping conditions,

measurements should be made on large samples of the material under

conditions such that heat transfer within the sample is large in comparison

to heat transfer between the sample and its environment. Tests, such as

the explosion temperature test, should be carried out to check the

scaled-up hazard predictions.

i
An experimental study on three thermally unstable materials, nitrocellulose (I) , m-azidobenzoic acid (II) , and azidotriphenylmethane

(III) , was carried out using a differential scanning calorimeter (DSC)

The procedure used is that of a proposed ASTM method of test for determining

Arrhenius kinetic constants from the variation of the temperature of the

maximum rate of heat evolution with scanning rate. Experimental results

for activation energies, AE, and prexponential factors. A; are: AE =

(I) 166.9 + 6.1, (II) 139.9 + 5.9, and (III) 145.3 + 4.4 kJ mol"^; .

log^Q(A,min"^) = (I) 18.0 + 0.7, (II) 15.1 + 0.7, and (III) 15.0 + 0.6.

A systematic error analysis clarifies and tends to support the test

procedure. The main drawbacks to the test are its non-applicability to

complex systems and the small temperature range of the data, which makes

the extrapolation of results to lower temperatures uncertain.

Key Words

Adiabatic calorimetry, azidotriphenylmethane, decomposition reactions,

differential scanning calorimetry, heat conduction calorimetry,

kinetics of heat evolution, m-azidobenzoic acid, nitrocellulose,

quantitative differential thermal analysis, thermal explosion theory,

thermal hazards, thermal instability.

ii
Foreword

This report consists of two separate parts. Part I contains the

conclusions and the main text, sections 1 to 5. The main results of this

study in context of previous work at NBS are given in sections 1 to 3. A

more complete and detailed summary of results is given in sections 4 and 5.

Part II is the documentation on which- the conclusions are based, appendices

A through C. Part II also contains a glossary of symbols. Appendix D, and

the references. Appendix E.

iii
Table of Contents

List of Tables vii

List of Figures ix

1. Introduction 1

2. Summary of Previous Work and Comments . 4

2.1. First Apparaisal of Methods for Estimating

Self-Reaction Hazards 4

2.2. Second Appraisal of Methods 5

2.3. Comments 6

3. Summary of Present Work and Conclusions 8

4. Thermal Explosions 11

4.1. Introduction 11

4.2. Models for Thermal Explosions 13

4.2.1. Explosion Temperature Test 14

4.2.2. Programmed Heating Test 16

4.3. Thermal Explosion Theory 17

4.3.1. The Isothermal (Uniform Temperature)

Sample 17

4.3.2. Criteria for Explosion 20

4.3.3. Extension to Non-Isothermal Samples and '

Complex Kinetics 22

4.3.4. Programmed Heating of Unstable

Materials 23

4.3.5. Conclusions 24

4.3.6. Definition of Thermal Instability and a

Thermal Hazard Criterion 28

iv
4.4. Thermal Explosion Experimentation and Self

Reaction Hazard Tests 31

4.4.1. Experiments on Explosions in Gases 32

4.4.2. Experiments on Explosions in

Condensed Phases 33

4.4.3. Conclusions Concerning Thermal Explosion

Experimentation . 36

4.5. Hazard Tests 38

4.5.1. General Description of Field 38

4.5.2. Thermal Sensitivity Tests 40

4.5.3. Adiabatic Calorimetry 41

4.5.4. Heat Conduction Calorimetry 43

4.5.5. Conclusions Concerning Hazard Testing ... 48

5. Experimental Work 54

5.1. Introduction 54

5.2. Summary and Assessment of Proposed ASTM

Test Method 56

5.3. Instrument Evaluation and Calibration 61

5.4. Experimental Work 65

5.5. Discussion of Test Results 38

5.6. Conclusions Based on the Experiments and

Recommendations for Future Work 102

Appendices and Ref erences •'(Contents of Part II)

A. Remarks on Initial NBS Study A. 1-1

A.l. CHETAH Program A. 1-1

A. 2. STULL-CRUISE Program and NFPA Hazard

Index A. 2-1

V
A. 3. Summary of the Results of the Initial Study .... A. 3-1

A. 4. Recent Developments A. 4-1

B. Thermal Explosion Study Notes B.1-1

B. l. Description of the Thermal Explosion Process. . . . B.1-1

B.2. Description of the Isothermal Model B.2-1

B.3. Steady State Analysis of the Isothermal Model . . . B.3-1

B.4. Nonstationary Analysis of the Isothermal Model,

f(ri) Constant B.4-1

B.5. Nonstationary Analysis of the Isothermal Model,

f(ri) Variable B.5-1

B.6. Temperature Gradients in the Material, Various

Geometries B.6-1

B.7. Autocatalytic, Multiple, and Heterogeneous

Reactions B.7-1

B.8. Thermal Explosions in Linear Heating B.8-1

B.9. Experiments on Gaseous Reactions B.9-1

B.IO. Experimental Work on Explosives in the U.S.S.R.

,

to 1970 B.10-1

B. ll. Hazard Evaluation Experimental Work ; B.11-1

C. Fundamental Paramter Tests C.1-1

C. l. Adiabatic Calorimetry C.1-1

C.2. Conduction Calorimetry Thermal Instability

Tests C.2-1

C.3. Simplified Measurement Theory of Quantitative

Differential Analysis (QDTA) and Differential

Scanning Calorimetry (DSC) C.3-1

C.3.1. Measurement Equations for QDTA C.3-1

VI
C.3.2. QDTA and DSC Equations for

Kinetics C.3-7

C.3.3. Measurement Equations for DSC C.3-14

C.3.4. Transfer Functions and Temperature

Calibration C.3-21

C.3.5. Determination of K C.3-27

C.4. Instrument Calibration . C.4-1

C.4.1. Main Experimental Results C.4-1

C.4.2. Analysis of Calibration Procedures C.4-12

• C.5. Error Analysis for Four Assumptions C.5-1

C.5.1. The Value of g(D^) at Maximum dri/dt .... C.5-1

C.5.2. Assumption of Constant Scanning Rate in

Evaluating g(n,) C.5-5

d

C.5.3. Correction to Assumptions 1, 2 of

Table 11 C.5-8

C.6. Closed Reusable Capsule C.6-1

D. Table of Symbols D-1

E. References E-1

vii
List of Tables

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7

Table 8.

Table 9.

Table 10.

Table 11.

Summary of High Pressure Cells for DSC

Applications

m-Azidobenzoic Acid, Calculation of T, . a Azidotriphenylmethane, Calculation of T^

Nitrocellulose, Calculation of T, . . . a Key to Tables 2, 3,

m-Azidobenzoic Acid, Kissinger Method. .

Azidotriphenylmethane, Kissinger Method.

Nitrocellulose, Kissinger Method . . . .

Summary of Results on Kinetics of Heat

Evolution . .

Comparison of Results

List of Assumptions (Proposed ASTM Test

68

74

75

76

77, 73

85

86

87

89

90

Method) 92

Table 12. Effect of Error in Assumption 1 of Table 11 . . . 94

Table 13. Effect of Error in Assumption 2 of Table

11 (Nitrocellulose) 95

Table A-1 CHETAH Parameters and Classification of Compounds

by Interpreters A. 4-4,

Table B-1 Induction Times and Reactant Consumption,

Explosive States, f(r)) is Constant B.4-4

Table B-2 (d0/dt) x (n^^) ] versus 0, Exponential

Approximation B.4-9

Table B-3 Results of Tyler and Wesley [39] B.5-12

Table B-4 Mechanical Hazard Test Survey B.11-5

viii
B.11-6

1 Table B-4a

* Table B-5

I Table B-5a

Table C-1

I Table C-2

.. Table C-3

f* Table C-4

H

Table C-4a

Table C-5

I

I

Table C-6

m

Table C-6a

Table C-7

I Xa.e C-Sa

I

Table ,C-8b

Table C-9

References, Comments, Table B-4

Thermal Hazard Test Survey

References and Comments for Table B-5

Conduction Calorimetry Tests

Details on Conduction Calorimetry Tests

Measurement Equations for QDTA

Baseline Treatment for Kinetics

Definitions and Comments, Table C-4

Equations for Simplified DSC Measurement

Analysis

Temperature Calibration and Time Constant

Study

Key to Table C-6

Energy Calibration

Approximate Equations for DSC, Formulation of

Equations

Approximate Equations for DSC, Solution of

Entry 2, Table C-8a

Correction for Assumptions 1, 2 of Table 11. . . .

B. 11-7,

8

B. 11-9

C.2-2

C. 2-3,4

C.3-5

.C.3-9 •

C. 3-10

C.3-19

C.4-4,5

C.4-6,7

C.4-11

C.4-15

C.4-16

C.5-11

ix
42

List of Figures

Figure 1 Sketch of Cross Section of an Idealized

Adiabatic Calorimeter

Figure 2 Sketch of Cross Section of an Idealized Heat

Conduction Calorimeter 44

Figure 3 Sketch of Cross Section of an Idealized

QDTA 46

Figure 4 Sketch of Cross Section of DSC Apparatus (Used

in Experimental Work) 47

Figure 5 Sketch of Cross Section o'f DTA 49

Figure 6 Sketch of Exotherm of Polyacrylonitrile 66

Figure 7 Sketch of Cross Section of Sample Containers ... 70

Figure 8 Ordinate Signal versus Nominal Program

Temperature, m-Azidobenzoic Acid 71

Figure 9 Ordinate Signal versus Nominal Program

Temperature, Azidotriphenylmethane 72

Figure 10 Ordinate Signal versus Nominal Program

Temperature, Nitrocellulose 73

Figure 11 Baseline Construction for Ordinate Signals .... 80

Figure 12 Plot of result 1/T^, m-Azidobenzoic

Acid 82

Figure !! Plot of log2_o

'^d Azidotriphenyl- ^

methane 83

Figure 14 Plot of versus 1/T^, Nitrocellulose ... 84

Figure B-1 Semenov Diagram B.3-2

Figure B-2 Semenov Diagram; f(r|) = (l~n) B.5-8

X
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

B-3 Sketch of 1/6 = a versus y=l/B, n=l, 3=0. .

t B-4 Sketch of 9 (9 for d9/dt=o) versus 5 (after

m

Merzhanov et al [42])

C-1 Sketch of Cross Section of QDTA Described by

Baxter [173]

C-2 Conductance Diagran for Figure C-1

C-3 Simplified Sketch of DSC-2 Output Circuitry . . .

C-4 Sketch of Melting Endotherm

C-5 Typical Plot of Onset- temperature of Fusion,

versus Scanning Rate, w

C-6 Plot of Static Temperature Corrections,

versus Program Temperature, T^

C-7 Estimate of from Exotherm of Freezing of

Supercooled Indium

C-8 Heat Capacity of Aluminum Containers

C-9 Sketch of Diametral Cross Section of Closed

Reusable Capsule

C-10 Sketch of Diametral Cross Section of Modified DSC

Holder Lids

B.5-14

B. 5-16

C.3-2

C.3-4

C. 3-17

C.3-24

C.4-2

C.4-3

C.4-9

C.4-13

C.5-11

C.6-3

XX
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1. Introduction

I

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i

The work reported here represents the third in a series of studies

carried out at NBS between January 1974 and August 1978 to develop a

clear definition of and quantitative test methods for thermal instability

of chemicals under bulk shipping (and storage) conditions.

This third appraisal is (1) a study of thermal explosion theory,

(2) an application of it to characterize test methods, and (3) an experimental investigation of the behavior of some thermally unstable compounds

using a commercial scanning calorimeter. In this appraisal, definitions

of thermal instability and self-reaction hazard have been developed.

The importance of thermal explosion theory as a guiding principle in

this field is presented. The usefulness of sensitive calorimetric tests

is explained. The experimental results of our studies using a scanning

calorimetry test method are given and the applicability and limitations

of the method are discussed.

This study was sponsored by the Office of Hazardous '-laterials

Operations, Materials Transportation Bureau, Department of Transportation

(DOT). That Office has supported these studies as a part of its own

long range program to improve both the systematization of its regulations

for the interstate shipment of hazardous cargoes and the emergency

response systems which can be implemented in the event of an accident.

At the present time, thermal instability is not used as a hazard

indicator, per se, by DOT, although thermal instability either is or can be

implicitly involved in the indicators "explosive,” "flammable liquid,"

flammable solid," or "flammable compressed gas." Thermal instability or

self-reaction hazards are currently identified and handled in two of the

shipper rules and regulations of Tariff Act 14 (i.e. as of October 1,

1979, the Hazardous Materials Safety Act, Public Law 93-633).

1
The first regulation, section 173.21, prohibits the shipment of any material

that can polymerize or decompose under conditions normally incident to

transportation so as to cause the dangerous evolution of heat or gas.

Such materials can be shipped when properly stabilized or inhibited.

The second regulation, section 173.51, forbids the shipment of explosive composi

tions that ignite spontaneously or undergo marked decomposition when subjected

for 48 consecutive hours to a temperature of 75®C.

Currently, the preceding regulation cannot be unambiguously implemented because there is no consensus agreement either as to the precise

definition or test(s) for self-reaction hazards. The absence of a clear

definition and unambiguous test procedure increases the probability of

failure to identify materials that are currently designated as safe

(i.e. under normal shipping conditions) but which can become thermally

unstable either under different packaging conditions or when an accident

occurs. Since the consequences of a thermal explosion can be as disastrous as the detonation of an explosive, there , is a clear mandate to

identify these borderline thermally unstable materials and the conditions

under which they can become hazards.

Thermal explosion theory is a well developed but complex branch of

physical chemistry. Experimental thermal explosion research has been

carried out extensively for many years. The applications of this research

are many and diffuse -ranging from high explosives to building materials.

The literature of this research, is scattered. We have attempted to

pull together those parts of thermal explosion work that are pertinent

to the thermal instability problem and have used this synthesis in

developing our conclusions. Roughly one- third of this report is devoted

to this exposition, which appears in appendices originally prepared as

working notes. This treatment may be useful as a starting point for

others

.

2
A summary of the previous two studies carried out at NBS for DOT is

given in Chapter 2 to provide some background for the brief stimmary of the

conclusions and recommendations of the current study that are presented in

Chapter 3. Chapter 4 contains a more detailed summary of our current

understanding of the implications of the concepts of thermal explosions

and tests for self-reaction hazards in terms of tests for and definitions

of thermal instability. Chapter 5 consists of a more detailed summary of

the experimental work. Detailed conclusions drawn from various parts of

the work are presented in sections 4.3, 4.4, 4.5 and 5.6.

Throughout the report there are many references to the appendices

where details are developed. But the essential parts of the arguments are

given in the body of the text. An attempt has been made to maintain a

consistent notation. A glossary of symbols used may be found in Appendix

D. The terminology is that used in our previous reports and corresponds

to common usage in the literature. The term "thermal explosion" refers to

the rapid large rise in the temperature of a material caused by the heat

liberated by a run-away self-reaction of the material. However, the term

"explosion" is less specific and depends upon the context in which it is

used. It can be a detonation, a violent rupture of a container, or a

violent fragmentation of a material. "Explosives" refer to materials

specifically designed to function by the substantially instantaneous release of gas

and heat.

The authors wish to acknowledge the extensive help they have received

from others. E. S. Domalski, G. T. Armstrong and E. J. Prosen, all of

NBS, have discussed and illuminated many points. Allen Duswalt, Hercules,

Inc. , has made suggestions and provided samples. Bruce Cassel, William

Brennan, and M. J. O’Neill, Perkin-Elmer Corporation, have helped us with

instrumental problems. The exacting task of organizing and typing this

report has been done by Miss Darlene Connelly, with precision, perseverance

and exceptional good humor. Corrections to the final draft have been made

by Mrs. Donna Whitworth with care and patience.

3
2 . Summary of Previous Work and Comments

2 . 1 First Appraisal of Methods for Estimating Self-Reaction Hazards

The initial NBS study [1] for DOT consisted of an appraisal of

whether or not some existing empirical criteria for estimating the

degree of self-reaction hazard might suffice for evaluation of this

hazard potential in a transportation context. The study also sought to

determine if the thermochemical and kinetic parameters needed to calculate

numerical values for these criteria for the degree of self-reaction

hazard could be estimated for materials where no detailed experimental

determination of reaction products or reaction rates has been made. The

particular criteria and thermochemical estimation procedures involved

were those in two predictive schemes: the CHETAH program [2-4] and

Stull’s adaption of the CRUISE program [5,6]. This work has been

reviewed by Domalski and Tsang [ 7 ]

.

A more concise summary is given in

Appendix A. 3.

The two main results of this initial study were:

(a) That at present self-reaction hazards should be evaluated by

experiment rather than solely by prediction schemes. The schemes were

judged inadequate for this purpose.

(b) More experimental sensitivity data are needed for borderline

hazardous compounds and monomers. Test methods to be used to acquire

these data were not specified.

What the methods should be is an important part of the problem (see

comments on pages 36, 37 and 76 of reference [1]). In particular, there

is a need for a more precise definition for self-reaction hazards and an

unambiguous procedure, e.g. a coordinated set of tests, for characterizing

the hazard.

4
2.2 Second Appraisal of Methods

In the second NBS study [8] for DOT, the appraisal of predictive

schemes started in the first study was completed. Three new tasks were

undertaken whose general aim was to determine if a (consensus) definition

of thermal instability and corresponding test(s) existed in the hazard

evaluation literature and to assemble the information needed either to

docximent or to develop such a definition and tests. These tasks were:

collection and evaluation of test methods and accompanying test data,

evaluation of accidental polymerization hazards and a survey and analysis

of thermal explosion theory and associated experimental work. A summary

of the second study has been published by Domalski [9]. The main results

of the second study were as follows:

(a) Predictive Schemes . Efforts to improve the thermodynamic estimation

procedures by trying to develop rules to predict decomposition products

appropriate to specific reaction mechanisms were unsuccessful. The

tendencies of the predictive schemes to overrank the hazard of stable

materials and to fail to identify polymerization reaction hazards, as

such, were more fully documented. A review of the predictive scheme

studies suggested that the ideas needed to develop a clearer definition

of thermal instability had already been identified in the initial survey

of thermal ignition theory (see p. 72, 73 of [1]).

(b) Thermal Explosion Theor;/ and Experimental Work . A preliminary

evaluation of review articles on theory [10-19] and some associated

experimental work (e.g. [20-24]) supported this conjecture.

(c) Test Methods . The array of test methods and test data that was

collected turned up nothing obvious in terms of a consensus definition

or tests for thermal instability. Therefore, methods were analyzed to
determine whether they measured primarily sensitivity to or power of

explosion. It was found that most thermal sensitivity tests could be

linked to some aspect of thermal explosion theory. A tentative grouping

of test methods was made according to their potential relevance in

developing a test procedure for evaluation of self-reaction hazard.

(d) Polymerization Hazards . Free radical polymerization of olefins,

initiated most probably by peroxides, was identified as being the most

likely polymerization process to occur inadvertently during transport.

2 . 3 Comments

During the present work the first NBS study has been reanalyzed.

This has been done in the light of current information and also to

answer comments made by others on the treatment of the CHETAH and CRUISE

programs.

First, the CHETAH program is an ongoing project of ASTM Committee

E-27. The analysis made in the- first NBS study may not be applicable

today. A current, experimental version of the program includes a novel

method for correlating hazards and thermodynamic parameters. Second,

the documentation for the earlier version of CHETAH makes it clear that

the predictive scheme was not intended to be a stand-alone method for

identifying self-reaction hazards, but is to be used as a first, screening,

step in some more elaborate test procedure. Third, while the first NBS

study criticizes the empirical criteria of the Chetah program because

they are based on parameters more directly related to explosive power

than to sensitivity, the, criteria do identify shock sensitivity.

These comments indicate that the objections to CHETAH raised in the

first NBS study may not be appliable to explosives. However, they may

still apply to the screening of borderline thermally sensitive compounds.

An appreciable fraction of these borderline compounds may not be shock

sensitive.

6
The relationship between the CHETAH criteria and explosive power

is discussed in Appendix A. 1. Earlier conclusions are revised in the

light of recent developments. The relationship between the thermodynamic

parameters examined by Stull (CRUISE) and the NFPA ratings is discussed

in Appendix A. 2.
3. Summary of Present Work and Conclusions

Thermal explosion theory has been examined for those features that

are applicable to an assessment of thermal instability. These results

are presented in terms of the explosion temperature test and tests

involving programmed heating.

A definition of thermal instability and one for thermal instability

hazards are presented that are suitable for use in a transportation context.

The existing hazard test methods have been analyzed in terms of

what they measure and their relationship to theoretical concepts. The

principles of adiabatic and heat conduction calorimetry are explained

insofar as they apply to the present problem.

An experimental study has been made on four thermally unstable compounds using a proposed ASTM test method, which is an application of

differential thermal analysis. The usefulness and limitations of this

method are discussed. Suggestions are made for extending this method.

The principal conclusions drawn from this work are as follows.

Both the material and its package must be considered in defining

a hazard potential test.

Thermal explosion theory is well established and is the appropriate formalism for the evaluation of hazard tests and for scaling up

laboratory tests.

Thermal instability can be defined in terms of thermal explosion

theory but this is insufficient. A hazard (i.e. potential damage to the

environment) must also exist. Thus, we define a thermally unstable

material-package configuration to be one; a) composed of substances that

decompose or react with evolution of heat, b) for which a thermal mechanism

of initiation of a temperature rise in the substances of the package

is present (e.g. a slight increase in external temperature), and c)

in which a rapid, large increase in the temperature of the substances
can occur. A thermal instability hazard exists if the decomposition

(or reactions) of the material-package configuration results in destruction

of the package or neighboring objects or poses a safety threat. The

definition of thermal instability is a generalization of the capability

of producing a thermal explosion in a material. The definition of a

thermally instability hazard incorporates the ideas that a thermal

explosion may not be hazardous and that decompositions (or reactions)

without a thermal explosion (e.g. evolutions of gases in an unvented

container) may be hazardous. These definitions are used to evaluate

existing tests.

Thermal instability tests are only part of the tests required

for hazard assessment.

Some key reqxiirements for thermal instability test(s) are

identified. An adequate procedure for determining thermal instability

can probably be developed through complementary studies using explosion

temperature and calorimetric tests. The tests would have to be carried

out with large amounts of sample under conditions that mimic bulk

behaviour. These conditions permit scale-up of test results for cases

where the formalism of explosion theory is not well developed.

The experimental results of applying differential scanning

calorimetry (DSC) in a proposed ASTM test method partially validate that

method. For the materials studied, nitrocellulose, m-azidobenzoic acid

and azidotriphenyl methane, our results agree with those obtained by

others using quantitative differential thermal analysis (QDTA) . One

other compound, polyacrylonitrile, gave ambiguous results because of the

deposition of products within the instrument. The principal reservations

concerning the proposed method are (a) non—applicability to complex

systems and (b) large uncertainty in extrapolation of the results to

lower temperatures.
The principles of operation of DSC and QDTA instruments have

been developed. To our knowledge this report is only place they have

been analyzed as a group in a compact fashion. As applied to the ASTM

method they are formally the same. Methods of extending the data reduction

procedures to take into account the extent of decomposition are presented.

Experimental modifications are suggested.

Scanning methods that use very small samples should be compared

with heat conduction and adiabatic calorimetry studies on large samples.

This should be done to evaluate the limits of applicability of the scanning methods in a hazard assessment procedure.

10
4. Thermal Explosions

4.1 Introduction

Thermal explosion theory and its application to hazard testing are

the subject of this chapter. The concepts of the theory are described

and experiments that validate it are discussed. The definitions of

thermal instability and a thermal instability hazard are developed.

This definition and the theory are used to analyze and classify selfreaction hazard tests in order to assess the usefulness of the concepts

in practice. Finally, a description is given of several calorimetric

tests that are used in (or proposed for) hazard evaluation. These are

state-of-the-art tests that require the use of theory in the application

of their outputs to hazard assessment.

These topics are covered in summary form with emphasis on key

concepts and conclusions. More detailed discussions are given in appendicies , with appropriate cross references here.

The organization of the chapter is as follows:

(1) Models of two types of explosion tests are introduced in

Section 4.2. The characteristic phenomena of thermal explosions are

described. These models are used in the detailed analyses in later

sections

(2) Section 4.3 presents the basic physical ideas of thermal

explosion theory, the approximations that are commonly used, and our

conclusions. It is based on a systematic treatment of those parts of

theory applicable to the present problem. That treatment was developed

because the principal reviews of the field [10-22] neither present the

theory in a concise form easily applicable to the present problem nor

use a common notation. Details are given in the first eight sections of

Appendix 3.
(3) Section 4.4 contains an analysis of experiments that validate

thermal explosion theory. It also characterizes practical hazard tests

in terms of theoretical concepts and what they measure. General conclusions are given. More details appear in the last three sections of

Appendix B.

(4) Section 4.5 is a description of four types of calorimetric

tests. What is measured is explained together with the necessary theoretical treatment of the data. The section is background for chapter 5

on our experimental studies. The material treated in section 4.5 is

presented in detail in Appendix C.

12
4.2 Models for Thenaal Explosions

The physical principles underlying the mathematical theory of

thermal explosions are simple and beyond question. Heat is generated in

a source (the decomposing sample) held in a container and this heat then

flows to the surroundings (the environment) which are held at a fixed

temperature. The rate of heat generation increases exponentially with

temperature, a common characteristic of chemical reactions, while the

heat flux (i.e. heat transfer rate) between the sample and its environment

is approximately a linear function of their temperature difference.

Lf the rate of heat generation can exceed the heat flux, a thermal

runaway or explosion may result.

The problems that arise in the mathematical treatment concern

specification of the behavior of the heat source, the heat transfer

within the sample and the heat flux to the environment. The rate of

heat generation depends upon the kinetics and mechanism of the chemical

reaction as well as on its exothermicity . The mode of heat transfer in

the sample may be conduction, convection, or a mixture of these. The

heat flux to the environment, if solely by conduction, depends (as does

heat transfer within the sample) on temperature gradients which are

controlled by thermal conductivities and heat capacities.

For various boundary conditions the behavior of a sample will

differ. The equations describing the temperature of a sample' as a

function of time are non-linear and can only be solved numerically for

the general case. Analytical solutions for limiting cases are developed

using various approximations. These must be justified for each class of

experiments.
But the approximate, analytical solution remains very important.

It provides a qualitative guide to the phenomena revealed in detail by

numerical calculations, it is the basis for classifying test methods,

and it provides guides for scaling up laboratoiry experiments to shipping

and storage conditions.

The exposition in the next section is developed with reference to

two idealized experiments, one of the explosion temperature test and the

other of programmed heating test of a material. These are described

below and used to introduce the principal terms used later. In the

explosion temperature test, the decomposing material is assijmed to be

homogenous. Thus, some details of the explosion temperature test, such

as the location of the initiation of an explosion, will not necessarily

apply to surface oxidation reactions.

4.2.1 Explosion Temperature Test

In this test an encapsulated sample, originally at room temperature, is immersed in a bath at a series of .fixed, higher temperatures.

The samples either explode or they do not. The limiting or critical

temperature of the bath dividing the explosive and non-explosive samples

is sought. The time to explosion increases exponentially as the positive

difference of the actiial minus critical bath temperature approaches

zero. The critical bath temperature depends on the size and geometry of

the sample and the physical parameters mentioned earlier.

In more abstract terms, and with some attention to the temperature

history of the sample, this test can be described as follows. A sample

and its environment are originally at a low, spatially uniform temperature, T^^. At zero time, the temperature of the environment is stepjumped (raised at an infinite rate) to a fixed high temperature, T^.

The sample temperature, T, rises as heat flows in. Depending upon the

value of T^ several things may happen.

14
When T is moderate and the sample is essentially inert, T rises to

o

T and remains there. If T is such that it causes a slight generation

0 o of heat in the sample, T rises rapidly to and then, much more slowly,

to a slightly higher temperature at which heat generation and flux are

balanced. The maximum temperature is at the center of the sample. As

the sample reacts the rate of heat generation decreases and, eventually,

T declines toward T . 0

The critical temperature of the environment, T^, is the highest for

which a balance between heat generation and loss can occur.

When T^ is slightly higher than T^, the temperature of the sample

rises quickly to T^, rises more slowly coward the critical maximum for

stability and Chen continues upward at a sharply increased rate. This

is the thermal runaway and is moderated only by exhaustion of the sample.

The maximum temperature at any time (when T > T^) is still at the center

of Che sample. (This picture is also used in the treatment of explosions

due to "hot spots".)

When T^ is substantially higher than critical, the outer portions

of the sample exceed critical conditions before Che center warms up to

T^. Explosions start in an annulus that approaches the edge as T^ is

raised. This is the "ignition regime" which is of importance for

explosives but is of less interest for borderline hazardous materials

(since they are not explosives or detonators) or for setting criteria

for handling.

The model of the explosion temperature test is applicable not only

to the. interpretation of laboratory experiments but also to accident

conditions involving fires and adiabatic compression. It is described

again in Appendix 3.1, developed in a simplified manner in 3.2 to 3.5,

and extended to complicated cases in 3.6 and 3.7.
4.2.2 Programmed Heating Test

The second test is used to examine the thermal decomposition of a

material when the temperature of its environment increases linearly with

time, T = T. + wt. (w is the rate of increase in the environment

o in

temperature and t is the time.) This is the analogue of the ASTM procedure discussed in chapter 5 and also applies to diurnal heating of

materials in storage. The model is developed in detail in Appendix B.8.

Briefly, as T^ is raised at a fixed rate, the temperature of the

sample follows it (with a small lag) as heat flows in. At some temperature the heat generation becomes appreciable and T rises above T^,

passes through a maximum and then, as the reaction is completed, decays

back to the (still rising) value of T^. In a properly designed experiment

there is no thermal runaway.

Chemical rate constants can be derived from a study of the departure of T from T^. The simplest method correlates the maximum temperature, T^, with the programmed scanning rate, w. The enthalpy of reaction

can also be obtained. Detailed analysis of the exotherm can provide

data oh the kinetic rate law.

Critical temperatures are not measured in this test. Instead the

rate constants and enthalpies, together with heat capacities and thermal

conductivities are used to predict the critical environment temperature

of the material under other conditions. The scaling rules used in this

process are based on thermal explosion theory.

16
4.3 Thermal Explosion Theory

The goal of the exposition here is to define thermal instability

within the context of thermal explosion theory. Validation of the

definition for use with experiments is covered in a later section. In

order to develop this definition and to provide a framework for understanding and assessing experiments of the previous section, a review of

the main points of theory is necessary. It is given in short form here

and in detail in Appendix B.

4.3.1 The Isothermal (Uniform Temperature) Sample

This case, the simplest to treat, assumes that the temperature

within the sample is spatially uniform. It corresponds to highly

efficient convective mixing or to a stirred liquid. Essentially all

descriptions of more complex cases can be developed in terms of (small)

extensions of this case if attention is restricted to the temperature

region near the critical point.. Assumptions are given in Appendix B.2.

The heat balance for the entire material is

C(dT/dt) = Q'(dn/dt) - h' (T-T^) (1)

where C is the total heat capacity of the material (excluding the container) , Q' the total heat liberated by the reaction, n, the fraction of

the sample reacted, and h' is an empirical average heat transfer coefficient

between the sample and its container. T is the temperature of the .sample

o container which is assumed to be constant and equal to that of the environment

of the container.

In general the rate of decomposition (or reaction) of the sample is

given: by the kinetic equation

dn/dt = f(n)k(T) (2a)

where f(ri) is an arbitrary but often simple function of the fraction decomposed and k(T) is the rate constant. For many cases the rate may be
considered independent of concentration and f(ri) held constant at a

value near its maximum, for the temperature range in question,

yielding a commonly used form

dn/dt = f(n)jj^k(T) (2b)

This approximation is most appropriate when Q' is high and the rate

increases very rapidly with temperature: an explosion occurs before

appreciable decomposition occurs, or while n is changing slowly.

The usual assumption made for the rate constant is that it has the

Arrhenius form

k(T) = A exp(-E/RT) (3)

although this is not necessary. Here A, E and R are the preexponential

factor (dimensions of time , the activation energy and the universal

gas constant.

Analytical treatments recast equation (1) into dimensionless form

f and introduce three parameters, S , Bi and B or a related parameter, B^,

which will be used throughout this discussion.

I The first, 5 , is the ratio of the rates of change with temperature

of the heat generation and heat loss rates on the right hand side of

eq(l) evaluated at T . o

5* = (Q'/h')(E/RT^)f(n) exp(-E/RT ) (4) o m o

f A thermal explosion, for this model, can occur only if 5 > 1/e, e

' 2 = 2.718...). When 5 < 1/e, T-T < RT /E. Since RT /E is usually — 0—0 o 0.001 to 0.01, the critical temperature rise is small...

In the working notes of the appendices the symbol f(n ) is used in place m

of f(ri) mm. f (ri ) is mathematically incorrect,

18
The cr-ucial importance of 5 is that it is a measure of sensitivity

to explosion and that its values near the critical temperature can be

predicted. This can be done for specific heat transfer conditions using

simple steady state heat transfer theory. An example is given in the

development of eq. B-31 and agreement with exact solutions is shown to

be about 10 per cent. This approach works because, just prior to explosion,

the material is almost in a steady state (dT/dt is almost zero).

The second parameter, Bi, is the Biot number for heat transfer. It

compares the efficiency of heat transfer between sample and environment

to- that by conduction within the sample. In general it is defined as

Bi = a^l/\ (5)

where a is the overall heat transfer coefficient per unit area of

o contact between sample and environment, Z is a characteristic length,

and \ is the thermal conductivity within the sample. For a sphere I is

the radius. Small values of Bi .correspond to efficient conductive heat

transfer and, hence, nearly uniform temperatures within the sample. As

Bi increases, thermal gradients build up within the sample.

The third parameter, B, is called the dimensionless adiabatic

temperature rise

B = (Q'/C) (E/RT^^) (6a)

3 is a measure of the total rise in T (for the sample) above T in the

T explosion temperature test when h is set equal to zero in eq. 1 (after

T has risen to 1^^) • More . exactly , B is the maximum value of the dimension^

less temperature rise, G,. where

0 = E(T-T )/RT ^ (7)

The related parameter, B^, is defined as

= (Q'/C) (E/RT^^) (6b)

19
1 I where T. is the value of T when 5 * 1/e, the critical value of 5 in

eq. 4. The criterion for a discemable thermal explosion in the explosion

temperature test is stated to be that the value of should be greater

than a number ranging from 5 to 100, depending upon the literature

source. (The preceding statement is ordinarily stated using B. Since

T^ must be greater than T^ for a thermal explosion to occur and B^ is

the maximum value of B under explosive conditions, B^ rather than B is

actually meant.) B^ is a measure of the power of an explosion.

We now return to the explosion test described in 4.2. Analysis of

the temperature-time and fraction-decomposed-time relations (Appendix

B.4) shows that the isothermal sample model based on eq. 2b approximates

more realistic cases provided the change in f(ri) is small prior to

f thermal explosion (e.g. for explosives) and provided 5 is only slightly

larger than 1/e. The warmup time (that in which T rises from T^ to T^)

I when 5 is close to 1/e is short compared to the "induction time" (that

2 2

time in which T rises from T to T + 2RT /E. T + 2RT /E is a rough

o o o o o ®

measure of the temperature of the sample at the onset of an explosion)

.

f The induction times become shorter as 5 (and T^) increase but under

I most conditions the warmup time is insensitive to 5 . The fraction of

the sample decomposed prior to explosion decreases as B^ increases.

T When the "ignition regime" is reached, 5 » 1/e, the warmup period is

no longer short compared to the induction time, due mainly to the decrease

in the latter. One result of this interplay of variables is that a plot

of the logarithim of explosion time versus 1/T^ is not linear and its

slope- is not a good measure of the activation energy.

4.3.2 Criteria for Explosion

The analysis has been extended to the more general isothermal sample

kinetic equation (2a), that is with f(ri) a. variable. This allows for

20
depletion of reactant. Here the sharp distinction between explosive and

non-explosive states disappears. Only numerical solutions are appropriate.

Details are given in Appendix B.5. Four types of explosion criteria

were examined for these more realistic cases and are the basis for later

conclusions. They are:

a) The value of T for which a one percent change in the parameters

o that characterize the critical condition (mainly 5’) can cause a

doubling in the maximum temperature that the system can attain.

b) An inflection point in the curve of temperature versus fraction,

of material decomposed; that is^ onset of upwards curvature in T

when T is greater than T^.

c) An inflection point in the temperature-time curve.

d) High sensitivity of the temperature-time or temperature-

• f fraction material decomposed curves to small changes in T^ or h .

This is the mathematical analogue to the experimental irreproducibility

found in replicate experiments when T^ is near its critical value.

For simple reactions with

f(n) = (l-n)"^ (8)

where n is a positive integer, and with greater than ten, all of

these conditions yield results that indicate that the critical value of

T 5 has the value l/ef(ri^). is a constant for a particular experiment.

f(n^) ^ (9)

In this expression f (n) is the maximum value of f(n). a., and a^ are

m . 12

weak functions of B^ and' can be approximated with constant values of 1

and 2.4, respectively. B.^ is given by eq. 6b where T. is the value of T

« o when o' = (l/e)f(ri^). Where thermal explosion phenomena are easily

discernable, criteria a) through c) give essentially the same critical

21
conditions for a first order reaction whila criterion d) classes the

material as more hazardous. When thermal explosion phenomena become

less discemable (1/B^ 0.05 to 0.1), the predictions of the range of

the non-explosive states for a first order reaction diverge rapidly (for

reasons discussed in Appendix B.5.). Criterion a) says that if B £ '^<20

all states are non-explosive,, while, for criterion b) , this limit is B^ ^ 4. Criteria c) and d) say explosive states occur well past B^ 4;

the range of non explosive states being larger for criterion d) than c)

.

(The predictions of the criteria also diverge for higher order reactions

even when explosions are easily discemable.)

These results mean that a single criterion, however arbitrary, must

be adopted for the comparison of theoretical treatments and for application to experiments.

4.3.3 Extension to Non-Isothermal Samples and Complex Kinetics

Heat transfer within the sample may be by conduction or by convection (forced or free) or a mixture of these. The temperature profiles

will be different and are very difficult to analyze, especially when the

sample 'geometry is irregular. The isothermal model is the limiting

approximation of these complex cases in which the temperature gradients

are small. If heat transfer is by conduction, the gradients are small

provided that the Biot number, eq. 5, is small. If this condition is

satisfied, then the isothermal model can be used up to the time a

thermal explosion does occur"^. The isothermal model can apply with

reasonable accuracy throughout the course of the entire decomposition if

heat transfer conditions are arranged so that a thermal explosion never

occurs or if B^ is very small.

"**It is still assumed the reaction kinetics are simple and homogenous and

that the environment boundary condition is a constant, uniform ambient

temperature. References to other boundary conditions are given in

Appendix B.6.
The expression for che critical parameter, 5 , of non-isothermal

but physically homogeneous samples can be written in the form of eq. 4

when heat transfer within the sample is solely by conduction and the

environment temperature is uniform. For example, for a sphere one

f f obtains eq. (4) if Q /h is given by

^ - |r^(0.101)(-^ + 1) (10)

Q is the heat generated per unit volume, X is the thermal conductivity,

and r is the radius of the sample. (The numerical values of 0.101 and

3.29 as well as the functional dependence on Bi are those predicted by

the extension of the isothermal model outlined in Appendix B.3. The

f values of 5 predicted by eqs. 4 and 10 agree with those of exact

treatments within 10%). Equation 10 or analogous equations for other

simple geometries and eq. 4 can be used to predict the conditions for

thermal explosions from laboratory test results. For this reason, eq. 4

can be viewed as a scaling relation.

The chemical reaction may be autocatalytic (promoted by products)

occur in two phases (heterogeneous) , or have a multiple step mechanism.

It has been shown by others that there are relatively straightforward

mathematical techniques for extending the analysis for critical conditions

of a single reaction to these cases using the isothermal model provided

we approximate the various f(Ti) as constants. The literature on this

topic is reviewed in Appendix B.7.

4.3.4 Programmed Heating of Unstable Materials

This model has been ' described in section 4.2 and is discussed in

Appendix B.3. Only the case of linear heating of the environment is

considered here although alternatives have been suggested in the fields
of thermo gravimetric and differential thermal analysis. The rising

temperature of the environment forces a similar rise in temperature in

the sample. There is a critical value of the heating rate, w^, that is

analogous to T^, the critical environmental temperature in the step-j\jmp

case. But the temperature of the environment at which the maximum race

of decomposition of the sample occurs (a peak on a temperature-time

plot) is greater than the critical explosion temperature, T^, of the stepjump case. This different behavior is due to the increased importance

of extent of reaction. At low heating rates more material can decompose

in a given temperature interval than at high heating rates. Numerical

values of this critical heating rate can apparently be predicted to

within the order of fifty percent or so by an approximate analytical

theory based on the assumption the material is always in the quasisteady

state (i.e. dT/dt = w) if w is less than the critical value.

4.3.5 Conclusions

(1) While the consequences of a thermal explosion may be different

for a material that is an explosive by design (e.g. TNT) and one which

is only capable of mild burning, the conditions for a thermal explosion

to occur are the same. The conditions are dependent upon the heat evolution

rate of the material as a function of temperature, the heat transfer

characteristics of the material, its container and container surroundings

,

and the fraction of the material that has decomposed. Thus, statements

to the effect that a material is thermally unstable or thermally hazardous,

while common (in hazard test literature) and convenient, are incomplete.

The amount of material and packaging conditions are an equally itiportant

consideration and can completely alter whether a thermal explosion can

occur for a given ambient temperature. It is also relevant to note that

24
the conditions that determine whether a thermal explosion occurs depend

upon the heat evolution rate as a function of the fraction of material

decomposed and temperature just prior to the explosion. Specific or

detailed mechanisms of decompositions are not, per se, relevant except

as they affect heat evolution rates.

(2) Statements to the effect that a discernible thermal explosion

does not occur if the heat liberated by a single exothermic reaction,

more exactly the parameter of eq. 6b, is below a certain range of

values are theoretically meaningless unless the criterion for a thermal

explosion upon which the statement is based is also specified. Practically, it appears that the statement is true if is less than 5 to

10. Statements to the effect that there are no critical conditions at

all if the rate of decomposition of the initial material depends upon

the fraction of material decomposed are misleading.

(3) The sharp simple mathematical distinction between an explosive

and a non-explosive state of a material occurs only in the approximation

that the rate of decomposition of the material is independent of the

amount of material decomposed, (i.e. the zero order reaction). The simple

distinction is lost when the variation of the rate of decomposition with

fraction of the sample that has decomposed is taken into account. This

loss does not mean, however, that critical conditions based on the zero

order approximation are invalid but, rather, that the conditions must be

modified. The investigations examined in the study indicate that the

modification of the zero' order critical values of the parameters that

dictate whether or not an explosion occurs is simply to multiply them by

a factor that takes into account reactant consumption. For reactions in

which n is a positive integer in eq. 8, this factor differs negligibly

from one, for our purposes, if B^ is large (i.e. greater than 100) since

25
f(n) equals one. For simple reactions, tlien, it seems reasonable, for

m

f our purposes, to approximate the critical value of 5 as l/(ef(ri).^)

unless the material is only weakly unstable.

(4) Identification of thermal instability . It is our conjecture

that further theoretical and experimental studies of the stability

criteria studies of single reactions, will show that the critical conditions

for a thermal explosion are close to- those of explosion criterion b),

listed in section 4.3.2. It is our proposal that criterion c) , an

inflection point in the temperature time plot when T > T^, be tentatively

adopted as the analytical condition for identifying thermal explosions

until it can be established that the less conservative criterion, b)

,

an inflection point in the temperature-reacted fraction plot, is a more

* realistic condition (i.e. in practice).

(5) It seems reasonable on physical grounds to conclude that if a

material decomposes by a sequential set of decomposition reactions, the

I T reaction which has the lowest critical temperature (i.e. Q , E, h for

each reaction in the sequence would be different) is the trigger which

determines whether or not a thermal explosion takes place. for the

"trigger reaction" may not determine whether the thermal explosion is

discemable. For example, for the trigger reaction may be small so

that only a minimal amount of self heating occurs. If, however, Che

liberated heat is sufficient to raise the temperature of the sample

above for another reaction for which B^ is large, a vigorous explosion

may occur.

C6) Suppose one rates the degree of thermal instability of a given

material-packaging arrangement by the magnitude of the value of (i.e.

t the ambient temperature, T^, when 5 equals 1/e). Then the degree of

T f instability will be more sensitive to changes in E than Q , h , or A.

26
is nearly linearily proportional to E/R but varies only as the inverse

' ' +

of the sxim of the logarithms., of Q , h , or A. For the same reason,

the thermal instability would be relatively insensitive to variations in

Major changes in the degree of instability of a given material are

possible only through reducing the thermal relaxation time of the material

by either reducing the amount of material or improving heat transfer

conditions for each package. Qualitatively, may be viewed as a

sensitivity criterion for a thermal explosion and of eq. (6b) as a

part of the factor that evaluates the damage potential of a thermal

explosion. The above comments mean that the sensitivity and damage

potential, as measured by B^ of eq. 6a, are not closely correlated.

(7) The time to explosion in either the step jump or linear heating

cases is not a measure of the sensitivity or power of a thermal explosion.

For materials decomposing by a single reaction, the time to explosion

may be very long if the decomposition is by an autocatalytic mechanism.

Yet, T^ may be low and B^ (the dimensionless adiabatic temperature rise)

large which makes this material dangerous from both a sensitivity and

power point of view. However, under accident conditions the time to

explosion can be viewed as a sensitivity criterion for a thermal explosion

t if 5 exceeds its critical value in the sense that some fraction of this

explosion time is the time that is available to take emergency response

action.

thermal relaxation time of the material. This formula accounts for the

better correlation of T^ with E than with Q’ observed in the first MBS

study for DOT.

27
4.3.6 Definition of Thermal Instability and a Thermal Hazard Criterion

It seems clear that thermal instability in a transportation context

can be defined as the capability of producing a thermal explosion. However,

this is too restrictive because the term thermal explosion applies only to a

limited group of self-reactions. (It applies only to materials whose

composition and rate of evolution of heat woxild be spatially uniform

through the course of a hypothetical experiment during which the material

is kept uniform during its self-reaction.) The remaining group of reactions,

heterogeneous reactions in inhomogeneous materials must be considered

since those that evolve heat show the same general type of critical phenomena

described in the explosion temperature test, and reactions of this type

(e.g. surface air oxidations) are common.

Also, the definition of thermal instability must take into account

the fact that any parameter which can alter the heat balance anywhere in

or on a package will have a critical value above (or below) which a large

rapid increase in the temperature of the package contents can result. Besides

the parameter we have discussed, ambient temperature, there are others

pertinent to the transportation situation such as radiant heating of a

package, or the insulation (i.e. heat transfer coefficient) between the

contents and surroundings of a package. Other modes of initiation of

decomposition woxild be local heating by shock or impact which apply to

more unstable materials.

To cover these various cases we propose the following definition of

thermal instability . A thermally unstable material package configuration

is one ,

(1) composed of substances that can decompose or react with the

evolution of heat,

(2) for which a thermal mechanism of initiation of a temperature

rise is present (e.g. a slight increase in external temperature), and

28
(3) in which a rapid, large increase in the temperature of a

material can occur.

For the purposes of defining a test for thermal instability, we

propose that condition (2) be modified to:

(2) for which a slight increase in a uniform external temperature

can initiate a decomposition or reaction.

Evaluation of the thermal hazard posed by a package involves additional

considerations. A material can decompose exothermally and "explode"

without a thermal explosion, as defined in section 4.2, having occurred.

For example, consider a material in a closed container that decomposes

with the liberation of heat and gases by a simple autocatalytic mechanism.

A thermal explosion does not occur until the fraction of the material

that has decomposed, n> is near 1/2. The hot gases could have ruptured

the container long before this value of n is reached.

It is also possible for a thermal explosion to occur and yet the

hazard be small. An example might be a case where the decomposing

material liberates no gaseous products, the adiabatic temperature rise

of the material caused by the decomposition is small enough that little

vaporization occurs, and no other decomposition reactions are triggered

in the material. The hazard of a spilled hot decomposing material of

this type might be dictated solely by cargo compatibility considerations.

On the basis of these considerations, it seems reasonable to state

the following:

a thermal instability hazard exists for a specific material-package

configuration at a given ambient temperature if a slight increase

in the temperature will cause the material to decompose and result

in destruction of the container or objects in the vicinity of the

package

29
The lower the Tnininrnm ambient temperature for which a hazard exists, the

more hazardous the material package configuration. In the absence of

other conditions this minimum ambient temperature will be the critical

temperature for a thermal explosion.

These definitions are proposed for three reasons. First, it covers

the entire spectrum of materials loosely described as being thermally

unstable. It applies to explosives, near explosives (e.g. organic

peroxides), rocket propellants, monomer with insufficient inhibitor, and

cellulosic materials which can undergo spontaneous combustion. (From

a practical point of view many materials in these classes are not hazards

as shipped. Wood is an obvious example. Also, explosives that may be

shipped that are thermally stable at ambient temperature but may become

unstable at elevated temperatures.) Second, the definition is consistent

with existing safety technology in storage and handling of explosives and

other energetic compounds [21, 24]. The third reason is that the physical

parameters needed to calculate the critical temperature in practical

situations are, in principle, identifiable. Also experimental procedures

to measure these parameters in small scale tests have been proposed.

Experianae with one such test is stimmarized in Chapter 5.

- 30
4 . 4 Thermal Explosion Experimentation and Self-Reaction Hazard Tests

A tentative definition of thermal instability has been given in

section 4.3.6. It would be identified as an inflection point in the

temperature time curve followed by a rapid rise in temperature of a

material (see section 4.3.5). The thermal instability hazard of a

material-package combination is also identified: the minimum uniform

ambient temperature that will cause the material to decompose and

destroy its container or objects in its vicinity. The latter definition

implies that either the temperature or pressure rise within the con-

‘tainer during the hazardous event.

Since these definitions rest upon thermal explosion theory, it is

advisable to test them against experimental work and to examine their

usefulness in terms of current self-reaction hazard testing. To do

this we proposed the following three sets of questions given below.

Answers are discussed in the remaining parts of this section.

(1) In what respects have the various aspects of thermal explosion

theory been verified experimentally? Have the problems associated with

heterogeneous reactions in solids or the heat transfer problems in real

decomposing liquids (free convection or stirring by gas bubbles, etc.)

been investigated experimentally?

(2) i'/hat is the current state of the art of self-reaction hazard

testing and is the proposed definition of thermal instability consistent

with this practice? If so, what other parameters or characteristics of

a material or material-package arrangement are used to characterize the

self-reaction hazard besides sensitivity to thermal explosion? 'I'Hiat

constitutes a complete test for thermal instability if the proposed

definition is correct.

31
The third question is based on our experimental work, reported in

chapter 5.

(3) In differential scanning calorimetry on thermally unstable materials, can the errors caused by temperature gradients in the sample be

reduced by a radical change in experimental conditions? Can this be

done without destroying the usefulness of the method? An error analysis

based on the theory summarized in section 4.3 suggested that an improvement would result if the effective thermal contact between the sample

and its container was kept large while the thermal contact between the

container and its environment was kept small. What are the merits and

limitations for this plan and what evidence for it is in the literature?

4.4.1 Experiments on Explosions in Gases

A limited number of experimental articles on gaseous systems were

reviewed and are summarized in Appendix B.9. In brief, the articles

indicate that the predictions of thermal explosion theory have been

confirmed for simple gaseous systems that decompose by a single reaction.

Rayleigh numbers must be less than 600 for purely conductive explosion

theory to apply. Measured critical temperature distributions and calculated

values of 5^ (i.e. using eq. B-31 with experimental values for the

various parameters and the critical value of the ambient temperature)

agree with those predicted by theory (i.e. in the case of 5^^, the value

given by eq. B-30 for infinite Bi and 3.32 for for the sphere).

Values of the energy of activation determined from critical pressuretemperature measurements agree with values determined by other techniques.

An exp-lanation has been given for the failure of the measured critical

pressure to vary as the nth root of the thermal conductivity of the

gaseous reactants, where n is the order of the reaction, that had bean

32
i

I

i

i

i

observed in many earlier investigations. More recent work has been

aimed at developing a stirred-flow reactor so that thermal explosion

studies can be carried out under conditions for which the isothermal

model applies.

4.4.2 Experiments on Explosions in Condensed Phases

A limited number of articles reporting experimental work in the

USSR on condensed phase explosives were reviewed (see Appendix B.IO.)

These articles were selected because they amplified a review article on

experimental work by Merzhanov [22] and cover a time span up to 1970.

•They describe a sequence of experiments that clearly is aimed at establishing

experimental conditions under which the isothermal model mentioned in

section 4.3 can be applied not only to the measurement of critical

conditions for thermal explosions but also to the rate of heat evolution

during the entire course of a decomposition reaction. The compounds

most often studied were two explosives, DINA (dinitroxydiethylnitramine)

and tetryl (N-methyl-N, 2,4,6 tetranitroaniline) . Both decompose

rapidly or thermally explode only in the liquid state. A brief summary

of the sequence of experiments is as follows.

(a) DINA and tetryl both decompose by a single reaction provided

gaseous products are allowed to escape (i.e. to prevent catalysis or

other reactions by products) . DINA decomposes by first order and tetryl

by simple autocatalytic kinetics. Kinetic parameters obtained by isothermal (i.e. constant temperature) differential calorimetry and weight

loss measurements (i.e. ‘thermo gravimetric analysis) agreed within their

combined uncertainties.

(b) Proof that thermal explosion theory applies to these compounds

was developed as follows. Experiments were first carried out on large

33
quantities of either DINA or tetryl (200-300 grams) under stirred liquid

conditions such that the (measured) temperature gradient in the liquid

is negligible. Measured and calculated values of the critical ambient

temperature, T^, the Semenov warmup of the material (temperature rise of

the material above the critical ambient temperature prior to explosion,

2

RT^ /E) and induction times to explosion (calcxilated by numerical integration) agreed within their uncertainties. The importance of including

the approximate effect of the changes in the volume of the decomposing

I f material as they effect h and Q in the above calculations and measurements was demonstrated.

(c) The conditions for the isothermal model to apply to explosion

measurements on unstirred liquids was shown to be that the effective

Biot number (eq. 5) must be equal to or less than one. Under this

condition, for example, a maximum temperature difference of 1 K within a

one gram sample was observed fjor a 10 K Semenov warmup. Also, critical

explosion temperatures for a one gram sample were kept near those observed

in stirred 200 gram samples by keeping a , the overall heat transfer co- -4 -2 efficient of eq. 5, small (i.e. order of 10 W«K*cm in the design of

the sample container-thermostat apparati. The thermal instabilities of

explosives were ranked by direct comparison of explosion temperature

measured in identical containers"^ (disposable glass Dewars in this

case)

.

(d) The design conditions outlined in (c) are necessary conditionsfor the isothermal model to apply to the entire course of the decomposition

in isothermal and scanning (i.e. linear heating) differential thermal

+ * * In this way h of eq. 1 is the same for all materials. Since h is small,

the behaviour of a large amount of material is simulated.
analysis measurements. Degeneration of a thermal explosion (i.e. reduction

of defined in eq. 6b) can be accomplished by dilution of the reactants

with a material of large thermal conductivity and thermal diffusivity.

This permits isothermal kinetic measurements to be made above the critical

explosion temperature of the material with minimal temperature gradients

in the material. An alternative procedure (which is currently believed

to be less effective) that can be adopted if is to be kept small is

the use of a metallic container of large heat capacity placed in good

thermal contact with the material, a then is the heat transfer coo efficient between a unit area of the container and thermostat. One is,

in effect, then including both the heat capacity of the container and

sample in C in the definition of in eq. 6b. Dilution alters C of the

material directly. This work confirmed our idea (cited earlier in this

section) about how to reduce errors caused by temperature gradients in

the sample when using heat conduction calorimeters.

(e) Some studies provided information about complex chemical

reactions. These include studies on autocatalysis by decomposition

products, heterogeneous reactions, modification of the rate of heat

evolution as a function of temperature due to the rate of heat evolution

caused by temperature due to vaporization of reactant in product gas

bubbles, the effect of free convection on the critical parameters o

(i.e. studies at large Rayleigh numbers), and the combined effect of

product gas bubble evolution and free convection on the critical paraf meter, o These studies can all be interpreted within the formalism

of thermal explosion theory. However they all introduce new extensions

of the art and suggest that additional details will be revealed by

future experiments.

35
4.4.3 Conclusions Concerning Thermal Explosion Experimentation

(1) Recent experiments on thermal explosions justify the use of

theory as a design parameter and a tool for predicting thermal instability.

(2) Studies specifically designed to determine the predictions of

thermal explosion theory have been carried out on unstirred gaseous and

stirred and unstirred liquid (i.e. at the onset of thermal explosion)

reactants. Experiment and theory agree where the reaction is simple and

known. Differences probably are attributable to complexities in reaction

mechanism or experimental apparati that have not been taken into account

(e.g. the former is our view of ref. [93]). Fewer studies of the t3rpe

mentioned above that deal with solids, which necessarily involve heterogeneous reactions, have been analyzed. Intuitively, it seems clear,

however, that any possible divergence between theory and experiment is

attributable again to complexity.

(3) The specific form of the effect of free convection on the

f critical parameters, 5 , can now be calcxilated if the Rayleigh number is

known (i.e. Appendix B.IO, ref. [114]). (Thus, for liquids, in which

the effects of gas evolution can be neglected, and for gases, the scaling

relation of a fundamental is known if decomposition occurs by a simple

machanism. ) The form of the effect of bubble evolution on the critical

parameters for a decomposing liquid is known at least in part (i.e.

Appendix B.IO, ref. [113]) and the nature of the "competition” between

bubble evolution and free convection (i.e. Appendix B.IO, ref. [115]) is

known. Variation of the critical parameter seems to be specific for

each liquid system. The effect of"

a

complex reaction involving a

simultaneous (i.e. competitive) first order and first order autocatalytic

decomposition on the critical parameter is reduceable to the single
autocatlytic reaction format (i.e. f(ri) = + ri)(l“n)). Qualifications

seem to be that the ratio of the first order rate constant divided by

the first order autocatalytic rate constant must be small (i.e. Appendix

B.IO, references dealing with tetryl, [105, 109]). In cases where

materials appear to melt with decomposition (Appendix C.2, ref. [149,

150]), the critical parameter is calculated from the maximum reaction

rate which occurs when "melting" is just complete.
4.5 Hazard Tests

4.5.1 General Description of the Field

To answer the second set of questions cited in the introduction in

section 4.4 (i.e. are our definitions of thermal instability and thermal

instability hazard consistent with the state of the art in hazard testing?),

a S3mthesis of what seems to be the state of the art in the self-reaction

hazard evaluation field (abbreviated SRHE below) has been attempted. In

the first part of Appendix B.ll, our view of the broad picture of the

field is stimmarized. This was constructed from work from such sources

•as government organizations (i.e. DOT, USA; RARDE, U.K. , and RVO-TNO,

Netherlands) concerned with safety regtilations in transportation, the

areas of handling and storage of industrial chemicals, fire research,

and the handling and storage of industrial and military explosives. The

spectrxom of materials involved in these articles covers explosives,

organic peroxides (near explosives), rocket propellants, and various

unstable materials important in commerce such as ammonium nitrate

fertilizers

.

A coherent picture of the SRHE field could be constructed by sorting

the tests according to the following criteria:

(a) Stimulus for initiation of decomposition: mechanical or

thermal.

(b) Relationship of the particular stimulus to transportation

accidents.

(c) Parameter measured: sensitivity to initiation of decomposition

or damage potential.

(d) Purpose: reproduction of accident conditions or evaluation of

physical properties of a material (fundamental parameter test).

38
(e) Scaling: evaluation of the difference in behavior of materials

in small scale tests and in bulk hazard conditions.

Details of the analysis are given in Appendicies B.ll, G.l and C.2.

Seven mechanical stimulus tests are sximmarized in Table B-4, ten thermal

and ignition tests in Table B-5 and five types of calorimetric tests in

Tables C-1 and C-2.

Two comments on the terms used above are pertinent before proceeding

to more general matters. "Damage potential" tests measure the amount

and rate of liberation of energy or gases associated with a thermal

explosion, deflagration or detonation. As such they are very closely

related to our definition of thermal instability hazard. "Fundamental

parameter" tests involve the important assumption that the algorithm for

predicting hazard characteristics is known.

Application of these criteria made it possible to understand the

otherwise bewildering array of. tests and to identify what each does.

The criteria should be appropriate for any proposed tests. However, this

study did not identify any single test that is sufficient for the

determination of self reaction hazards posed by a wide variety of

materials .

This is recognized by workers in the field. The self-reaction

hazard rank of a material is constructed from a hazard profile based on

a number of test results. Tests of both damage potential and sensitivity

to decomposition by both thermal and mechanical stimuli are involved.

Tests on commercial chemicals are carried out on samples in various

physical foirms and with different amounts of chemical impurities in

order to assess the effect on hazard rank due to these variables. Tests

of the effect of the container are carried out since the degree of
confinement can affect greatly the hazard rank. There are indications

that the effect of the container will or can be of decisive importance

in ranking the hazard of borderline unstable materials. Also, the

concept that a reduction in the hazard rank can be obtained by appropriate packaging (which is known to apply to explosives) , evidently

applies to borderline hazardous materials also.

The current trend is to develop and incorporate more fundamental

parameter tests results in constructing hazard profiles. This type of

test can eliminate debates over the unambiguous meaning of "accident

•condition" tests as for example, the PVT test (No. 11 in Table B-5)

.

This trend does not imply that there is a trend to eliminate "accident

conditions" tests. For example, full scale tests, e.g., bonfire tests,

are necessary (and sufficient) to resolve ambiguous hazard rankings

associated with difference between small scale tests and bulk conditions.

It was of interest to note that at least one experienced investigator

felt the problem of attempting to improve the precision of test methods

and results is compounded by the present lack of any quantitative data

on thermal or mechanical stress in accident conditions.

4.5.2 Thermal Sensitivity Tests

Nineteen types of thermal sensitivity tests have been analyzed in

order to determine what would constitute a sufficient single test or a

series of tests comprising a test protocol for thermal instability. No

single test now being used is sufficient. Three of these tests are noncalorimetric, but are in' the fundamental parameter class: the thermal

surge test, the self accelerating decomposition temperature test (SADT)

,

and methods that determine the relationship between critical temperature

and sample dimensions. They are summarized in Table B-5 and discussed

in Appendix B.ll.

40
The thermal surge test is the pulse heating of a small, fully

enclosed sample. Time to explosion and kinetic parameters are obtained.

There is poor agreement with other techniques. Problems of interpretation limit its acceptability as a fundamental parameter test. The SADT,

no. 15 in Table B-5, detects self heating in a material sized and packaged

as it will be shipped. It has a major advantage in that no scale-up is

required. Use of it has been restricted to organic peroxides. The

results are a conservative measure of instability because the temperature of onset of seif heating may be less than critical. The third set

of non-calorimetric tests, critical temperature versus sample dimensions,

is important because it provides the link between laboratory scale tests

and bulk conditions. These tests show that thermal explosion theory

provides a suitable formalism for scale-up.

Of the remaining sixteen tests, three are applications of adiabatic

calorimetry (Appendix C.l) and thirteen of heat conduction calorimetry

(Appendix C.2). The principles of these calorimetric methods are described below.

4.5.3 Adiabatic Calorimetry

Here the approach is to eliminate heat loss from the sample and its

container. To do this the temperature of the environment is made to

track that of the sample. The temperature of the sample is measured as

a function of time. A typical system is shown in Figure 1. Approximate

temperature equality is maintained either by heating a shield E electrically,

or by forcing an externally heated gas through the shield and around the

sample container 0. The first method is used in the adiabatic storage

test and the accelerating rate calorimeter (ARC)

The second is used in

the adiabatic self-heating test. Pressure effects are measured onlv in
Figure 1. Sketch of Cross Section of an Idealized Adiabatic Calorimeter

A, difference thermocouple for controlling Tg to be equal to Tq; E,

adiabatic shield; H, electrical heater for controlling adiabatic shield

temperature; L, calorimeter supports; M, decomposing sample; 0, sample

container; T, temperature of sample, T^, temperature of adiabatic shield;

Tq, temperature of sample container; temperatures of sample, container,

and adiabatic shield assiimed to be uniform.

42
Che ARC, which has been used primarily for liquids. The other two tests

use unconfined samples, but this does not appear to be a basic requirement.

Rate constants and time to explosion can be extracted from adiabatic

calorimetry tests. This requires careful interpretation of the results.

If the kinetic mechanism is simple, rate constants can be extrapolated

to ocher temperatures. If Che mechanism is complex, or if appreciable

decomposition occurs before explosion, interpretation of the results can

C be ambiguous, unless separate kinetic studies are made.

In principle, these adiabatic calorimetry tests can model the

'thermal behavior of either a small mass of material in the center of a

bulk shipment or chat of a sample and its container. The nearly automatic

elimination of temperature gradients in the sample is a major advantage

Chat adiabatic calorimetry has over heat conduction methods.

4.5.4 Heat Conduction Calorimetry

Thirteen tests involving heat conduction calorimetry are summarized

in Appendix C.2. In heat conduction calorimeters, the heat liberated by

the decomposition of the sample is measured directly by allowing it to

flow CO its environment only through a heat flow meter. The temperature

of the environment is independent of chat of the sample. It may be held

constant (isothermal mode of operation) or be varied with time (programmed

heating or scanning mode)

The basic design principle (see [157]) is shown in figure 2. The

heat flow meter is a thermopile placed between the outer surface of the

sample holder, S, and the inner wall of a block, S, whose temperature is

controlled. The sample holder has within it the sample, 2, and its

container, 1. The thermopile output, A^, is linearly proportional to

Che race of heat liberated or absorbed by everything within the sample

holder
Figure 2. Sketch of Cross Section of an Idealized Heat Conduction Calorimeter

A]_, output of thermopile; E, block; H, electrical heater for controlling

block temperature; S, sample holder; Th, two elements of thermopile:

different wires of thermopile; 1, sample container; 2, decomposing

sample; T^, controlled temperature of block; Tg, temperature of sample

holder; T]_, temperature of sample container; T 2 , temperature of sample;

all temperatures assumed to be uniform.

44
A thermal twin of the sample calorimeter (i.e., same thermopile,

holder, container and inert reference material of the same heat capacity),

mounted in the same block, often is used to cancel out variations in the

sample thermopile output due to unwanted temperature fluctuations in the

block. The differential output is linearly proportional to the differences in heat liberated or absorbed by the holders and their contents.

Extremely sensitive single and twin heat conduction calorimeters are

used in tests 1 and 2, respectively, of table C-1, Appendix C.2. To the

extent that these types of calorimeters meet their design principle they

have no calorimetric error associated with different geometrical locations

of heat sources inside the sample holder.

Three simplified versions of the twin heat conduction calorimeter

are applicable to hazard testing. These are "quantitative differential

thermal analysis" (QDTA) , "differential scanning calorimeter" (DSC) , and

"differential thermal analysis" (DTA) . They are usually run in the

programmed mode. They can be used for tests on samples that either absorb

or evolve energy.

In QDTA the thermopile of each conduction calorimeter is replaced

by a heat leak path and a single thermocouple, as shown in figure 3. A

temperature difference is measured. The heat leak must be calibrated.

In this type of calorimeter, the energy calibration source must be both

physically and thermally in the same location as the sample container if

calorimetric error is to be avoided. Provided the latter condition is

met, the output of the differential thermocouple is equivalent to the

differential output of the thermopiles of figure 2.

DSC is a compensated version of QDTA. A typical instrument is

sketched in figure 4. Heat is supplied by electrical heaters in the

sample and reference holders to keep the temperature of both of them at

45
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the same temperature and equal to a program temperature. The net difference in electrical power supplied to the two holders is measured.

The major difference between DSC and QDTA is the electrical feedback

circuitry, to which the term compensated refers, and which the name DSC

does not convey. Both commercial QDTA and DSC instruments are considerably less sensitive than the heat conduction calorimeters mentioned

above (see table C-1)

.

In simple DTA (differential thermal analysis) measurements are made

of the absolute temperature of the sample and the difference between it

•and that of the reference material. All are driven by programmed control

of the temperature of the block, E. A sketch is shown in figure 5. For

kinetic studies this method has the advantage over QDTA and DSC in that

the sample temperature is measured directly. If the design is such that

the temperature of the sample is kept tmiform, this method can be made

calorimetrically equivalent to heat evolution kinetic measurement involving either a QDTA or DSC measurement.

4.5.5 Conclusions Concerning Hazard Testing

These conclusions are based both on material in earlier sections

and in the Appendices.

(1) Determination of the thermally instability hazard produced by

thermally unstable materials is the aim of all the thermal sensitivity

tests with the exception of flash point and ignitability tests. The

critical temperature for thermal explosions is correctly identified as

the maximum safe storage and handling temperature (i.e. where applicable

- see section 4.3.6) in' many but not all fundamental parameter tests

(e.g. see tests 15, 16 of table 3-5, Appendix B.ll and tests 3, 4, 5b. 1,

5d.3 of table C-2)

.

In the latter cases, the recommended maximum safe
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handling temperature generally turns out to be at or below the critical

temperature. Thermal explosion theory is an explicit feature of the

interpretation and application of thermal sensitivity tests that are

advanced state-of-the-art for hazard evaluation (e.g. references and

discussion of test 14, 17, table B-5; Appendix C.l; tests 1, 2, 5b. 2,

5c, 5d.3, 5d.4 of table C-2)

.

(2) Thermal sensitivity tests are only part of the tests required

to construct a realistic hazard profile of a substance to rank its self

reaction hazard in a regulatory transportation context (i.e. see dis-

’cussion in first part of Appendix B.ll, ref. [122, 124, 123]). This

conclusion is consistent with work on general problem of safe handling,

processing, and transport of materials (e.g. ref. [116, 128, 129]). It

is clear that mechanical sensitivity and one or more types of damage

potential tests (e.g. to measure rate and amount of gas evolution,

determine the possibility of deflagration or detonation under confinement

etc.) are required. This confirms and amplifies our conclusion in ref.

[ 8 ].

(3) The thermal sensitivity tests can be organized into "accident

condition" tests and "fundamental parameter" tests. Those not involving

local heating have the general purpose of determining the minimum uniform

ambient temperature required to initiate a thermal explosion (unless

gaseous decomposition product ruptures the container first -a possibility

that can be calculated) . The accident condition tests do this by direct

measurements of the critical explosion temperature of the bulk material

either in its shipping container (e.g., the SADT test, or in a thermal

equivalent of the container (Warmestalagerung test). The fundamental

parameter tests do this indirectly by measuring the heat evolution

50
kinetic parameters (E and A) and in some cases the other characteristics

of the material (X, thermal conductivity) or decomposition reaction (Q,

heat of decomposition per unit mass of reactant) needed to calculate the

critical temperature. The fundamental parameter tests assume that the

scaling relation (e.g. eq. 4 and some geometrical heat transfer variant

of eq. 10) is known. This assumption is discussed in conclusion 3,

section 4.4.3.

(4) The fundamental parameter tests can be classified as calorimetric tests (Appendix C.l and C.2) or direct explosion temperature

’measurements (tests 14, 17, table B-5, Appendix B.ll). Some of the

calorimetric tests explicitly involve the determination of Q and X (i.e.

Appendix C.l, the adiabatic self-heating test determines Q, A, C, and X;

Appendix C.2, test 5c. determines Q; test 5d.3 determines Q and X).

(5) Three calorimetric thermal sensitivity tests (i.e. ref. [148],

Appendix C.l, and tests 4 and 5d.5 of table C-2) determine the amount

and rate of liberation of gases during the decomposition. Gas liberation

is an important parameter for assessment of damage potential and a

sensitivity test with this capability becomes equivalent in this regard

to the fundamental parameter test 12 of table B-5. Since another parameter important in damage potential is Q, it seems clear that a thermal

sensitivity test procedure should measure both Q and gas liberation even

though neither may have an important effect on T^.

(6) The explosion temperature test developed by Bowes (see discussion,

test 17, table B-5, Appendix B.ll) has the important advantage that it

addresses the problem of determining E and A under the equivalent of

bulk conditions. It takes into account so called "model distortions"

(froathing and foaming etc.), and addresses the problem of the scaling

relation as a function of sample size. The difference in rate constants
determined by this method and by isothermal calorimetric techniques on

benzoyl peroxide, calorimetric values of E were 10 to 20% greater,

emphasize that all fundamental parameter tests must attempt to measure

E, A, Q, etc., under conditions simulating bulk conditions. This

necessarily implies conclusions (7) through (9)

.

(7) Small scale explosion temperature tests to check calculated

values of or evaluate E and A from should be run under conditions

when the effective Biot number is small and is known (i.e. see ref.

[104] and Appendix B.IO). This conclusion constitutes a major objection

'to the thermal surge test (test 14, table B-5) apart from the fact it

determines the onset of container rupture rather than thermal explosion.

(The other objection to the thermal surge test, that it gives values of

E in error by 50%, may be due in part to incorrect data analysis, see

Appendix B.ll). The explosion temperature test devised by Rodgers [149] ,

(see Appendix C.2) apparently does not meet this low Biot number requirement.

(8) Small scale measurements of E and A in calorimetric measurements

must, in general, be run under conditions of minimal Biot number. This

is automatically fulfilled in adiabatic calorimetry but usually is not

an explicit part of heat conduction calorimetric measurements.

(9) The predictions of the critical temperature made from parameters

measured in calorimetric tests should be checked directly by explosion

temperature measurements as specified in conclusion (7) . If the Bowes

method (discussion, test 17, table B-5, Appendix B.ll) is used, E and A

should be checked with calorimetric measurements. Since explosion

temperature measurements are time consuming, it seems clear that it is I

expedient to run the calorimetric measurements first. The explosion I

temperature tests must evaluate the effect of sample size. This should 1

be mandatory if the material decomposes in the solid state (in which »

52
case particle size is important) or if there is a possibility that

reaction mechanisms are affected by sample dimensions (e.g. diffusion).

(10) It is clear from studies on explosives (see Appendix B.IO

references dealing with tetryl, [105, 109]) that possible catalysis

and/or alteration of heat evolution mechanisms by gaseous reaction

products when kept in contact with reactants dictate that calorimetric

tests must have provision so that they can be run under closed, small

free volume, conditions. This is explicitly provided in one adiabatic

calorimetric test (ref. [148], see Appendix G.l). This should be a

requirement if maximum hazard or unvented container conditions are to be

determined.
5. Experimental Work

5 . 1 Introduction

An experimental study of the decomposition of thermally unstable

materials was carried out to identify what the various detailed components

and specifications of an experimental test procedure should be in order

to rank thermal instability as defined in section 4.4. The specific

test method that became the focal point for the work is a test method

proposed by ASTM Committee E-27 on Hazard Potential of Chemicals. It is

an application of differential thermal analysis. The method has the

•advantage that it has been proposed by workers experienced in the field

of self-reaction hazards and direct interlaboratory comparisons were

possible. The materials studied included some of those used in an ASTM

round robin test of the method"**.

In effect, three studies were carried out in parallel. The first

study was an analysis of the operation and capabilities of the commercial

DSC apparatus (Perkin-Elmer , Model DSC-2) used in the second study, the

actual experimental decomposition work using the proposed ASTM method.

The third study was an evaluation of the limitations and possible extensions of the test method in context of its use as part of a test procedure

for thermal instability.

The test method and the assumptions underlying it are discussed in

section 5.2. The operation and calibration of the instrument are summarized

in 5.3. Experimental results are given in section 5.4 and discussed in

5.5. Conclusions and recommendations for future experimental work are

collected in section 5.6.

'We are indebted to Dr. Allen Duswalt of Hercules, Inc. for supplying

the test materials.

54
The discussion and analysis of sections 5.2, 5.3, and 5.5 and the

related appendices C.3, C.4, and C.5 apply to extended arguments that

were developed from a study of number of literature sources. These are

the ASTM test method documentation [158], the critiques or reviews of

ref. [152, 159-165], and three texts [166-168].

55
5 . 2 Sunmary and Assessment of the proposed ASTM Test Method"^

The purpose of the proposed test method is to determine the kinetic

parameters of materials that decompose with liberation of heat in order

to evaluate their potentials to undergo thermal explosions. (Reaction

of the material with the surrounding atmosphere, e.g., autooxidation,

could be considered to be within the scope of the method, although this

is not specifically stated.)

The method is an application of differential thermal analysis based

on a procedure introduced by Murray and White (169) and extended by

'Kissinger (170). Later analyses of it by Ozawa (158) are cited in the

test procedure. The basic method was developed for the study of endothermic

decompositions and phase transitions.

In this method samples weighing a few milligrams are heated at

various constant rates, w, in a series of runs. The temperature at

which the maximum rate of decomposition occurs, T^, is determined for

each run. The sets of data points (w, T^) are processed to give the

parameters of the Arrhenius equation, k = A exp(-E/RT).

The method is stated to apply to materials that decompose by a

single exothermic reaction for which the simple general rate law

f(n) = (1 - n)“ (8)

is applicable. n is the order of the reaction and is greater than or

equal to zero. The method is specifically stated not to be applicable

to reactions that are autocatalytic or to decompositions in which the

material undergoes phase transitions in the same temperature range as

that of the decomposition. That is, at least any process that shows a

complex exotherm is to be excluded.

"^This description of the method is based on the third draft of the proposed procedure.

56
The data analysis is based on the kinetic model

dn/dt = f(n)k(T) = f(n)A exp(-E/RT) . (11)

The maximum value for the observed exotherm and that for the rate of

reaction, dri/dt, are assumed to coincide. Eq. 11 is differentiated with

respect to time to identify this maximum and rearranged to yield

g(n^) i -(df(n)/dn)^ = (k"^(dk/dT)(dT/dt))^ = (wE/ART^^)exp (E/RT^) (12)

The term dT/dt is set equal to the programmed scanning rate, w.

Assuming that g(n^), E and A are constants, the basic equation of

the method is obtained by differentiating eq. 12

d Zn w/d(l/T^) = -(E/R)(l/(1 + 2(RT^/E))) = -E/RD (13)

The activation energy, E, is extracted from the slope of a plot of

logiQW versus 1/T^ using the formula

E = -2.303 (R/D) ( log^Qw/d(l/T^) (14)

where D is defined in eq. 13. An iterative procedure is used starting

with an initial value D = 1.051 and a value of T, near the middle of the

d

experimental temperature range.

The preexponential factor. A, is obtained by applying eq. 15

A = (wE/RT,^) exp(E/RT.) (15) d d

to each data point pair and the mean value is determined. This equation

is a rearrangement of eq. 12, with the assiimption that S(O^) = 1.

The kinetic parameters are checked experimentally in an "ageing”

run. The temperature at which 50% decomposition would occur in about

one hour is calculated, assuming a first order reaction. A sample is

aged for that period and then scanned. If the total heat evolved is one

half the amount that would be evolved by the sample without aging, the

kinetic analysis is considered to be verified. This ageing experiment

extends the range of measurements to slightly lower temperatures.
The experimental apparatus is required to be a QDTA or DSC, as the

terms are defined in section 4.5.2. The procedure states that commercial

instruments manufactured by Perkin Elmer (DSC) and DuPont (DuPont 900 or

990 including the DSC module) are suitable. The sample is kept small to

minimize temperature gradients within the sample. A sample weight resulting in a heat generation rate of 8 mW or less is stated to be satisfactory, apparently in terms of the above commercial apparati. The

sample also is to be placed in good thermal contact with and in a container with which it will not react. The atmosphere around the sample

should represent the conditions of usage and either a hermetically

sealed or high pressure container is required for volatile samples.

The method described above has been applied in our experiments, due

allowance being made for instriimental corrections.

Several aspects of the test method and the model merit discussion

here. Others will be covered .later.

1. The Arrhenius rate expression is assumed. This is used widely

in kinetics, with a high degree of success. It is appropriate for the

analysis of data over a limited temperature range, but may not be adequate

for long extrapolations. Deviations from this rate expression, substantial non-linearity in the iln w vs. 1/T^ would be evidence for failure.

But experience in kinetics indicates that very precise data are needed

to show such deviations for simple reactions and that the more common

cause of non-linearity is a change in mechanism with temperature, i.e.,

a change in the form of f(ri).

2. The calculation of the p-reexponential factor, A, using eq. 15

requires g(n^) in eq. 12 to be constant and equal to one over the experimental temperature range. This assumption is examined in Appendix C.5.1.

58
There it is shown to depend upon dT/dt for the sample being equal to the

programmed scanning rate during the entire exotherm. Assuming the

scanning rate to be constant, g(Tl^) is shown to be unity or close to it

for reactions of simple order and to vary slightly with reaction order.

(It may deviate appreciably from unity for autocatalytic reactions.)

This analysis yields an extention of equations 13 and 14 which are used

to calculate E. Again, the corrections are small, when the simple rate

law, eq. 8, applies.

3. The heating rate for the sample, dT/dt, cannot equal the

scanning rate, w, exactly during the exotherm. The errors introduced by

this self -heating are discussed in Appendix C.5-2 and an upper bound on

the error in E is developed. Briefly, for reactions of interest here,

the error in E would be less than 2% for a temperature rise due to selfheating of less than 0.2 K.

4. No specification in the test method addresses the required

constancy of dT/dt during a scanning run (see above) except possibly the

peak power limitation of 8 mW. In any event, some requirement needs to

be identified. It is clear from the critiques by Garn (see [159], p.

82) and Sharp ([160] section III, A. 1) that the assumption of a uniform

sample temperature is or can be a major source of error. Thus, also

identifying precisely how the peak power specification limits this error

seemed to be of paramount importance.

5. Understanding the physical basis for the temperature lags in

the test method is important, although these lags apparently are small.

They are relevant to calibration of the temperature scale of the instrument and are related to the use of metallic melting point standards in

determining the transfer function of the instrument (i.e. correction for

time lags in the ordinate or energy signal)
6 . Ozawa [158], Garn ([159], p. 85), Sharp ([160], II, B.) and

Wendlant ([166], p. 189) indicate that errors of varying magnitudes can

be made in assuming that the peak of a DTA curve coincides with the

maximum in dr|/dt. (Much of the comment is based on the work of Reed, et

al ([172]). Although the error may be small for QDTA or DSC [158], the

assumption of coincidence could be violated.

7. Methods are needed to determine a) whether the reaction is

complex (and how to interpret the data in that case) , b) how far outside

the experimental temperature range the data may be extrapolated, and c)

the form of f(ri) for both simple and complex cases. Ideally these

methods should be independent of assumptions about the constancy of

dT/dt (an essential feature in refs. [161, 162]) and about the form of

k(T). These assiimptions can be avoided when the reaction is not complex

if ri and dr|/dt are measured. This is discussed further in section 5.6.

60
5.3 Instrument Evaluation and Calibration

Two types of instruments, QDTA and DSC, are permitted in the procedure studied here. It is important to determine the conditions under

which they yield the same result, how the data should be reduced in each

case, and, in general, how they operate. Neither measures the temperature

of the sample directly. Heat losses, electronic circuitry and definition

of the temperature calibration all are involved in interpretation of the

instrumental output. Errors may be introduced by approximations in the

models

The theory of operation of QDTA and DSC instruments is given in

Appendix C.3. That section presents (1) energy balance and signal

response equations, (2) methods for determining dri/dt and the temperature

of the sample (when the reaction is simple) , (3) methods for calibrating

the energy and temperature scales of a DSC, and (4) the effect of time

constants on the signal. The calibration of the instrument used here is

presented in Appendix C.4 and then the determination of the transfer

function of the instrument is discussed. Qualitative results of these

analyses are given below.

The simplified QDTA system that was analyzed has the sample and reference containers and holders in the same compartment. It was found

that this lack of isolation introduces negligible erro-r provided that

the corresponding holder and container are in good thermal contact.

It was found that the form of the measurement equations for the

QDTA and DSC were formally identical in terms of the ordinate or energy

signarl of either instrument after it has been corrected for instrumental

lag. This was expected and is consistent with ref. [175].

61
Two methods for determining ri, dri/dt, T, and the heat"*" of decomposition were worked out in terms of the energy signal, the heat capacity of

the sample and sample container, and the heat transfer coefficient

between sample and sample holder. These are summarized in Table 22,

Appendix C.3. The least restrictive assumptions for acceptable performance compatible with parameters that can be characterized experimentally

are that the temperature of the decomposing material and its container

be the same and uniform. Provided volatile products are brought into

good thermal contact with the container before they escape, the container

can be either open or closed. (If results differ under these open and

closed conditions, the mechanism of decomposition is different when

reaction products escape.) Keeping the thermal contact between a unit

area of the sample and its container large keeps the difference between

the temperatures of the sample and container small. Conversely, keeping

the thermal contact between the sample container and the sample holder

small helps to reduce temperature gradients in the sample. At least one

necessary qualification is that the decomposition is or mimics a homogenous

reaction (so eq. 11 applies -see Appendix, B.6).

The calibration of the energy scale of our DSC (determination

of the calibration constant, K) was made with the known heats of fusion

of indium and tin and the known heat capacity of Al^O^ (NBS Standard

Reference Material 720)

.

K was independent of temperature within the

uncertainty of the measurements. This is consistent with the design and

specifications of the instrument.

Depending upon circumstances, either the enthalpy or energy of decomposition at a specified reference temperature.

L 62
The temperature scale and time lags of the instrument were determined

using the indium and tin fusion points. The static temperature calibration was determined from the intercept of a plot of the so-called onset

temperature, > versus scan rate, w. The slope of the plot was

interpreted as suggested by Flynn [191]. Equations are worked out in

Appendix C.3.4 to support his interpretation. The assxmptions were 1)

the sample and sample container temperatures are uniform and constant

during melting, 2) the sample and sample container temperatures are

equal, 3) the ordinate transfer function of the instrument is a first

order differential equation involving the time constant, T^, of the

differential power circuit, and 4) the sample holder temperature lags

behind the program temperature by the amoxint t w where T is the time

constant of the average power .circuit and w is the nominal scanning

rate.

An upper bound of the time constant of the differential power

circuit was determined from the exotherm of partially supercooled indium

in a cooling run.

Equations based on the assumptions of the T^^(w) analysis are

presented for evaluating the varying temperature of the sample during a

scanning decomposition run. They show that the temperature calibration

procedure prescribed for the ASTM kinetic method correctly accounts for

the temperature corrections apart from self heating provided the following

conditions are met: 1) T^w must be small in comparison to (t^ -i- T^)w

where is the time constant of the sample container and 2) the heat

capacities of the samples in the temperature calibration and decomposition

runs are small in comparison to those of the sample containers. (For

other restrictions see Appendix C.3.4.)

63
The heac transfer coefficient between the sample container and

I holder, is needed to determine the temperature offset of the sample

and its container due to self heating. This can be calculated from the

leading edge of pure metal endotherms as both Flynn’s [191] and our

analyses of the versus w plot show. However, this approach is

not correct if the second assumption of the T^^(w) analysis does not

hold during melting. A test for the validity of this second assximption

was developed and applied to the experimental endotherms for indium and

tin. The test showed the temperature of the sample and its container

T are equal for the purposes of calculating

A more exact analysis of the DSC energy balance and electrical

power feedback equations was carried out. Results to date are: 1)

assumptions 3 and 4 of the versus w analysis are valid to a good

first approximation, 2) the condition for the time derivative of the

sample holder to always equal w is that given by O’Neill [176] provided

certain measurable non linear terms in the equation relating the program

and holder temperatures are small, 3) the ordinate signal correctly

accounts for the rate of heat liberated or absorbed by the sample provided certain measurable non linear terms in the equation for the ordinate

signal are small. The first result provides support for the validity

for the temperature corrections in section 5.4 and the error analysis in

section 5.5. The condition of the second result holds for both the

melting point and decomposition studies. Thus, essentially all the

measurement equations we have formulated which involve this assumption

should, be correct in this regard. The third result confirms an argument

given by O'Neill [178] and extends it by suggesting what to measure if

there is doubt. Applied to our studies, the nonlinear therms in the

third result are negligible.

64
5.4 Experimental Work

Preliminary experiments were run on a sample of polyacrylonitrile

(supplied by A. Duswalt, Hercules, Inc.) and three analytical reagent

grade oxidizers: ammonium nitrate (NH^NO^Cc)), potassium persulfate

ammonium dichromate ((NH^)2Cr20^(c)) . The purpose of

the experiments was to become familiar with the problems in studying

decomposition reactions. Samples were run in a variety of containers

including heremetically sealed disposable aluminum containers (Perkin

Elmer Part 219-0061) with a burst pressure of 2-2.5 atm. The instrument

was run with the standard block lid with a flow of dry N

2 gas of roughly

20 cc/min.

A smaller exotherm was observed for NH^N02(c) in a sealed container

than that expected. This was apparently caused by lack of confinement

and/or volatilization of the material. In the case of polyacrylonitrile,

vaporization of the decomposition products caused a large change in heat

lost by the sample holder. Products condensed on the ceiling of the

sample holder chamber. This evidently resulted in a change in the emissivity of the surface and thus increased the rate of radiant heat loss

by the sample holder. To illustrate, the difference in the steady

output signal of the instrument between the initial and final isothermal

temperatures of an initial blank (i.e. empty holders) scanning run was

preset at 0.2 mW. The observed difference for a scanning decomposition

run was 4.3 mW, see fig. 6. The heat loss was progressively altered in

an unknown way to a final value of 4.1 mW. Since the rate of heat

generation was only on the order of 5.8 mW, it was evident that this

extraneous emissivity effect had to be reduced. The results on NH^NO^(c)

indicated that a sample container with a higher burst pressure than 2

atm and a temperature range with an upper limit of 500 C was needed.

65
-AO

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4J 0 3 3 H u 4-1 3

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lu 3 to 31

CJ 1 3

4J

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ft

fti i

ft

ftiIII' I
The high pressure cells that have been used in DSC or QDTA studies

are summarized in Table 1. One reusable and two disposable containers

are available commercially; only the disposable containers fit into the

sample holder, with holder lids in place, of our instrument (0.790 cm

I.D. and 0.32 cm inner depth). A reusable container was designed that

would fit into the existing holders in order that holder lids could be

used"*". A prototype cell fabricated from 750-X inconel appears to be

satisfactory. It was pressure tested to 80 atm and changed weight less

than 5 ug after a first cycling between ambient and 500 C in a dry

nitrogen atmosphere. The gold disc seal must be replaced every two or

three runs. A preliminary run of the decomposition of polyacrylonitrile

confirmed previous observations that the difference in isothermal

signals was due to a change in heat loss by the sample holder. Because

it was clear that further time was needed to perfect, characterize, and

replicate the high pressure cell for routine use, experiments using the

cell and studies on polyacrylonitrile was stopped. We mention this

development work because it was clear at the outset of the experimental

work that a closed reusable cell was also needed to indicate maximum

hazard conditions (see conclusion 2 of section 5.6). Comments on the

needed further development and testing of the cell is given in Appendix

C.6.

The instrument can be run without holder lids provided the same reusable

containers are used in calibration of the energy and temperature scale

of the instrument. The containers must have the same emissivity and the same thermal contact with the sample holder in both decomposition runs

The possibility of systematic error due to temperature gradients is,

however, much greater than when holder lids are used.

67
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68 I1^ ii

II

ii I
Nitrocellulose film, m-azidobenzoic acid, and azidotriphenylmethane

(supplied by A. Duswalt of Hercules, Inc.) were sealed in their containers

in air. Two types of containers were used. Crimped but open aluminum

sample containers (Perkin Elmer part 219-0041) were used for nitrocellulose as shown in fig. 7b. No emissivity effect, as described for polyacrylonitrile, was observed and scanning runs were carried out with the

standard block lid and 20 cc/min of dry nitrogen flushing gas. In the

case of both m-azidobenzoic acid and azidotriphenyl methane samples were

placed in hermetically sealed cells (Perkin Elmer part 219-0062) . Good

thermal contact between sample and cell was promoted by sandwiching the

sample between the bottom of a cell and an aluminum disc forced into the

cell as shown in figure 7a. (This procedure was recommended by A.

Duswalt.) In the case of both m-azidobenzoic acid and azidotriphenylmethane, small emissivity effects were observed. However, they were

eliminated by using a modified block lid supplied by the instrument

manufacturer. The purge gas flow is substantially greater than in the

standard arragement and deposition of products on the lids causes less

change in radiant heat loss by the sample holder.

Sketches of exotherms for m-azidobenzoic acid, azidotriphenylmethane,

and nitrocellulose are given in figures 8 to 10. The results for the

ASTM test are summarized in tables 2 to 4. The various symbols and

their meaning are given in table 5. The nominal exotherm peak temperature, T^ (nominal), in column 4 is the instrument program temperature

corresponding to the exotherm peak of the observed output signal, 0, in

the decomposition run minus the output signal, 0^, in a subsequent blank

run when only reaction products are present in the sample container.

The uncertainty for column 4 is estimated to be 0.1 to 0.2 K at minimum

and on occasion may have approached 0.5 K to 1 K.

69
70 s

CJ <-l

cfl }-• E

0) S-l c <u •H

0) ucO {2 U CO

•* T3

cO "H O rH • * )-l T3 0) •H C

iH tH

cO .H U

rt cs E O

i-i o

(U " u X

cn

CO •"

cn

(U • •> d

t3 *h

CO •Hd

;-l rH

U

(U

d

d 5-4

O

•H oj

y

d

d

u •H

y

d

d rH o U CLi o d E

(U o d • y CO r—1

du. * E • E d-3 y d hJ d- -H w •H CU S E

<4-4 Sj 3 d o y rH CO d d d

•H jO o d T3 d

•H 4-1y

H

4J

d 1-t CO

O o d o

(U

y

y

cn s y t'. o

CO rH 3 y

CO a.

y o E a d 33

CJ CO ' rH

<4-( ' S •* o 3 y ••

A

d rH 3. js •H d. 3 o E E y

4J 3 d

<U i-( CO

u

-i4i

d

y

C/3 00

d

dJd •H • y -H d

-H CO

y

d o d

<U

y a- o

V4 cn 6 y 3 o

00y

•H

d

y

X

p44

r T3

u I IfIIII

I. II
-320r 3 • •1 3 0)

3-1

O

a

03 cd t3 • ^ CO 1—1 O CO iH rH eg Cd 3 m c O to 3 o •H

03 4J CO X) M

•H 0) r—i i-H a IM Ov O

<! 03 cn

M

M

U o B C

•H 3 o O O •H u O N 4-< C 3 0)

(1) 3 rH 4-1 3 1 Cd O •H 3 CJ T3 3 •H •tH •H 4J a •d N 3 M c O o 1 O a o • #4 3 rH CNI

««

U

03 3 3 dJ

3-4 3 4-1 rH 3 34 3 XI

4J 3 34 Cd

cO 4-1 4J

3-1 3 60

03 34 3 9k -H m a 3

03

U 3 H 3 • * 9k O Cl. a 34 3 X

cO 3 0)

3-4 TS

60 1—1 O CD O O 3 0) M -3 3 CO Ph N 3 9k 1—1 34 CO

ns 6- 3 CJ 3 a TS o

•H 3 34 a 3 O 4-1 o 3 •H 25 3 3 d

•H 34 CJ

U3 o 3 34 9k u

to 3

3-1 3 u

03 •H • 9k 0) > 3 0) XJ

4J rH 4-) f-H 3 cd o

n3Oa 3 O (f) • 9k 60 13

•H 3 o>

a

urt t—1 4J IX cd o

03

a C! CM

4-3 3 •H 3 3 'd

O 3 u • *ri 3 Q o

T) 3 U 1—

C II O & •H

4J • 3 OJ •rl 000)

00

•iH

r-l QO

C • a. c 3 u a 03

rt

X

W CJ

o3

•> ••X

4-1 PQ

pq

M o ‘4-4

'< ?! Ii

']

71
-zoI

Sil NO ‘ TVN-OIS SlVNiailO

•oo

•0 ic

<\

lu

CC3H<

OCm

CL o E O lu

Jo

h o

*o

MJ<5Zoz o

*os;

*0

<M>0

<uc

rt J=

4-1 <uc

(U J2

et,

)-i 4-1 o

•HN<

II

r. 4J

1-1 -H

(Va c

-a -H 3 o to

4-1 u

0) 3

f-l xl

o- o a M

cfl OCO C C O

•H *H

4J P 1-i <U CO c o

•rl CU

rt s O X) o

<r

to 4J 3 (X H tu a. u

M

X M-l u

u

rt

m

u

u

0) o C >4-( o

(U

r e

iKi cd

CO

to LO r-Ho O 1-1 4J 4Jcoo a td o c

to

'H

«i 3 3 CO z 333

•H T—1 a 3

0) 4J < 33

Cd

CO

W •H 3 acdC (L 33 3 i4 •H

nd

§* PQ

a3 O

uo o

or

(L 3Ha3 a3

,3

m 1-1

«. 3 O 33

CO

OJ 3 3 •r 3 r—i

00oM

tdc

•Haoz

CO3

COV4

tu>

cd3

00

•rl

c/d

td3

•rl Xl

V4 O

CU

l-< 3

00

•H

Pt4 0 3 ^ X 3 OO 3

<0 XI 3 -1-1 iH •rl 3 0 3'' O 3 00 3 cn o mo

D. 3 a 33o ca 33 O 3 3 b >3 3 3 U 3 N 3 3

T3H 3 3 3 3

•H XI3O 3 O iH .« (- 3

•rl3

in 4J cn 3 M

ro o ? 3

•HXI - 3O

CJ 3 3 3 3 3 0 3-3 3 •H 3 g 3 3 O 3 3 3 3 3 ••• J3 M-l O S 3 3 o a O 3

XJ 3 0 C >s 1-1 CM 3 3 0-1^

3-

a

O

«•

a 3 • H 3 M O 3

72 I II IIIIIIIIIIfIII1i[
A,

empty

containers

in

holders;

li,

same

as A

except

sample

in

sample

container;

C

same

as B

ex

cept

products

in

sample

container;

I,

change

in

ordinate

scale;

Z,

recorder

zero;

scanning

rat^

10K

min

^

from

373

to

573

K;

one

chart

unit

=

0.0720

mW;

other

conditionssee

exp.

16,*table

4;

ordinate

signal

controls

same

for

A,

B,

C.
Table

2.

m-Azidobbnzolc

Add,

Calculation

ofT ^ • B Ou <n rH c e fn ON CPs o

•ft (U lOm • Om h o

•o

a . H

•H O. ^ <n fH w a ON cs CM cs

m

w* S <n C H

O

l-i • o • m 00 u • • • • • tj *0 VI 0^ CO o H O 00 CO SO lA wu -4* +BH

/“S• iA CO

Vi

m so 00 fv • m

H VI ^ • • • • • e

<3

O o 9 O 9 9 ft O + + + + + a 5HMH<

ty> CM?

oo+

CM o% eo

oo r«« • • • + + +

vOoo oCM o

vO01 o<n 01

00o01 B31 o lo pH o CM o 09 o <n pH O O r-^ 1 •» rH • • • • • O o .c IH 9 9 9 a • + + + + + o m p^ r% <n u a irt CM O o ' • * • • • •* o o O o o 1 • . 1 1 •» 01 o ovO

eomo+

«»

CM

a fH CMm P^ o B . CM Os o pH o •• o ft i4 • • • • • a < S 00 ON pH o Vi o 00 so m a B -4 pO a

eoB • 4’

T3 « U a CJ V a

00 c P^ CM cn ON fH PA so CA .s

X

a « -H cn o cn vo U*> CM pH lA •H

rH

a 00 sO o cn 00 CM ON o pH

O

B •n • o. iJ • • • ^ • • • • • • a B • s s «H 00 pH • O CO O 00 pH 00 3 a u a o CM 00 CM CM CM a. 3 to U CM o a u < B *

ft CM

a

M 00 CM CM pH -vO

a

as CM

p sn - * «k «k lA pH Vi • > • «H • ^H CM CM m pH CM p a 00 • a O • o • « • • • • O u •H 9 ai CM O rH O m o CM O pH cm a S4H aa p

Vi s6 a pH • cv o o o o o • 3S cn m <n o cn u U

cn 0 a • • • • * U-l X

4J 4Juu 4H

u 4)

CL.

C CMC CM

C CM

C CM

C CM

a

X <U sO 0) vO 0) sO 0) sO (U

O

a

a

w > o > O > o > o > o 3 o o o o o o b * • • * p a SM CM • lA - p> p ON p pH • > ^ oa -4 O >4* O •4 U lA CJ +

74
Table

3.

Azldotrlphenylroethane,

Calculation

ofT

.+ • a a. <n fn •O' e a os CM

4) •A ITI

tL,

H

H VI <a ou

00dI

00

CM• oI 0d1 od+ 01

0) OS S3H

Ui H<

'j- <n? ?

00?

00? eo

I. o o o m U 3 yO cn rM a K • • • • c H O o o 9 1 1 1 1 1 3 3 ao O fn O o CM o o o o o o 1 . -1 os • • • • o o Ji o o o o • 1 1 1 o

OS 1 lA c*> rv u a lA CM O ' »U ^ OS • • • • •m o o o o os 1 l_ 1 3

'' 3 a ) rv CM rv rv o o lA CM iH O 1 . 'H OS • • • • o o O 9 o 1 + + + + m m 1^ , • A ac ^H Soe

00s u

<ue

o> <H (0 O. VI . e c

VI

a o 3 CO U

CA oo iri •• • • • • o

fH CM Vi o Oc 00 «

1/^ in «»

m

00c • •o • <8 u

<U vO r«* cn CM

X

»A vO CM CA vO CM o o e •o a c

oo 1^ 00

d 3 0) CM CM CM <A Q. C »

00 CM CM

a 03 «M • f-4 • • • CM 00 os • vH • lA • u • « O • o • Sb • • O 0) 00 V 4) 3 OS CM O 1-H O lA O CM CM 4J

4)a

M-4

<8 • U vO

<a ^

Q. o o o O o • z <*3 <A CA <n b VI 0 a • A m A A •44 vt U 4i U 4J 4) o. C CM e CM C CM C CM «

»< 0) 0) sD 01 sO

V vO 0>

a

U] > o > O > o > o 9 o o o o o Vi • A A A CO >VI OH • 00 • • CA • > CA 09 CA CQ CA A lA U + U

75
Table

4.»

Nitrocellulose,

Calculation

ofT 0 CL PI «n P'S fn P4 a a id r»* r*i0 CM o ^ 0

au

H

»Am lAm so e • u a ^ V S H

PI

tn

r«* «n

<*> <n

<*> tn o«» 0a

'O

<o > b

*0

h

HOU

CN«T

OOS

p»

eo•»

00

so«»

PO<«*»

b

H

b aii 4 ou o

«ne ?

•o

e<« e*4 1OCM

gMH<0 Sftf 01 o • ^ id Jl

U| 9 ?bra

*o -5 Stf H eoc

os?

lO

sO<n

•O’ •o01

Ot?

•o

soOI

som01 oo

C4

PO01

soe ?

ee

PO

so•O01 i01

PO

so0d1 01

soO

PIo01 o0®

d

u

sb

X 0 "H 0 ^ 5 ® a Oi

a. o •< PO

OS

a P4O OS

O

W o o P4 m en o • P4 • • • • o 'O PO o o d •03 • id «b

-o so <A lO • o <n p< - so • • » • e • 0 fb o o lA P4 b

os OS CO r*»9 •O 0 • •o -O u a o.

ao

X 0 4) *0 a 0 O 0 b

03 0 OS os ^ CM •O 00

O p» CM ^ b sO S 0 -H ov a^ CM ^ sO o O sO b O P-* 0 o ao so oo lO o 'S’ ao «A 00 O so 0 U • O. w a • a e P4 sO •b sO r*. b sO ^ CO CM <« s **

SJ 0 0 CM CM CM CM CM CM 3 cn u 0 0) •o 00 •»

p«

CM 0-4

so

b • • O 0 ao u oo 0 • b O0 ?

ee so » CM • ^ * A * »A • b « • ^ - b lA <H tA «-4 CM ^ 0 ^ • 0 o • o • o • * • 3 • • • H 3 ee CM O CM O b o so o P4 O b O 0

No. o o o o O

O

a • 0 -T b sO 0 O

O. • ob

CM CM CM CM CM CM 0 1 « u

r

o. QOe-M 00 .b 03

b 00 0-4 03

b 99H

X <U '•T 0 '0* 0 -T <u ^ 0 -T 0

tal ee o oe o s a ee o ee o sae o 3 +• o o o a o a • m • • A * C9 C CM • ao <• * en • r*b • ^ • > ^

•b < b •<b < b < b < PI < + w III

76 I <
Table 5. Key to Tables 2, 3, and 4

Column

No.

Column

Heading

Items

left to right, top to bottom

1

2

3

Expt. No.

w,R ,R ,S

’ a e

Wt. (mg)

Experiment Number

Reg-standard block lid; VentVented

blocklid

Inlet pressure, dry nitrogen, P.S.I.

A,B,C; At ^ ^ from A, B, or C of fig. 19

Type of sample container (Perkin Elmer

part No.), 0062 fig. 6a; 0041, fig. 6b

w, scan rate, K*min^

R^, sensitivity, meals ^

R^, chart recorder setting, mV full scale

S, chart speed, cm min ^

Sample weight, mg. in air

Sample container weight, mg in air

4

5

T , (nominal)

d Temperature, peak of 0-0°. See column

5 note.

K(0-0 ) /h, 0, observed ordinate signal at T, (nominal),

o m is a decomposition run;

0^, observed ordinate signal at T^ (nominal),

blank run with decomposition products in

sample container;

f h, heat transfer coefficient between

Is

sample container and sample holder

K, calibration constant

K(0-0 ) , the maximum in K(0-0 ) o m o

77
Table 5. Key to Tables 2, 3, and 4

Column Column Items

No . Heading left to right, top to bottom

I 6 -wC/h,

Is w scan rate, K*s ^

C heat capacity, aluminum container

7 K(0-0“)/h^g 0° ordinate signal at T^ (nominal)

,

blank run with no sample or container

in sample holder. Sum of columns

5, 6.

8 -T w

a

T time lag of temperature of sample

holder temperature behind program

temperature.

9 At

STAT ATstat temp. corr. of program temperature

if sample holder temperature held constant

10 AT(corr.) Sum of columns 7, 8, 9

11 T,(corr.)

d

Sum of columns 4 and 10

12 Initial temp. Initial temperature of scanning run; Ambient to initial at 160 K/min.

13 Final temp. Final temperature of scanning run.

78
0 and 0° of (0-0 ) are computed by subtracting the isothermal base

o m

line signals from the corresponding observed signals at (nominal) as

shown in fig. 11. The isothermal base line signals of each pair of runs

at a given scanning rate were constructed by passing a straight line

through the isothermal signals taken at the beginning and end of each

run. The nominal program temperatures at the beginning and end of both

decomposition and blank runs were the same.

T, (nominal) is converted to T, (corrected) , in column 11, which is

d <1 the temperature of the sample. The formula used is

I T^(corr) = T^ (nominal) + (K(O-O^)^ - wC)/h^^ - x^w +

This is developed in Appendix C.3.4. (see eqs. (C-22a,b). The second

term on the right of eq. 16, given in column 7 with its components in

columns 5 and 6, is the temperature difference between the sample and

f container (assumed to be the same) and the sample holder. K(0-0 ) /h. o m Is

is the correction for self heating. The determination of the energy

calibration constant, K, is given in Table C-8, Appendix C.4. The heat

capacity, C, used here is taken from other results of expt. 16

C = 0.900 + 0.67 X lO”*^ (Tp - 380)Jg"^,

which is within 5% of that of the actual containers used in the experiments (see Fig. C-7, Appendix C.4). An estimated , typical, value is

* used for h^^^, the heat transfer coefficient between sample container and

sample holder. Its uncertainty lies between 5 and 10 mW and probably is

closer to the former.

The term X w, in column 8, is the time lag of the apparent mean

temperature of the sample holder behind the programmed temperature. X

is from the calibration studies using melting solids. Table C-6, Appendix

(16)
• 4% c ^

•H o —*> +

lVNi>IS 3iw;oyo

V4

(UC -H

•H

cO -UO

4-1 JS !-» (S 00 <U O -H N a COU M

d) iJ (U -H CO T3 a i-i a **0

CO cs

O

CO

H <u z

<to

a:

CLro — c o

•H 4J

CO

Mo

S']

UJ

CJ cO CJ 3 J3

•3 J-i O "3 C

V4 3 <4-4 C3 4J h- cO 4J r-l 3 0-0 3 3 O- CsO O 3 *H UJ X ^ M 3 1.1 , o. X 3

s:

CO 3 4J3

-H3

UJ

cn 3 CN 3 > 3 H *H •H 1- cOc

•3 3 V4 Cu 4Ja

00

g 3 O 3

•H 3 4.1 Oc/3

0) 4J

COc

•H•a

S-i o

V4o

cO oO

CJ iH

CO •« • • g CQ CO

>-i S-i 0) 0) T3O

(U

1-1 3

4.1 3

.. -I >-• H 3 3 C O- « oO g uo

4.1 Ci-i O c C- 3 -H 3 o g >H 3 4J •H 3 3 _

4J 3 3 3 a 3 00 4J 3 3 3 -H 3 !-l

!-i •H 3 3 00

4J •H

O 3 Sj 3 -3 '.- c 3 4.1 J.I 0< o 3 3 0 o •H 3 rH 3 T-) « 3 3 4J "3 O 3 3 3

t4

O

O HI

f—4 o 3 3 H HI 31 3 3 O 3 3 3 3 14-4 -H 3

S3 rH

O

Ou 3- D- >—1 • 3 , 3

fH 3 *H + "H rH rH -H 4J a, 3 3 -H 3 g 3 3 3

>4 3 3 3 *H 3 3 ^

00 jj •

t-4 « •> 3 CM

Fi14 pa Cn -Q

H

80
C.4. At , in column 9, is the correction added to the nominal proSTAT’

grammed temperature to obtain the apparent mean temperature of the

sample holder when it is held at a constant temperature. Values are

from Fig. C-8, Appendix C.4.

It should be noted that the sum of columns 5, 6, 8, and 9, which is

given in column 10, is analogous to the temperature lag described in the

ASTM test so long as the product of the time constant of the output

signal of the apparatus and the scanning rate, w, is small in comparison

to column 10. It should be noted again that column 10 assumes that the

maximum decomposition rate is identical to the maximum power output.

The error seems to be small and is discussed in section 5.5.

Plots of the data in terms of the ASTM test method of data treatment

are given in figures 12 to 14. The length of the horizontal bar of the

cross corresponds to 1 K (i.e. + 0.5 K from the center). The straight

line through the points was drawn visually.

The slope of the log^^^w versus 1/T^ plot was determined taken from

the graph and also from a least squares fit of log.^w versus 1/T,

10 d

assuming only 1/T^ has an error. E and A then were calculated by the

ASTM test method. To obtain some unambiguous statistics, A and E were

calculated simultaneously by a least squares fit of the natural logarithm

of both sides of equation 12 to the data (see Appendix C.5.I.). The

results of these calculations are given in the computer program printout

in tables 6 to 8. Results of the three methods of calculation are

summarized in table 9. An example of the variation of the results with

various assumed orders of reaction is given in footnote 6 of table 9.

81
82

Figure

12.

Plot

of

log^^W^

versus

1/T^,

m-Azidobenzoic

Acid
Figure

13.

Plot

of

log^^W^

versus

1/1^

»

Azidotripheny

Inethane
84

Figure

14.

Plot

of

log^^^W^

versus

1/T^,

Nitrocellulose
Table 6. m-Azidobenzoic Acid, Kissinger Method

j/

KISSINGER METHOD

N0.DATA PAIRS 5 ASSUMED ORDER I

I ter. NO. INTERCEPT SLOPE SIG^t + B

1 2.930I5E-IA 5.62738E-2 2. AA363E-5

2 -3.95729E-5 5 * 9 A 1 A5E-2 2* A 5333 E

3 -3.95339E-5 5 • 9 A 1 A 6E2. AA022E-5

A -3.95339E-5 5.9AI A6E-2 2.AA022E-5

e= 33AA5.3 STD DEV= 1 Al A. 7

LOGA = 15.0937 STD DEV= .675103

A 1.25507E+15 A>MAX 5.93977E+15 A>MIN 2.65121E+1

SCAN RATE T^ OBS T^C3S-CAL

20 A 8 8 . A -.56385 1 .

1 0 A30.5 .7 A2A62

'

5 ATI..75 1 1 A3

2.5 A61 .

- 1 . A6 569

1 .25 A5A.8 . 5AA325

VARIANCE COEFf ICIENTS

A* .22303 B* .258326 A-E = -8. 376A2E- A

I I Iter, no.: Iteration no. in least squares fit of y

= a-b[X + 2 in(6/y ) - in(l-4>) ]

I ° I to data, y = y - y^, y

= 1000/T^, y^ = 1000 (T/T^) , a and b, see below, X = X - X^,

X = log w, X^ = (log w) ,

(j) = 2(l-n) (b/Y-3(b/y)^)

Intercept-; -y^ + b (Zn a + Zn b - 2 Zn y^ - ^ ~ 1000 R/E, a = 1000 A

Slope; b

Sig**2: variance

E: activation energy, cal mol

STD DEV: Standard deviation

log A: log^^A, A in min ^

-1

85

r
^able 7. Azidotriphenylmethane, Kissinger Method

KISSINGER METHOD

N0.DA-^A PAIRS 4 ASSUMED ORDER I

I ^er.no. INTERCEPT SLOPE SIG=*=4'2

1 1 .79559E-14 5.41452E-2 8.70438

2 -2.46568E-5 5.72146E-2 8.66893

3 -2.46464E-5 5.72 143E-2 8.61955

4 •2 . 46464E-5 5.72148E-2 8.61955

E = 3473 1 .3 STD DEV= 1284. 18

LOGA= 14.9969 STD DEV= .581776

A 9.92713E+14 A^MAX 3.73965E+15 A^MIN 2.

SCAN t>A*^E T> OBS T>OBS-CAL

20 510.4 -9. 4451 4E-2

10 500.6 -•322462'

5 492.5 .88607

1

2.5 482.3 -.469953

VAnlANCE COEFFICIENTS

A= .279212 B= .519192 A-B=- 1 . 07 72 3E-3

86

' j

599 82E+ 1

4

IJw

Ul
Table 8 Nitrocellulose, Kissinger Method

KICSIMGEH METHOD

N0.DATA PAIHS 6 ASSUMED OHDEH 1

I'^EP.NO. INTEPCEPT SLOPE SI G**2

1 1 .96Se0E-l.347563 1 .774S9E-5

2 •2.69946E-5 4.98097E-2 1 .79447E-5

3 -2.69660E-5 4. 9S095E-2 1 . 78 70c E

4 -2.6966CE-5 4.98095E-2 1 . 78 7 06 E

E= 39895.4 STD D£V= 1467.91

LOGA= 17.9597 STD DEV= . 686506

A 9. I 1393E+I7 A^MAX 4.42803 E+18 A^MIN 1.37532E+17

SCAN PATE T, OES T> OBS-CAL

20 492.9 1.44745

20 490.8 -.7553

1 0 433.4 -.306 1 1

5 475.7 -.423 19 7

2.3 468 -.780094

1.25 462.4 .830827

VAPIAMCE COEFFICIENTS

A* .182948 B* .187949 A-B“-4 . 99445E-4

87

W

CU

Ul

(J
5.5 Discussion of Test Results

A comparison of the results on the three compounds run successfully

by us is given in table 10 along with values determined by other laboratories"*"

using the ASTM Method ane one result for nitrocellulose plastic run by

adiabatic calorimetry in an open container. Values of E and log^^A agree

within their combined imprecision at the 95% confidence level. It is

also apparent that the imprecision in our results is roughly a factor of

two, or more, greater than laboratory A’s results. This may be associated

with a number of factors discussed below but at this stage our relative

inexperience with the method must also be taken into account.

The agreement of the nitrocullulose values obtained by DSC and adiabatic

calorimetry is comforting but of little impact because of the possible

differences in samples.

Intercomparison of the DSC and QDTA results may be made by constructing a log w vs. 1/T^ plot for each using the parameters in Table 10 and

eq. 12. For m-azidobenzoic acid, where the kinetic parameters show the

largest difference, the T^ predicted for the results of Lab B agrees with

ours at w = 1.25 K/min, but is higher at w = 20 (493 vs 489 K) . This

probably is acceptable for an interlaboratory comparison involving different methods. Difficulties arise, however, when the two rate constants

are extrapolated. At 323 K (50“C) , our experiments yield k = 0.3 x 10

^

min ^ while those of Lab B yield k = 1.6 x 10 ^ (with, of course, over-

. lapping error bounds). This spread may be too large for establishing safe

storage conditions unless a large safety factor is applied.

"*"Values for laboratory A and B were kindly made available to us by A.

Duswalt. These values were run through our least squares program for a second validation of the program and the results are given in table 10.

They agree well with the program results given to us by A. Duswalt.

I I Standard deviations must be multiplied by t in table 10 to obtain

imprecision.

88
Table

9.

Summary

of

Results

on

Kinetics

of

Heat

Evolution

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Table

10.

Comparison

of

Results

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The calculation of rate parameters as a function of reaction order

(footnote 6, Table 9) shows no variation. In part this is due to the

small experimental temperature range. But is is also consistent with the

model. Numerical integration of eq. 9 showed that T^ changes very little

with reaction order and that q, has a characteristic value for each reacd

tion order that is nearly independent of scan rate. For m-azidobenzoic

acid, T, decreases about 2 K as n changes from zero to three at both w = 5

d

and 20 K/min. The change in between these scan rates is less than 0.3%

of its value. On the other hand, the reaction order can be determined

from aging experiments at several temperatures.

We now turn to an analysis of possible systematic errors in our DSC

experiments, based on the assumptions of the method. These are listed in

Table 11. To our knowledge, the corrections developed below have not been

applied in the QDTA studies. Thus the proper intercomparison is that in

Table 10.

An estimate of the correction of the ordinate peak to the conversion

peak, assumption 1, is worked out in Appendix C.5.3. This is

T, (dri/dt max.) = T,(corr.) - w(t +

u d s d

where Tj(corr) is the sample temperature at K(0-0 ) and T is a o m s

(17)

Cl +

a. (tables 2, 3, 4)

Is Is Is

where C^ and C

2 the heat capacity of the sample container and sample, respectively. The second term on the right side of eq. 17 is the shift in

the program temperature (the correction in the self—heating term is

negligible for our experiments) . For our measurements it is smaller than
Table 11. List of Assumptions

(Proposed ASTM Test Method)

Assumptions concerning T^ and W

1.

* The maximum dri/dt occurs at K(0-0 ) o m

2. dT/dt equals dT^/dt = w, in eq. 12 at the maximum dri/dt, T^ is

nominal program temperature.

3. Deviations of dT/dt from dT^/dt = w during entire decomposition

run do not cause g(D^) to differ signif iciantly from a constant

value of one.

Assumptions concerning temperature

4. Sample temperature is uniform.

5. Sample container temperature is uniform.

6. Sample and sample container temperatures are equal.

Assumptions concerning temperature measurements

f

7.

* h^^ is constant for disposable cells of same type

Assumptions concerning nature of reaction

8. Reaction is simple and obeys general rate law.

9. Reaction is not autocatalytic, partially inhibited, or does not

involve melting with decomposition.

*Assumptions apparently a part of the ASTM method but neither is required

for the validity of the first two equalities of eq. 12. These two equalities

require one to determine T, and (dT/dt) .. d d

92
the measurement error since the time constant of the sample and its con-

» tainer is small (i.e. I = C/h, of column 6 of tables 2, 3, and 4) and

s is

the time constant of the ordinate signal detection (or measurement) system

is also small (i.a. T, < 0.01 min, Appendix C.4). The changes in E and A

d —

based on the estimate of eq. 17 are given in table 12. E, A and K all increase slightly, but are still within the measurement error. The error ,

would be far from negligible for a sample container of large heat capacity

and a reasonably accurate correction would have to be worked out.

The error in assumption 2, the rate of change of sample temperature

being equal to the programmed rate (at the maximum) is small so long as

the maximum in dn/dt and the DTA peak nearly coincide. The magnitude of

this error is worked out in Appendix C.5.3 (see eqs. C-47, 48). It is

,dT

”ls "is

(18)

-2

where a„ is the curvature of the ordinate signal in WK at K(0-0 )

. In

z o m

appendix C.5.3 it is also shown that the magnitude of (dT/dt) , at K(0-0 ) d o m

is

“is

(19)

Thus, to be consistent the values of T^(corr) of tables 2, 3, and 4 should

be used with eq. 19 if we assume the maximum in the ordinate signal corresponds

to the maximum in dn/dt and eqs. (17) and (18) should be used to calculate

T, and (dT/dt) , at the maximum dn/dt. d d

The values of E, A, for nitrocellulose from table 10, the ASTM method,

are given in the first row of part b of table 13. The values in the

second row are also for the ASTM method but without expt. number 12, which

93
9A
Table 13. Effect of Error in Assumption 2 of Table 11.

(Nitrocellulose)

(a) Correction for scanning rate

Experiment w. K/min

nominal corrected

18 20 20.578

16 10 10.089

13 5 5.008

17 2.5 2.503

31 1.25 1.250

(b) Changes in rate parameters

E, kJ mol ^ Log^A, A

. -1

in min

T (dT/dt) Value std. dev. value std. dev

at

^max Ord. w 166.9 6.1 17.960 0.69

^max Ord. w 173.1 4.6 18.668 0.52

max Ord. eq. 19 173.2 4.6 18.672 0.52

3

max dn/dt eq. 18 181.0 6.3 19.565 0.7

Taken from table 10; based on all nitrocellulose experiments; T calcula

ted using eq. 16.

2Nitrocellulose expt. 12 not included in calculation of this and lower

rows. T calculated using eq. 16.

T calculated from the sum of eq. 16 and eq. 17.

95
has a large self heating correction. The values of the second row are the

"base" for the error estimate. Experiment 12 was not used in the other

calculations. The values of E and A in the third row are for both dT/dt

and T of the sample at the maximum ordinate signal. The values in the

fourth row are for both dT/dt and T of the sample at the maximum dri/dt.

The estimated correction for E and A for these five experiments is less

than their standard deviations. As for assumption 1, the corrections for

assumption 2 increases E, A and K.

Assumption 3, that dT/dt for the sample is equal to the programmed

temperature scan rate, w, throughout the exotherm cannot be met exactly.

The increment to the scanning rate due to self heating can be derived from

the exotherm and is its derivative with respect to time. It shows the

characteristic maximum and minimum of the derivative of a "line" with

finite half-width. The effect of self heating on g(n^) is considered in

Appendix C.5.2 where it is shown to be order dependent and vanishes for a

first order reaction. An upper bound is derived. The sign of the correction

is indicated by the bound

_ E (apparent)

1 + e(n-l)

where

e <

_ K(0-0 ) E o m

2 ' RT, h, d Is

For a reaction order n > 1, E decreases due to the effect of self heating

on the constancy of g(n^). This is opposite in sign to the assumption

that g(Tl^) is constant for the rate law if n 1 and dT/dt for the sample

equals w. For activation energies such as are found in our experiment.

96
the error in E will be less than 2% if the temperature excursion is kept

below 0.2 K. (This is the repeatability desired for the ASTM method.)

This upper bound estimate may be too conservative to be useful and must be

improved by numerical calculations. Some of our experiments had larger

temperature excursions dnd they increased with T^.

Assumption 4, uniform sample temperature, is crucial for the theory

of the method. Fortunately, the error associated with it seems to be

negligible in our experiments. This error is estimated below.

Using nitrocellulose as an example, one can estimate the critical

explosion temperature T^, in our apparatus for a 1.5 mg sample using the

measured heat of decomposition, 2.19 kj gm a heat transfer coefficient

of 20 mW/K, E of 167 kJ mol and log^^A min ^ of 17.96. One obtains a

value of 491 K. This yields a ratio of 0.301 s/K for f^/w where ^2 is the

dimensionless scanning rate defined in eq. (B-64) of section B.8. Assuming

sample and sample container are in much better thermal contact with each

other than the container is with the sample holder, one has for y* (defined in eq. B-67) a value of '>X).l. From ref. [165], fig. 5, one can see

that if w is 20 K min the ratio |A9^|/|A9^|, corresponding to of

0.1 and of 0.1, is of the order of 2 to 2.3 from eq. (13) of that

reference. This ratio is AT /AT. where AT. is defined in eq. B-68b of

section B.8. maximum difference in temperature between the

center and surface of the sample during the decomposition runi Since

AT^^^ is independent of Biot number and fig. 5 of [165] applies to infinite

Biot number, the aforementioned ratio provides an upper bound on AT so m

long as, one is willing to approximate the reaction as being homogenous and

assume heat transfer is solely by conduction in the sample during the decomposition.

97
Assuming the sample is in the form of a thin sandwich, = 1 in eq. B68b, and estimating the thermal diffusivity from data in ref. [146] table

2-1

k, as 0.0018 cm s , AT. is -93 r2 Kelvin for w equal to 20 K/min and a

’ xnt

half thickness of the nitrocellulose film of r in cm. AT. ^ is 0.004 K

lllu

if, for example, one assumes r is 0.006 cm and thus AT^ is of the order of

0.01 K. Since AT is proportional to w, one can see by analogy with the

m

results of table 12 (i.e. the change in T, was 0.64 K at an w of 20 K/min

d

that the error for the example cited would be totally negligible.

It is not clear, at present how accurate the above analysis is so it

must be regarded as tentative"^. One should note that the estimated error

is small because our experiments were carried out with small sample

thickness and not because the sample weight is small. The estimated

effective Biot number is also quite small. For the example we are using,

the effective Biot number, Bi, can be estimated as

Bi

a r

oA

h. r Is

AA

’ 2

mA

7

A is. the area of the sample which can be expressed in terms of its density

(taken as 0.82 from ref. [186]), p, its mass m (.0015 g) and an assximed

2 ' thickness r (.0126 cm, the actual thickness of the film), h^^ is the

heat transfer coefficient (.02 W/K) between the sample container and

holder.

For example, the dimensionless critical scanning rate, assuming the decomposition is a first order reaction is 1.3 from the numerical result

cited in Appendix B.8. Accordingly, for the example given, the critical

value of w would be 260 K/min! In principle a thermal explosion should

not occur for this example situation in our instrument! Until this and

other consequences of theory are examined experimentally, one must treat

the calculations with caution.

98
Qualitatively, the error analysis associated with assumption 4 confirms

the proposed ASTM procedure for preparing samples. The analysis also

suggests that one should also run the experiment with smaller values of

f h- . There is a limit to the effectiveness of this procedure, as disIs

cussed in Appendix C.l, because thermal loading of the sample container

I produces temperature gradients in the sample even if h^^ is zero.

For small containers made of materials of large thermal diffusivity

and conductivity, assumption 5, uniform sample container temperature, will

contribute little error. Assumption 6, equal sample and container temperatures, is crucial for the validity of and must hold for the entire decomposition experiment. This would be important even if the sample temperature

were measured directly. In addition it implies that the gaseous decomposition products escape quasistatically and at a temperature equal to the

container. It implies that very good thermal contact is maintained between container and sample throughout the decomposition. Probably none of

our experiments fulfilled this condition adequately. The error in meeting

assumption 6 is unknown.

The error associated with assumption 7, the same heat transfer

coefficient for all similar cells, can be appreciable but it seems probable that procedures can be developed to eliminate it. One possibility

is to place the sample container on a smooth lightweight platform made

from a material of high thermal conductivity. If the heat transfer coefficient between the platform and sample holder is, for example, 0.1 of

the coefficient between the platform and container, variations in the net

coefficient and holder will be '^'0.1 of the variation in the coefficient

between the platform and container* I^ effect, the platform is a thermal

shunt

99
Another possibility is that the heat transfer coefficient can be

determined via a time constant analysis of the transient in starting a

scanning run with and without the sample container. It has the drawback

that one must have a reliable and sensitive method of separating out the

effect of the time constant of average power circuit (3.6 sec) as well as

that of the difference power circuit 0.5 sec). The possibility that

a method of data analysis can be devised may be possible by application or

extension of the analysis of Appendix C.4.

Procedures are available for determining if the reaction is simple

(assumption 8) and, if so, determining f(ri) of eq. 11. These are discussed in ref. [165], section III. Procedures for "separating" complex

reactions under certain circumstances are discussed in ref. [152] as

well as procedures for determining the kinetics of decomposition outside

the temperature range of the actual observed exotherms. The general problem of obtaining heat evolution kinetic data for systems that melt with

decomposition has been studied by Rodgers [207] and Dorko, et al [208-

211 ].

In sxjmmary, the systematic error analysis indicates that the errors

in the kinetic equation used to analyze the data in the ASTM method are

or can be arranged to be kept small provided the simple general rate law,

eq. 8‘, applies. Also at least some techniques are available and probably

more (or better ones) could be developed to determine when eq. 8 does not

apply. The systematic errors associated with carrying out kinetic measure

ments by calorimetry of any type are more difficult to assess. We conclude (some what to our surprise) that there is some reason, albeit

theoretical, that the sample temperature is probably uniform in our experi

100
ments. Also it is possible that an experimental technique to detect nonuniformity of the sample temperature can be devised. However, efforts to

assess the error in the assumption that the sample and sample container

temperatures are equal have not been successful. While it seems reasonable

to conclude that one can avoid this assumption by measuring the temperature

of the sample, this is, in a sense, the same assumption in disguise. The

assumption to be analyzed is then the equality of the temperature detector

and sample

It is difficult to see how the necessarily limited temperature range

of -the data obtained by this method will ever provide sufficiently precise

values of E and A to permit reliable extrapolation to lower temperatures

even if the decomposition mechanism does not change. While there are

ageing techniques that permit measurements at lower temperatures, they are

indirect and time consuming.

101
5 . 6 Conclusions Based on the Experiments and Reconmiendations for Future

Work

1. Intercomparison of DSC and QDTA results with those from other

methods are needed. Emphasis should be on studies at lower temperatures.

The low sensitivity of commercial DSC and QDTA instruments makes them

usable only at relatively high temperatures. The exothermic reaction at

high temperatures may not be the dominant heat producer at the lower

temperatures characteristic of shipment and storage. That is, the mechanism

may change with temperature.

One protection against this possibility is to build into the proposed

ASTM test method the lower temperature aging tests used by Duswalt [24].

They extend the range of validity of the scanning method results. They

could be extended to even lower temperatures, but would become very time

consuming. An alternative approach is desirable.

A second question is whether or not procedures that use very small

samples can mimic bulk hazard conditions. The DSC and QDTA tests use a

system consisting of a thin layer of material sandwiched between a metal

container and cover —adding appreciable heat capacity. Correlation of

the results on such systems with bulk samples requires that the properties

of the material itself be extracted and then interpreted using thermal

balance equations. The validity of this approach is more in question for

marginally unstable materials than for explosives, for which see Rodgers

et al [23,149].

Because of these questions the DSC and QDTA methodology should be

tested against heat conduction and adiabatic calorimetry using larger

samples. Lower temperatures can be used in the latter two methods; how

102
high temperature data should be extrapolated can be explored. Predictions

of explosion temperatures from all four types of experiments can be compared using methods developed by Bowes [119,120].

Heat conduction calorimetry is included in the proposed comparisons

because of its flexibility. Both isothermal and programmed studies can be

made, including non-linear programming, which may have advantages. Also,

the sample is in relatively poor thermal contact with its environment

(i.e., is quasi-adiabatic) which should promote uniformity of temperature,

an important goal that is discussed later.

It cannot be ruled out that these intercomparisons would show that

testing of marginally unstable materials should be based on the explosion

temperature test, although that is time consuming and highly empirical.

2. Provision should be made for the use of both open and closed

containers in DSC and QDTA tests for thermal instability hazards.

Closed container tests can indicate maximum hazard conditions: both

with respect to pressure rise and catalysis by gaseous products. Ideally,

a measure of the pressure rise should be incorporated, because this is an

important hazard parameter [148,150,155,156]. But this will require

substantial development. In the absence of a pressure measurement two

types of closed containers can be used. One is the reusable high pressure

cell discussed briefly in section 5.4. The other is a high implosion but

low burst pressure cell. This could be used in an apparatus in which the

sample holder can be pressurized (e.g. DuPont DSC apparatus).

Studies with open containers (e.g. [104,212]) are needed to verify

hazards associated with catalysis by products and to determine whether or

not vaporization of material (absorbing heat) mitigates the hazard.

103
There are problems in experiments with open containers that must be

studied. First, good thermal contact must be maintained. Second, the

gases should be lost nearly quasistatically if all of the liberated heat

is to be measured. The perforated lid in contact with the material may

be suitable for controlling these features. Third, condensation of

products outside the sample container can lead to changes in instrument

sensitivity as observed in our work. (See discussion of emisivity effects

in section 5.4.) Here the preferred solution is to flush gaseous* products

out of the system during the experiment, instead of introducing the

assumption that emissivity effects are proportional to the extent of

reaction [213].

3. The sample temperature in a DSC or. QDTA experiment must be

controlled to match the assumptions of the method. This means, that the

sample and its container should be at the same temperature and that

gradients within the former should be minimized.

Identity of sample and container temperatures is required in the

model and would be necessary even if the temperature of the sample were

measured directly. The control circuitry responds to and the output circuitry records the history of the combination, A spring-loaded lid (perforated to allow escape of gases in an open container experiment, e.g. a

screen [214]) should be effective for solids, even when there is appreciable change in volume during the reaction. Introduction of an inert

liquid film between sample and container might be an improvement [215,216].

Increased surface area for heat transfer (perhaps fins on the lid) could

be used for gaseous reactions. Finally, direct measurement of the temperature difference between sample and container should be attempted, although

the small size of samples used in DSC and QDTA experiments make this

difficult. Any of these variants must be tested experimentally before

being adopted for a test procedure.

104
Uniformity of temperature within the sample is necessary because the

temperature is measured indirectly and it must be associated with a

property of the material that is sensitive to temperature. Temperature

gradients in the samples used in our experiments probably were unimportant.

They can be reduced by decreasing the efficiency, of heat transfer between

sample container and its holder relative to that between the sample and

its container.

There are limitations on this approach, as discussed in Appendix C.l

for adiabatic calorimetry. Thermal loading due to the heat capacity of

the container can introduce gradients even when the heat transfer between

sample and container is high. The use of thin samples counteracts the

effect, as may be seen in the experiments of Merzhanov [165]. Partial

thermal isolation of the sample container may enhance the possibility of

thermal runaway. This would influence the choice of scanning rates (which

should be less than the critical value) . The optimum arrangement must be

found experimentally using tests with various known heat transfer efficiencies .

4. More detailed data analysis is desirable. This would assure

that the measurements fall within the limitations prescribed for the

method.

The proposed ASTM test uses only the peak temperature for the decomposition and the nominal scanning rate. Analysis of the exotherm can

yield Q, the rate law and heat transfer data. The actual scanning rate

for the sample can be obtained. Possibly, simultaneous analysis of all

runs in a series can provide information on the functional form of k(T)

[165]. These data are all of interest for the interpretation of aging

experiments
Efficient application of more detailed analysis will require digital

data logging and computer processing if the extended method is to be more

than a research tool.

5. Improved accuracy and precision would result from measurement of

the heat transfer coefficient between the actual sample container used and

the sample holder.

The present practice is to use an average heat transfer coefficient

for disposable containers based on ones drawn from the batch used in the

experiments. We conjecture that the heat transfer of a container can be

determined either by a thermal shunt or from analysis of the starting

transients in a scanning run, but these must be tested.

106
A. Remarks on the first NBS Appraisal of Methods for Estimating

Self-Reaction Hazards

A.l. CHETAH Program

A precise summary of the CHETAH program is given in ASTM Publication

DS51 [3]. The material on pp, 1, 6-9, and 96-115 were foxind to be

particularly relevant in the review of the first NBS study [1].

The purpose of the program is to provide a quick, low-cost, screening

technique to identify materials that may be a potential explosive hazard

([3], p. 6) prior to further laboratory testing ([3], p. 8). The program

was not intended to be the sole test of whether a chemical is hazardous

([3], p. 8). The procedures for estimating thermodynamic parameters

(solely from chemical composition) that are used in evaluating the explosive hazard were put into the program in order to have some means of

screening the estimated 100,000 new chemicals encountered each year in

research and development in the chemical industry prior to measurement

of their thermochemical properties.

In the original CHETAH scheme, a material that is an explosive hazard is one

which’ is shock sensitive. The latter is defined to mean the material

decomposes, with or without detonation, either upon impact by a falling

weight, explosion of a No. 9 blasting cap, or a 50 g pellet of tetryl

([3], p. 99). A combination of the first three of the four empirical

criteria used to estimate the degree of explosive hazard was capable of

identifying all 83 shock sensitive compounds in a group of 218 compounds.

However, roughly 12% of 'the 135 insensitive materials were categorized

as being shock sensitivie. The CHETAH program and other evaluation

schemes are still undergoing development (see [3], p. 101 for proposed

future work and [25] for an example of this work).

A. 1-1
CHETAH was developed by some of the members of ASTM Committee E-27.

This committee also is concerned with standardization of nomenclature

and tests for the energy hazard potential of chemicals. This hazard is

defined as the degree of susceptibility to ignition or release of energy

under environmental conditions (see [3], p. 96, and [2]). This definition

is equivalent to the preliminary definition of thermal instability

adopted in the first NBS study. There the hazard potential is called

explosive sensitivity. The latter term is used in work dealing with

explosives (see [10], p. 1-8). The three main empirical criteria used

in the evaluation of the degree of the energy hazard potential in CHETAH

are based on the (maximum) enthalpy of decomposition, enthalpy of combustion

minus enthalpy of decomposition, and the oxygen balance. Unfortunately,

these parameters are not directly related to explosive sensitivity or

the energy hazard potential. Evidence in support of this conclusion is

as follows.

For explosives, it seems well established that the enthalpy of

decomposition per gram of material is primarily an indicator of the

damage potential of an explosion, called explosive power (see [10], p.

5-24) and is only indirectly related to explosive sensitivity. Examples

in support of this conclusion are the distinction between primary and

secondary high explosives (see [10], p. 1-4; [26], pp. 801-802; [27], p.

4.9-15), the direct relation of the heat of decomposition with explosive

power (see [10], Chp. 3; [26]), and the indirect effect of enthalpy of

decomposition on impact sensitivity implied in current theories of

impact sensitivity (see [10], section 12-3).

For explosives, it also seems clear that oxygen balance is a measure

of explosive power. Oxygen balance correlates with ballistic mortar

'^. 1-2
test results and enthalpy of decomposition [28]. Price indicated in

1955 [26] that the ballistic mortar test is only a qualitative measure

of explosive power. However, the more recent discussion of reference 4,

section 3-9, indicates the maximum work measured in the ballistic mortar

test correlates well with the enthalpy of detonation for partially

confined charges determined calorimetrically. Evidently, the test

determines the maximum work for explosives with near zero oxygen balance

with the least error (see [10], p. 3-3).

Thus, although the parameters used in the hazard criteria in CHETAH

are not directly related to sensitivity, the empirical criteria themselves

are indicators of explosive sensitivity, simply because they succeed in

identifying sensitivity to shock.

For the purpose of setting regulations, the first NBS study concluded

that this distinction, or apparent inconsistency, is important because

it meant that there is no mechanism (definitions, criteria, etc.) for

identifying materials that are shock insenstive but thermally unstable.

Thus, it appeared that CHETAH could not serve as a screening method for

what is assumed to constitute much of the class of materials of interest

(i.e. borderline hazardous materials). This topic was explored further

in the second study [8].

A. 1-3
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A. 2. STULL-CRUISE Program and NFPA Hazard Index

Stull’s first publication on the scheme [5], deals only with the

correlation of thermodynamic parameters with the NFPA hazard index [27].

In ref. [5], it was found that the parameter (X^-X^)/X^ correlates well

with the NFPA hazard index. The subscript o refers to oxidation and d to

decomposition. The quantity X was either enthalpy, maximum temperature"^,

or maximum pressure"**. A plot of versus X^ can be divided into

sectors consisting of materials having the same NFPA rating for about 80%

of the total (50) compounds considered. The agreement between the sorting

of materials when X was enthalpy, temerature, or pressure was close (i.e.

about 95%) . The preference of temperature for X seems to be based on the

fact that the values of T -T, versus T, came very close to forming single

o a a straight line for the compounds considered. It seems useful to note that

apart from the fact that (or T^, P^) is computed from products determined by a modified equilibrium method in the CRUISE program, the parameter

(AH^-AH^) /AH^ is the parameter used in the second criteria of the CHETAH

program.

The NFPA reactivity ranking scheme is essentially a qualitative

hazard index appropriate to fire fighting conditions. It contains both

sensitivity to explosion and explosion power criteria required for any

realistic hazard index. ’’Explosive power” seems to be associated primarily

with the violence of the reaction of the material with water and the words

’’detonation” and ’’explosive” are used in describing a material as susceptible to explosive decomposition or reaction. ’’Explosive sensitivity”,

in this scheme appears to be related to strength of thermal or mechanical

Under adiabatic conditions at constant volume.
stimulus and the associated ambient temperature-pressure conditions

required to initiate the decomposition (be it explosive or otherwise)

On this basis, one would judge the differences between categories 0, 1,

and 2 and the differences between categories 3 and 4 to involve differences

in sensitivity to explosion by self-reaction. The difference between

categories 2 and 3 would clearly involve differences in explosive power

due to self-reaction. A useful introudction to the NFPA as well as other

consensus hazard ranking schemes related to self-reaction hazards is

given in reference [25a].

(

A. 2-2
A. 3. Summary of the Results of the First NBS Assessment

The validity of the hazard rankings predicted by the two predictive

schemes (CHETAH and STULL-CRUISE) was first assessed by comparing predictions with explosion sensitivity assigned on the basis of experience

for groups of compounds (e.g. primary explosives or detonators, secondary

explosives, materials that can self-polymerize, and relatively stable

compounds). In a more detailed assessment, the numerical values of the

three different molecular parameters involved in computing the degree of

hazard in the predictive schemes were compared with sensitivity test

data for a group of known explosives. The three molecular parameters

were the experimental values for the enthalpy of decomposition, oxygen

balance (i.e. defined by the composition of the material), and the

activation energy of decomposition as estimated from bond dissociation

energies. Each of these parameters was separately correlated with the

sensitivity data from Impact, thermal surge, and explosion temperature

tests. This was done because the different test methods gave somewhat

different sensitivity rankings for the compounds.

The results of these two assessments were as follows:

(1) The CHETAH predictive schemes overrank the explosive sensitivity

of compounds that experience indicates are "safe" and both predictive

schemes do not clearly identify a polymerization hazard, as such. Both

of these two findings were verified in more extensive comparisons in the

second DOT study (see [8] p. 69-97). Both findings are in accord with

how the CHETAH hazard criteria were devised (see section A.I.), and the

second finding would be consistent with the fact that neither scheme

automatically (i.e. without operator intervention) considers polymeric

decomposition products.

A. 3-1
(2) Comparison of the reaction products produced by either predictive scheme with available experimental data showed reasonable agreement

for organic explosives but poor agreement (either in species or amounts)

with data or experience for non-explosives and polymerization reactions.

The products predicted for non-explosives lead to more negative enthalpies

of decomposition and this accounts for part of the tendency towards too

high a hazard ranking for "safe" materials. A preliminary examination of

some alternative rules for predicting decomposition products that produce

better agreement with experiment indicated some improvements might be

possible. However, there is no simple substitute for experimental decomposition

data.

(3) The sensitivity test data for explosives showed a rough correlation with activation energy, some correlation with oxygen balance, and

essentially no correlation with enthalpy of decomposition. The relatively

clear-cut correlation of the data with activation energy was found to be

consistent with what one might expect from thermal ignition theory. That

is, the greater the decomposition rate constant, the greater the material

sensitivity becomes. Since the preexponential factor was relatively

constant for the compounds studied, the sensitivity should Increase as the

activation energy decreases. The absence of a clear correlation between

sensitivity data and the enthalpy of decomposition is in accord with the

fact that latter is primarily a measure of explosive power. That the

oxygen balance showed some correlation with the sensitivity data was

unexpected. A possible reason for this correlation was that, for the

compounds examined, the overall rate constant decreases as the amount of

oxygen increases because the relative amounts of unsaturated hydrocarbons

A. 3-2
A. 4 Recent Developments

Treweek et al [25a] have addressed the problem of providing the user

with a single predictor of sensitivity as opposed to the four indicators

of the CHETAH program. They have introduced the concept of the "interpreter" by means of which the several indicators are combined. The

technique used is an application of pattern recognition, in which a

series of independent parameters are correlated against some class

variable, here impact sensitivity or reactivity. The interpreter does

not provide a detailed relative ranking but instead places the materials

in broad classes, such as non-sensitive and sensitive. The interpreter

concept is very pertinent to the regulatory problem of decoding, broadly,

how various chemicals should be handled.

Two types of interpreters have been developed. The first correlates

sensitivity to impact with the four CHETAH parameters plus one additional

parameter, - the number of peroxide bonds in the molecule. Peroxides

are a class that is not easily characterized by CHETAH. Two classes

are used: non-sensitive and sensitive. The second interpreter correlates

the NFPA reactivity rating and the similar U.S. Coast Guard self-reactivity

ratings with the four CHETAH Parameters and two other for olefinic

monomers - enthalpy of polymerization and a bond dissociation energy

difference. Three classes are developed: non hazardous, hazardous

decomposition or polymerization, and explosive. Both interpreters are

stated to be applicable to organic compounds with three or more carbon

atoms

The performance of the interpreters is very good in the test cases

presented. For the impact sensitivity interpreter, 105 compounds were

assigned correctly to their classes and four were over estimated.

A. 4-1
Using this interpreter 127 other compounds, for which the impact tests

are considered are less certain, were assigned correctly.

The self-reactivity interpreter classified correctly 33 out of 34

substances for which two consensus ratings agree, toluene-2,4 diisocyanate

being under estimated as not hazardous. Eleven other materials were

considered, for which either the two consensus ratings disagreed or there

was only one. Ten of these were classified "correctly”.

We have applied these interpreters to the compounds whose CHETAH

hazard ratings were developed in the first and second NBS Appraisals

[1‘8]. The results are listed in Table A-1. The four CHETAH parameters

are shown (by classes) first and then the impact and self reactivity

interpreter results. Compounds in the table that were used by Treweek

et al to develop the interpreters are marked by an asterisk. Using the

data referenced and conventions given in the notes to the table we obtained

the same classifications. Compounds are listed by sensitive functional

groups, with those substances believed to be non-hazardous last.

If C^ and C^ compounds are ignored, the performance of the interpreters is very good. Nitriles appear to be overclassified by the

impact interpreter. The CN group may behave as an (inert) pseudohalogen instead of an energy-rich grouping. Polyalcohols and sugars

are the only classes overclassified. This can be traced to their low

oxygen balance. It is easy to exclude these classes.

A different estimation method, "the L-System" has been developed

by DeHaven [25b]. It combines thermodynamic and rate data to calculate

the temperature at which dT/dt is a maximum. There is a clear relationship between the derivation of the method and simple explosion theory.

DeHaven tested the system against the list of 80 compounds used by

A. 4-2
Stull [6] in developing his "Reaction Hazard Index", compared the

results against the NFPA ratings for these compounds, and developed a

correlation between the L-system and NFPA. Agreement between the

systems to + 1 (on the NFPA 0-4 scale) is considered good performance.

On this basis, the prediction was poor for only 3 compounds. This is

appreciably better performance than for the Reactive Hazard Index when

it is scaled, separately, to give the best fit to NFPA ratings. A

comparison of this sort on this compounds has not yet been made for

the CHETAH interpreter.

A. 4-3
Table A-1

CHETAH Parameters and Classification of Compounds by Interpreters

Compound CHETAH Impact

1—

Self-•React

mercury fulminate MHHH S E * silver azide MHHH S E * lead azide MHHM S E * azoethane HLLM S E * dimethylglyoxime HMMM S E

nitroglycerine HHHH S * E k

ethylene dinitramine HHHH S * E k

trinitrotoluene HHHH S E k

nitromethane HHHH S E

ethyl nitrate HHHH s * E k

diacetyl peroxide HHMM s E * peracetic acid HHHM s E * ethylene oxide HLLM s E k

propargyl bromide MHMM s * E

1 , 3-butadiene HLLM s H

isoprene HLLM s H

styrene MLLM s H k

vinyl acetate MMLL I H k

vinyl chloride MMMM s H k

vinylidine chloride MHHM s H k

acrylonitrile HLLM s H k

methyl acrylate MMLL I H

methyl methacrylate MLLL I H

ethylene HLLM s H k

1-octene MLLL I H

n-octane LLLL I N

dimethyl ether MLLL I N

ethylene glycol MMML I E

glycerol MMMT. I E

pentaerythritol MMML I N

ribose MMMT, I E

sucrose MMML I E

ascorbic acid MHML I E

formic acid MHHL I E

acetic acid LMML I * N

methyl acetate MMMT. I N

ethyl acetate MLLL I N k

ethyl chloride U'lML I N k

acetonitrile HLLM s k N

malononitrile HLLH s E

A. 4-4
Compound

adiponitrile

benzonitrile

pyridine

pyrimidine

formamide

acetamide

N , N-dimethyIformamide

urea

glycine

e-caprolactain

triethylene diamine

Table A-1 (Continued)

CHETAH (+) Impact

(-H-)

HLLM S

HLLM S

HLLM S

HLLM S

MHML I

LMMM I

MLLL I

LHHL I

MHML I

MLLL I

MLLL I

Self-React. ^

N * N

N

N

E

N

N

E

E

N

N

"^The four CHETAH parameters are given in the order maximum enthalpy of

decomposition, enthalpy of combustion minus maximum enthalpy of decomposition

oxygen balance and "Y" (scaled square of the first parameter). H = high,

M = Medium and L = Low Hazard.

The impact interpreter classes are I = insensitive, S = sensitive.

The self reactivity interpreter classes are N = non hazardous, H = hazardous

or polymerizable, E = Explosive.

*Used in constructing this interpreter. Numerical values for CHETAH

parameters were taken from ref. [1] Tables 1-5 and ref. [8], Tables 2-13,

18-22. Oxygen balances for the following compounds and incorrect in refs.

[1,8] and were recalculated (ref. 1, pg. 13): nitromethane, vinyl chloride,

acetic acid, vinylidene chloride, vinyl acetate, methyl methacrylate and

methyl acrylate. Bond dissociation energies for saturated analogs of

olefinic monomers were provided by W. Tsang (personal communication)

Those- for C-H on the more highly substituted carbon were used. The

^ enthalpy of polymerization of CgH^^(g) was estimated as -20 kcal mol

In the self-reactivity interpreter -AH (poly) per mole was used.

A. 4-5
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B. Thermal Explosion Study Notes

B.l. Description of the Thermal Explosion Process

The temperature- time sequence of events occuring when identical

samples (i.e. composition, amount) of a thermally hazardous material at

some low initial temperature, are suddenly exposed progressively

higher environmental temperatures, T^, has been described in the second

NBS hazards assessment (see [8], p. 122-123). The description is repeated

here with some additional comments for sake of completeness.

In each experiment, the sample is placed in a container constructed

from a metal having a large thermal conductivity. A thermocouple is

imbedded in the material to monitor the temperature of the sample. After

the sample, container, and thermocouple have come into thermal equilibrium

and have an initial temperature this assembly is suddenly immersed in

a constant temperature bath held at some constant temperature T^ greater

than Experiments differ in that T^ is held at progressively larger

values. It is found that, if T is less than some critical value, T.,

the temperature, T, of the material first increases rapidly from T^^ to

T^. Then T slowly rises slightly above T^ due to self-heating to a maximum

value that is still near T^. Finally, T decays back to T^ after the

decomposition of the material is complete.

If an experiment is now carried out with T^ slightly higher than this

critical bath temperature, a thermal explosion occurs. After T has increased slowly to a value slightly higher than T^, T now abruptly (i.e.

relative. to the time required to slightly exceed T^) starts to increase

rapidly. This signals the onset of the thermal explosion. T rises to a

very high maximum value (assuming the sample container and thermocouple
are not destroyed) and again decays back to after the decomposition of

the material is complete. Both the magnitude of the rapid rate of temperature rise and the maximum temperature of the sample in the thermal explosion increase as the total heat liberated per unit volume of the

material, Q, by the decomposition reaction increases. However, the fraction of original material that decomposes prior to the explosion onset

decreases as Q increases. The fraction of material decomposed is of the

order of 5% or less for explosives but may be 20% or greater for cellulosic materials. Notable exceptions are those materials which, while

liberating large amounts of heat, decompose by an autocatalytic mechanism.

In this case the fraction of material decomposed prior to the explosion

onset may be as large or larger than those for cellulosic materials.

The T versus time, t, curves for materials having small or large

values of Q are similar when T^ is near but greater than its critical

value. That is, in contrast to the non-explosive situation, the curvature

of T changes sign twice (negative to positive, then positive to negative)

rather than always remaining negative. However, the curves differ in that

for materials having a small Q, the time for the explosion onset to occur

becomes less distinct (i.e. less easily identified) and the maximum

temperature rise above T^ is less.

Experimentally, it is found that when T^ is very near the critical

value, T^, it is difficult to reproduce in separate experiments either the

length of time it takes to explode (which increases very rapidly, the

nearer T is to its critical value) or, in fact, the actual occurrence of

o the explosion itself. This phenomenon, which is called part of the

critical phenomenon, is due to the fact that only a slight perturbation in
initial conditions (i.e. amoxint of sample, heat transfer to the container,

etc.) or the bath temperature can shift the material from a explosive

to a non-explosive condition or vice versa. The other part of the critical

phenomenon is the seemingly abrupt and very rapid rise of the temperature

to a very large maximxjm value (see [12], Chapter 6).

If experiments are made with a test tube of larger diameter, and,

hence, a large amount of material, the critical value of is lowered.

Hence, this critical ambient temperature is not a property of the material.

If experiments are carried out in which a number of thermocouples are

imbedded in the material, so that one can also monitor the temperature

distribution of the material as a function of time, one finds that for

non-explosive conditions the maximum temperature of the sample occurs in

the center of the material. The temperature of the material decreases as

one progresses from the center to the test tube wall (at T^) roughly as an

upside down parabola. For values of T^ slightly above the critical value

it is found that the rapid rate of temperature rise first occurs in the

center of the sample. That is, the explosion is initiated at the center

of the material. As T^ is progressively raised to values much higher than

the critical value of T^, the explosion is initiated at positions in the

material progressively closer to the walls of the test tube - ultimately

at the material-test tube interface. This phenomena is called a progressive

transition to the ignition regime. In the ignition regime, the surface of

the material reaches an explosive condition while the temperature of the

material at the center of the test tube has risen only slightly above
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rx 'xS«s«4Silb

i*a^,»ia»#t^‘iSP*/;^:l^

'i^l

i!yi 1'!*M
B.2. Description of the Isothermal Model

The simplest model used to describe thermal explosion phenomena is

one in which the temperature distribution in the sample is ignored and

the temperature of the sample is described by some average value T.

While this model cannot explain "fine grained” effects such as the

transition to ignition, it is regularly used by most investigators in

the field of thermal explosions (theory or applications) either to

explain the results of more realistic models qualitatively or as an aid

in analyzing both the latter and experimental results. The model will

apply to some laboratory or industrial stiuations quite closely (i.e. a

well-stirred liquid or gas, see [13], p. 18). The five assumptions of

the model are as follows.

1. The temperature of the material, T, is uniform.

2. The reaction takes place ion one stage (i.e. simple kinetics)

These two assumptions mean that the fraction of the original material

reacted, r), at any time t is given by:

In eq. B-2, A is the preexponential factor, E is the activation energy,

and R is the gas constant. A and E are assumed to be independent of

temperature.

"**The temperature dependence of k^ need not be of the Arrhenius form in

order for critical conditions to exist.

and is irreversible (i.e. far removed from equilibrium).

B-1

where f(ri) is some arbitrary function of ri and k^ is the specific rate

constant and is a function of T alone. We shall assume"^ k^ is expressed

in the Arrhenius form:

k^ = Aexp(-E/RT) B-2

B.2-1
3. No phase transition occurs prior to explosion (i.e. melting,

as usually occurs in explosives)

4. The container of the material remains closed and the rate of

heat transfer, between the sample and container depends only on

Newton's Law.

These two assumptions give for the heat balance equation for the

entire material:

T where C is the total heat capacity, Q is the total heat liberated by the

I reaction, and h is an empirical average heat transfer coefficient between

the sample and its container.

f 5. The characteristics of the reaction (A, E, Q ) and the physical

f properties of the sample (C, h ) are independent of temperature

and time over the ranges of interest. (This would exclude

diffusion limited kinetics in eq. B-1 but does not require

homogenous kinetics, per se.)

The last assumption is often justified by saying one is neglecting the

t I temperature dependence of C, h , and Q in comparison to k^ which seems

reasonable. In practice, this is justified provided one restricts the

analysis both to a range of value of T close to T^ and also to the initial

stages of those decomposition reaction in which gases are evolved (otherf - wise. one would not be able to assume h is constant over even a short time

or temperature interval)

The main problem in solving eqs. B-1 to B-3 as well as the corresponding equations for more detailed or realistic models of a self-reacting

system is that their general solution must be obtained by numerical

B.2-2
integration. To carry out these integrations efficiently, various dimensionless variables have been introduced. Becoming familiar with at least

some of these variables is important not only to understand the physical

meaning of the results of the numerical computations but also to cull out

the useful practical results of approximate analytical solutions of eqs.

B-1 to B-3. The variables are commonly used in almost all papers concerned with thermal explosion theory or its applications. A selected set

of these variables is used here.

The analysis of eqs. B-1 to B-3, as well as those of more complicated

models, may be grouped roughly into two categories; steady state theory

and non stationary theory.

In steady state theory, the following hypothetical situation is

posed. Suppose f(ri) is a constant, usually its maximum value, which

we will call f(ri ) (i.e. the reaction mechanism is zero order kinetics),

m

The questions asked are: a) At what values of T are the rate of heat

generation, q^, equal to the rate of heat loss, q^?* and b) ’'That is the

stability of these equilibrium states? In the model cited above:

» Q f(n^)Aexp(-E/RT) m

B-4

h'(T-T ) o B-5

By stability is meant the following. Suppose T is displaced by an

infinitesimal amount from its equilibrium value. If T always returns to

its equilibrium value whether the displacement is positive or negative,

the state is said to be stable or subcritical. If T returns to its

equilibrium value when T is displaced in one direction but not the other,

the state is called metastable (as at the critical value of T in the

o previous section -this is called the critical state) . The state is said
I f to be unstable if the initial values of Q ,f(n ),A,E, and h (and the

m

value of T ) are such that q, can never equal q„.

o 1 2

In non-s tat ionary analyses, the more complicated problem of understanding the temperature- time solutions of eqs. B-1 to B-3 is investigated in

terms of the results of the previous steady state analysis. Non-stationary

analyses can be classified into two groups. The first is the simpler case

where reactant consumption is ignored (as in the steady state analysis)

This applies to a first approximation to the events occuring prior to the

onset of an explosion in materials having a large heat of decomposition

per unit volume of the material. The second group is the more difficult

case where the effect of reactant consumption or material decomposition on

f(ri) prior to the onset of explosion cannot be ignored. This is the case

of interest to DOT.

Because it is impossible to understand the results of analyses in

this last category without first understanding the results of the other

simpler steady state analysis and non stationary analysis neglecting

reactant consumption, a brief summary of each type of analysis is given

below in sections B.3. and B.4. using the simple model already described

above

B.2-4
B.3. Steady State Analysis of the Isothermal Model

This analysis was originally developed by Semenov (see [29]). Plots

eq. B-5 versus T are usually referred to a Semenov diagram whatever the

temperature dependence of and q^. A Semenov diagram for values of T

near T (the only case of interest) is given in the figure on page 125 of

o ref [8] in which q^ is plotted for various values of in analogy to the

results of the explosion temperature experiment described in section B.l.

and in the second NBS hazard assessment. It is simpler to analyze a

Semenov diagram by considering to be constant and to examine the

I situation as q^ varies due to changes in h . A diagram of this situation

t is sketched in figure B-1 for a material for which Q is positive. One

f can see that for a large value of h , case (1) , q^ and q^ intersect at

I only one temperature and the state is stable. In case (2), h is

less than in case (1) . q^ and q^ intersect at two temperatures"*", and

only the state at temperature (Tj^)2 stable. Assuming the material is

originally at some temperature, less than or equal to T^, T increases

f only to (T^)^. As h is decreased to case (3), and T approach T and

^ ~ 3c Q C

at this point both q^ = q^ dq^/dT = dq^/dT. One can see this state

is metastable (i-.e. increase T slightly) and corresponds to the critical

f condition. Smaller values of h will result in a thermal explosion.

Solving eqs. B-4 and B-5 for the critical condition, one obtains, using

simple algebra and assuming 4RT^/E is less than one (which applies to the

cases of interest)

Because exp(-E/RT) versus T flattens out at high temperatures, there

will actually be three temperatures at which q 2 intersects q]_ in case

(2), and in case (3) ^2 will also intersect q^ at a very high temperature.

For values of h near the critical value, case (2), Ti and are near

Tj,. The other intersection temperature is very large and is approximately

equal to the higher intersection temperature for case (3) which is T +

Tq(E/RT )^. The latter value is far outside the validity of equations B-1

to B-3.

B.3-1
t

/

Figure B-1. Semenov Diagram

» i^ate of generation of heat by sample; q , rate of heat loss of sample; T, temperature of sample, Tq, temperature- of sample en- vironment; T at intersection

q^_ and q 2 , case (1); (To) 9, (T ) 9 at intersection of q and q case (2); T . T at tangency of I and

case (3) . ^ c ^1 ^2

B.3-2
Tj, = T* + T^[RT*/E....] B-6

and

I I 6=6 B-7

c where

6* = -[1 + RT /E...]; e = 2.718... B-8

c e‘ o and

I 6 = ^

Af(n) --» exp(-E/RT ) B-9

h “rt

2 °

o

T. is the value of T computed from eq. B-7 using eq. B-8 and B-9.

« o

» t Values of 6 greater than 6^ will result in always being greater than

t so that a thermal explosion occurs. 6 is a dimensionless variable

that is analogous to that Introduced by Frank-Kamenetskii (see [12],

Chapter 7) in connection with a more complex model than that being dist I cussed here. The value of T when 6 = 6 , is very close to T. since E is

c c * much larger than RT (i.e, if E = 35 kcal/mol and T is 100 ®C, T-T. is 8 o o “C).

t The physical Interpretation of the parameter 6 in eq. B-9 is that it

is the ratio of dq^^/dT divided by dq

2 /dT, both being evaluated at the

ambient temperature, T^. Both of these derivatives must be positive (i.e.

materials that decompose with absorption of heat are not hazardous) and

dq^/dT must increase more rapidly with increasing T than dq

2 /dT in order

for critical conditions to exist. It is clear that the latter requirement

was automatically satisfied by assuming k^ can be expressed in the Arrhenius

form with both A and E having positive values and by assuming ^2 increases

linearly with T, the simplest case.

B.3-3
Critical conditions can also arise naturally whenever mimics the

Arrhenius form over a limited temperature range. (The dependence of

and at values of T much higher than will, however, determine the

I I maximum possible value of T that can be attained if 6 exceeds *5^.) To

illustrate, it is first useful to digress and introduce the dimensionless

temperature 6 by expressing in terms of T-T^ as follows:

k^ = k^ exp [0/(1 + 30)] B-10

‘’o where

k^ = Aexp(-E/RT^) = 1 /t^

0 = E(T-T )/RT^

o o

B-11

B-12

3 = RT /E

o B-13

IcT^ is the rate constant when the temperature of the material equals T^.

Since k^ has the dimensions of reciprocal time, it is often expressed as

‘'o the reciprocal of a reaction time for which we use the s3nnbol T^. (It is

evident that since 0 is (T-T^)/T^3 where 3 is quite small, 0 will be small

only if (T-T^)/T^ is small.) Using eq. B-10, one can write the rate and

heat balance equations in terms of the dimensionless temperature, 0, as

= kj, f(n)exp[0/(l + 30)] B-14

^o

where

and

dt

B =

pdn Q

®dT X

q

= Bk^ [f (n)exp[0/(l

*0

E

RT

o

_5_ =

2 C

X = C/h

q

1^ Y

+

B-15

30)]-— f(n„)] B-15a

6

’ “

B-16

B-17

B.3-4
One can see from eq. B-15 that B or 1/y (another symbol often used) is the

maximum rise in 0 above 9=0, A0, that can be produced by decomposition

(i.e. adiabatic conditions). In this case, the thermal relaxation time,

Tq, is set equal to infinity and eq. B-15 yields A9 = B since Af) =

1 .

Returning to the question of what the temperature dependence of

must be to have critical conditions that are the same as when has the

Arrhenius form, consider first the case where the argument of the exponential in eq. B-10 is replaced 9, the so called exponential approximation. exp(9) differs from exp(i|^), where ijj is 0/(1 + 60), by at most 20%

t in the range of 9 from -2 to +2 when 6 = 0.05. The value of 5^ for the

exponential approximation is the same as that given in B-9 except that 6 =

RT^/E must be set equal to 0 and 0^ = 1 in Eq. B-6. Alternatively, suppose

2

one replaces exp (ij;) with 1 + a^0 + a20 where a^ = e-2 and a^ = 1, the so

called quadratic approximation (see [13] p. 27). This mimics exp (ijj) over

the range of 9 from 0 to 2 but is less than exp (ip) by 10.5% at 0 = + 2

(if 6 = .05). The critical conditions are the same as for the exponential

f approximation. In both of these examples, 6 is given by

6 = Bk,^ f(n) T = Bf(n^)T /t B-18

T m q m q r

o which is the form of eq. B-9 in terms of the new variables.

As to the question of what form k^ must have in order for a thermal

explosion to occur (if conditions are correct) , one can see that the

critical phenomenon does not naturally arise if exp(i|;) is replaced by a^ +

a20 where a^ and a

2

are arbitrary but positive. Thus, we can infer that,

qualitatively, the conditions for a substance to be potentially thermally

» • unstable are that Q be positive and that 5 , which can be written in

B.3-5
in the form of eq. B-19, increase very, rapidly with for an appreciable

range of temperatures.

6* = [dq^/dT]^ /[dq2/dT]^ B-19

o ^o

For example, critical conditions will be exhibited by heterogeneous

reactions which are diffusion limited provided this limitation applies

only to the higher portion of ambient temperature regime of interest. An

example of a Semenov diagram analysis of this type of mechanism is given

by Frank-Kamenetskii (see [12], Chapter 9) for the case where

dn ^ dt + (|> B-20

where k^ is of the Arrhenius form and (j) is a constant (i.e. (}) is approximately independent of temperature). At small values of T, dri/dt depends

on T as k^ while at high values of T, dr|/dt is independent of T. A short

physical description of this type of problem in context of an oxygen

combustion reaction is given by Thomas (see [15] p. 68).

The effect of the mechanism of the reaction, reactant consumption,

and previous thermal history of the material on the critical conditions is

contained in the quantity f(Tl^j^) of eq. B-9. If f(ri) is simple

f(n) = (l-n)"" B-21

where n is greater than zero, one can see f (ri ) = f (0) = 1. Thus, qualim

tatiyely, the effect of reactant consumption for simple mechanism will be

to increase the value of T in eq. B-9 and, hence, T in eq. B-6 at the

o c critical condition. In other words, if a real material is in a supercritical state (one which results in a thermal explosion) , the temperature

always slowly rises to some value above T^ given by eq. B-6 when f(r|) is

constant before an explosion occurs. An alternative way of obtaining this
result is to note that is lowered in the Semenov diagram so that the

point of tangency of and q

2

is shifted to a higher value of T for n > 0

as compared to ti = 0. Clearly, a partially decomposed sample that self-reacts

according to the simple scheme of eq. B-21 will be less dangerous than a

newly prepared sample of the material.

Unfortunately all materials do not decompose by this simple kinetic

scheme, although portions of a dri/dt versus r| plot may show this behaviour.

A very great number of materials decompose by an autocalytic mechanism that

may be nominally represented in its simplest form by the equation

f(n) = (n + UqXi - n); i

In this case fCu ) occurs atu =(l-ri)/2 (if ri =0, this a first

m moo

2

order autocalytic reaction) and f(ri mo) = (1 + U )/4. Clearly an older

sample of a material decomposing by eq. B-22 will be more dangerous than a

2 freshly prepared sample (when n = 0, f(0) = ri^; < (1 _ /4 if < 1) •

This is usually the case for many of the known explosive materials.

It is important to note that the initial stages of the decomposition

of many materials may involve two or more exothermic reactions with which

f are associated different values of Q , A, and E. Analysis of the critical

conditions proceeds in essentially the manner as described above but are

considerably more complex, as even the simple problem studied by Bowes

[31] illustrates.

As it stands, eq. B-9 or B-19 is not as informative as one might like

I because h is an empirically determined heat transfer coefficient and the

dependence on sample size, geometry, and thermal conductivity is hidden in

I f the quantity in the expression for Q /h . To indicate how this dependence

may be qualitatively built into eq. B-9, one may proceed as follows.

B.3-7
Suppose the material is a continuous liquid or a solid "chunk** of

material having a thermal conductivity X and a symmetrical shape similar

to its container with a characteristic dimension r (i.e. for a sphere, r

would be the radius) . Then for this solid one may write

i + ih h

B-23

h o where h is the heat tranfer coefficient of the material if its outer

surface is in perfect thermal contact with its container and h^ i,s theheat transfer coefficient of the material and container in the limit when

the temperature of the material is always uniform, h and h^ can be expressed as the product of the surface area of the material, S, and heat

transfer coefficients per unit area, a and a : o h = aS; h = a S o o B-24

Provided heat transfer in the material occurs solely by conduction , one

can express a approximately in the form

a = Xg^/r B-25

where g^ is a function of geometry alone provided a, a^, and S are independent of temperature and time. Equation B-25 accounts for the temperature gradients in the material due to heat transfer in the absence of self

f heating. Expressing Q as the product of the heat liberated per unit „

f T volume of the material, Q, and its volume, V, we have for Q /h :

1

.) . SXgi ra^

+ 1) B-26

h o Equation B-26 requires that Q be the same at every point of the material

f (or that Q be interpreted as the volume average of Q ) and that g^ in eq,

B-25 be approximately the same when self-heating occurs. V/S can be

written in the form

B.3-8
V

S = ^^§2 B-27

where g

2

is a function of geometry alone (but different from g^) provided

Q and V are also independent of temperature near T^. The quantity ct^r/X

is called the Biot number, Bi, which characterizes the relative effect of

the internal heat transfer in the material and heat transfer between the

material and its environment. (As Bi get large, the sample approaches

perfect thermal contact with its container.) Thus, one may write

4 = (§ + 1) h

X Si Bi

This permits us to write eqs. B-7, B-8, and B-9 in the form

6 = 5

where

and

1 ®1

6 = i — (1 +

c e g^

= 9l^

RT _

E •••-'^Bi + g,

expCO

)

B-28

B-29

B-30

B-31

The form of eqs. B-29 to B-31 makes 6 of eq. B-31 identical to the

critical parameter for the one dimensional models that take into account

temperature gradients in which heat transfer takes place only by conduction

(see section B.6.). Comparing eq. B-30 with the results of these models

at infinite Biot numbers yields values of g^/eg^ of 0.88, 2.00, and 3.32

for the infinite slab (half thickness r) , infinite cylinder (radius r) and

sphere (radius r) . From thermal relaxation time considerations"*" one' would

predict that the corresponding values of g^/eg^ are 0.91, 2.13, and 3.63.

The thermal relaxation time, see eq. B-1, can be written in the form ^ where K is the thermal diffusivity. Comparison with the longest

thermal relaxation times of the corresponding geometries, when aQ is

zero, gives the values of g-^/eg2 cited. This is the first approximate

method discussed by Thomas [32].

B.3-9
Since equals 1, 1/2, and 1/3 for the infinite slab, infinite cylinder,

and sphere, respectively, one can also obtain values of for each onedimensional geometry. Using the thermal relaxation values of g^/eg^ and

g. values calculated as indicated, one finds that 6 for the infinite slab

and cylinder given by eq. B-30 agrees within 7% with the exact values (see

-4

[31]) for all Biot values greater than 10

Other methods besides thermal relaxation times may be used to estimate 5^ for the geometries mentioned above as well as other geometries

(see [13], section 3.3, and Appendix B.6).. While a detailed check of all

known results has not been made, it seems probable that 6 can be estimac ted by methods involving an extension of the isothermal model to about 10%

or less"*".

In order to draw qualitative conclusions, it is somewhat easier to

I use the critical parameter 6 of eqs. B-8 and B-9 with eq. B-9 being

replaced by eq. B-31a:

+ 1) Af(n„)^ exp(-

O

B-31a

Equation B-31a indicates that, for a continuous material undergoing a

single exothermic homogenous decomposition reaction, there is, for any

given Biot number, a maximum critical size for each ambient temperature,

assuming it is constant and uniform, that must not be exceeded if a thermal

I explosion is to be avoided. Using 5 as an indicator of the instability

of the material, one can say that this critical size is approached at a

f rate (i.e. by 6 ) proportional to the first power of the diameter if Bi

In the literature, the dependence of 5^ on Bi is often approximated as being independent of geometry. For the three symmetrical cases considered

this dependence would be given by g]^ = e for eqs. B-28 and B-30.

Bi/ (Bi + e) usually appears as BiZ exp [+(Z-2) /Bi] where Z is Bi'- + 4 -

2

Bi. (see eq. 3 of ref. [14]). The latter matches 5^ to about 10%.

B.3-10
proportional to the first power of the diameter if Bi is small and the

second power of the diameter if Bi is large in comparison to g^. The

former case would apply to liquids or gases where convection (free or

forced due to gases liberated in the decomposition) dominates heat

transfer due to conduction. Small Biot numbers correspond to the physical

situation where the temperature difference between the surface of the

material and its container is much greater than the temperature gradient

in the material"*". Reducing the Biot number by reducing r or by reducing

have opposite effects on the thermal stability; the latter increases

the thermal instability of the material in the sense that it lowers the

critical temperature of the material.

The preceding simple extension of the simple model, which has been

discussed by many authors (see [14] section 2), works primarily because

prior to the onset of a thermal explosion, the temperature gradient in

the material is small. The important step in the analysis is the expression in eq. B-25 which is a statement of the fact that rate of loss of

heat of a symmetrically shaped chunk of material in perfect thermal

contact with its container, q^, is given by

q„ = £5 X4> (VT) dS „B-32

^ S

where (VT)^ is the gradient of T at and normal to the surface of the

material. One obtains eq. B-25 by dividing by S and replacing (VT) by

s

(VT)s = (T-T^)/r B-33

where T and T^ are the average and surface temperatures, respectively of

the material.

From eqs^ B— 32, B— 33, and B— 24 one has from steady state considerations

that \g]_ (T-Tg) S/r = a^S(T^-TQ) where T is the average temperature of the

material and Tg is the_temperature of the material adjacent to the

environment. Hence, (T-Tg) / (Tg-T^) = Bi/g^^.

B.3-11
Neither of these relations will necessarily apply to a material made

up of solid particles. Our literature survey has not turned up any

detailed analysis of this case -one of practical importance. Inclusion

of the effect of the sample container also has not been worked out in

detail (though the success of the previous extension of the simple model

suggests it can be treated in an analogous fashion"*") . All of the preceding

comments indicate that theoretical solutions are still needed to provide

a complete set of relations to predict the behaviour of materials in

shipping containers from laboratory test measurements bn small samples.

+

The result would be of the form

1

6 .= - — (1

c ® §2

RT t X g

)Bi [Bi(-^r +

E

^ '• ^r X g

1 ) + + 1 )]'^

where t is the thickness of the container which has a thermal conductivity,

X and a heat transfer coefficient per unit area, a , to the external

environment, gj, is analogous to g^ of eq. B-25 except that it applies to

the container. Thus, if the container has a large thermal conductivity,

small thickness, and a large heat transfer coefficient with respect to

the outside environment so, that (t^ X g]_)/(r X^ g^.) « 1 and g]_ 0.^/ 0.^ «

1 the effect of the presence of the container on 6^, is negligible. In

effect, the container becomes the environment of the material.
B.4 Nonstationary Analysis of the Isothermal Model, f(ri) Constant

An approximate idea of the relation between the time required for a

thermal explosion to take place (when the material is in a supercritical

state) and the parameters determined in the previous section can be obtained by neglecting the consumption of material due to decomposition on

f(ri). As such, the comments apply only to real explosives and then only

approximately. As in the previous section, one assvimes that f(n) in eq.

B-14 is replaced by a constant value; we shall use f (ri^) . Rearranging and

integrating eq. B-15a, one can write for the time, t°, to explosion that

is defined to occur at temperature T^;

9

d0

Bk f(n ) -^9.

r Q Tq in[exp(^ ^ gg) ]

B-34

where

RT

Tz e o B-35a

in

RT

7 (hn

- B-35b

’ 1 1

6 > —, or = e-A; e > A > 0 (a supercritical

state) B-36

is the initial temperature of the material at zero time when the

material is abruptly exposed to some higher ambinet temperature that

will cause the material to explode. 9^ is negative. The quantity

(Bk )“^ is called the adiabatic induction time, T , for reasons given

o later. Thus

r = B-37

In the following discussion we shall consider the variation of t° due to

changes in T the integral in eq. B-34, and f (n ) in that order.

m

B.4-1
It is important to note that the value of the integral is related to

T because from eq. B-18 we have

6 = f(ri' )t /t

'm q' CO

f and for explosive states 6 must be obey eq. B-36. (thus, in supercritcal

or explosive states, x^/f(n^) must always be less than e times

Thus, t° is a function of the four parameters T^/f(ri^) or t^, the ratio

T /f(ri )t , 9. , and 9 . It will be seen below that t° is relatively oo' q» xn’ e

^

insensitive to the choice of 9^ so long as it is large in comparison to

one. Thus, for fixed and 9^^, the greater the supercriticality of the

f t state (i.e. the greater 6 above 1/e, the less 1/5 below e, or the

greater A in eq. B-36), the smaller t° becomes because both Tggand the

integral in eq. B-34 decrease. This situation would apply to the explanation for the decreasing time to explosion for progressively increasing

ambient temperatures in the explosion temperature test in section B.l.

Also, this suggests that activation energies based on time to explosion

mesurements that assume

An t° = Jin T + aoo 1

where a^ is a constant independent of T^ will be subject to some error -

t because a^ changes with T^ through 6 in the integral of eq. B-34. This

has been verified in detail by Zinn and Mader [33] for more complex models

that include the effect of temperature gradients in the material.

In order to understand the effect of the integral in eq. B-34 on t°

in more detail, consider, first, t° for explosive states near the critical

state -when A is small in eq. B-36. From the section B.4. we know that

the sample will explode near 9^ = 1 so that 9^ should be some value greater

than one when d9/dt is quite large. It is convenient under these cirB.4-2
cumstances to break t° into the sum of the time, t'^ required to heat the

sample from T.^ to T (i.e. 9. to 0) and from T to T (i.e. 0 to 9 ): ^

--

t° = t° + t?

q 1

B-38.

where

T 9. « j. ±n d0

•q f(n ) 'e

m

[exp [ -0 9— ; 0 = >0

“] + —r ] 1-69 <S

B-39

X 9

t° =

®

i f(n ) ^0

m

d9

r / 0 X 0 1

O

B-40

Looking first at eq. B-39, t° can be estimated"^ for “0^^^ much

greater than 2 as:

T 2 O <» r r ^q f(n_) ^*^0 d9 , -0.

in.

m [1 + (a^ - A)9 + a^(0 ) ] — + <5 iln(-^) - (6 )"e) i

-9. » 2, e < 0.018, a, = 1.8628, a. = 0.2055

in — 1 2

Values of t° for 9 . = -10 to 9 = -2 and 9 = -2 to 9 = 0 are listed in

q in

the first two lines of table B-1 for A = 0.5 and A = 0. Warming of the

sample from 9^^ to -2 is accomplished almost completely by heat transfer

from the environment. The value of t° for the latter temperature interval

f depends only on the thermal relaxation time (i.e. from eq. B-18, 5 T^/f (ri^)

= T^) . Warming of the same from 9 = -2 to 0 involves both self heating

and heat transfer; neglect of the former would result an infinite value

of t° for this temperature interval. t° divided by T /f(ri ) decreases

q q

oo' ^ 'm as A decreases because,- at constant X /f(n ), this requires that X

<» m ^ q

decreases (see again eq. B-18). Thus, the heat transfer to the sample

increases, which shortens t . q

+ .

' . ’ ’2

^1 ^2 determined by fitting exp[—9 — 0.05 (9,) ] with^a quadratic

at 9 =1 and 9=2. The error in the integral from 9 = 0 to 9 = 2 is

0.02 X^/ f (r| ) . The value of e is p(-2) where p(X) is defined by Flynn

and Wall (see [6], eq. 13c).

.-41

B.4-3
Table B-1

II

ll

Induction Times and Reactant Consximption,

Explosive States, f(n) is constant

more

explosive

less

explosive

II

II

0.in

to 0 Parameter 0.5 0.25 0.1 0.05 0.01 0

-10 - 2 0.72 0.59

- 2 0 do 0.92 0.79

-10 0 do 1.6 1.4

0 2 t°/x /f(ri )'^ 2.9 4.6 8.1 12.2 29.5

0 00 do 4.3 5.5 9.0 13.1 30.5

-10 0 1.5 0.5

0 2 7.6 12.5 22.4 33.8 81.4

0 2 n°(B = 50) .15 .25 .45 .68 >1

0 2 n°(B = 300) .025 .042 .075 .11 .27

0, « - 2 . in*

quadratic approximation

A = e - 1/5

II

B.4-4

1
Using the quadratic approximation, one can estimate for t^:

-29-Y ^ II

t° ^ y

= 2 - A; Z = (4 - yY'

^ m'' When A becomes small"*^, t? becomes for 9 greater than one:

X ®

o n

'i f(tl„) A

m

Numerical values of t./(T /f(n„)) for 0^ = 2 and 9

-» ® are listed in

table B-1 in the fourth and fifth lines, respectively. The latter set

of values approaches the value of eq. B—43 as A decreases.

As the state gets less supercritical (i.e. as A decreases) t°

becomes progressively less dependent on the exact value of 9^ (compare

corresponding columns of the fourth and fifth lines) as eq. B-44 predicts.

B-42

B-43

t?/(T /f(n )) increases as A decreases for the same reason that t /(t /f(ri„))

1

' ' oo' ' 'm'' m ' Q °° m

decreases. It should be noted that t°/t? also decreases as A decreases;

q 1

the statement in the literature that t°/t? is negligible applies only

when A is small regardless of how big 9^ may be. At A = 0.5, the minimum

error in assuming to be zero is 38%. The value of A = 0.5 actually

corresponds to only a 7 K increase above the critical bath temperature,

T^, in the explosion temperature test described in section B.l, if T^ is

500 K and E = 35 kcal/ mol. It should be noted that if the time to

explosion is divided into two parts and one of these is computed for the

temperature range where self-heating is negligible, then this range must

"*'The conditions for eq. B-43 are A << 2(9 -1), /~A « 2(9 -1), and 5 << 1.

If 9 = [(Tq)^ - (Tq)^^q]E/RTo then 2 £n(l+69 ) - Tq 9 / (1+69 ) = £n(l-A/e)

or approxi^tely [ (Tq)^ - (Tq)a^q] - RT^/E)£n(l-A/e) . The effect of

changing 6 when T^ is constant xs:

t(^o>A - (^o>A=0^

2

1

0.5

C^Ja/^'"oo)a=o

0.26

0.63

0.82

45 K

16 K

7 K
be below 0 = -2. Thus, boundary conditions of the form ri = 0 at 9 = 0

do not approximate reality well and the error in this approximation is

greater as A increases in the explosion temperature test.

Times to explosion of different materials can be correlated to a

good first approximation by the comparing them at the same value of 6

and 9 . . Then the times to explosion are proportional to the thermal

xn

I relaxation time, T . (From eq. B-18 T /f(r| ) equals T /5 , so each term *q ^ oo m q

in the right side of eq. B-40 and eq. B-41 is proportional to T^.)

Since from eq. B-31, we have

^ cr g g

X

q

-T—^(1 +

^

Bx

2

the product of (crg^) should be the same at small Bi or t®X/ (cr g^)

should be the same at large Bi"^. c is the heat capacity per unit volume

of material.

The values of t® calculated from eq. B-41 and eq. B-42 are lower

bounds on the actual values of t“ when f(r|) is changing according to an

order decomposition mechanism provided the value of 9^ corresponds to a

value actually reached by the material before d9/dt = 0 when f(r|) is

not constant. Since t? equals t 9 /[(I - 9 )f(ri )] when A = 0, 9 =9

X 00 e e m e c

I will be less than one when 1/6 = e if f(ri) is not constant. From the

material reviewed in the following section (see in particular fig. B-4)

this limits the utility of eqs. B-41 and B-42 to values of A (and B)

that are not too small. ^

This assumes that all the materials are continuous solids, and, that up

until 9^, heat transfer is by heat conduction alone.

B.4-6
With this restriction in mind, one can use the corresponding expjfessions for to estimate upper bounds for the amount of initiaX

material that has decomposed prior to an explosion in the case where

f(n) is not constant. From eq. B-14 one has letting ip = 6/ (1 + 36)

o Substituting eq. B-15 into the left side, rearranging, and integrating

one has

0

1 r ® exp(ij;)d9 _ _o o ^ B

Q

exp(ij))-0/6' ^q ^i

in

where is the reactant consumed during the time t and n . is the

q q 1

reactant consumed during the time t?. An estimate of ri° for -9. >> 2

1 q m

o 1 r ^2 r - ^q

(e - A)9

[1 + (a^ - A)9’ + a^o')^]

-]d9 + 5 (0.50)}

where a^ and a^ are the same as for eq. B-41.

Using the quadratic approximation, n? can be estimated as

n° = ke +

~ ^ kn[9^ + (A - 2)0 + 1] +

(2 - A) (e - A) iBe _ ^ 2 e e 2 i

which for small A becomes

fl? ^ - 1) + — } I ^ ® ® ® /A"

Numerical values of and n? divided by B are listed in the fifth and

sixth lines of table B-1 for 0 =2. The numerical values of r|? for B

e ' 1

=50, (a cellulosic material) and B = 300 (an explosive) are tabulated

in the last two lines. -It is evident that r)°/ri° for small 9 is not

q X e negligible for large A -an assumption of analyses that use the initial

Equation 3-42, which involves the quadratic approximation, yields 1.3

for 0 » 1 and A = e because the quadratic approximation underestimates exp(-E?RT).

is

1-44

>-45a

l-45b

B.4-7
conditions of n 0 at 9 = 0. Also the statement that reactant consumption

is negligible prior to an explosion applies only to explosives (i.e.

when B is large) and even then only when 9^ is small.

t T is called the adiabatic induction time because if 1/5 = 0 (i.e.

GO

I h » 0 or 1/t = . 0) then eq. B-40 becomes for an order reaction (i.e.

q

f(n^) ” 1) (see [13]).

m

t° - T^[l - exp (-9^) + 23 [1 - (0^ + l)exp(-0^)] + 6^....]

» T ; 9 » 1

oo» e

It is to be noted if at the temperature T^, the material is isolated from

its environment, is decreased to Q^/B because of of the heat generated

by the decomposition goes into heating the sample. On the other hand;

Q

I n is now increased to -9, /B if 1/5 is zero from zero time onwards

q in

because heat absorption form the environment has been eliminated.

The dependence of induction times on the definition of the temperature,

9^, when an explosion takes place is mathematically somewhat arbitrary

although t° becomes essentially independent of 9^ so long as it is much

larger than one. In practice, explosions are detected by rupture of a

container or some other physical event which corresponds to a temperature

when d9/dt or 9 becomes large due to self-heating . One can see that a

large d9/dt always requires 9^ be greater than one regardless of how

explosive a state the material is in (i.e. whether A is small or near e)

from table B-2. This is in true in spite of the fact the feature of a 9

•versus t curve that distinguishes explosive from subcritical states, an

inflection, moves to progressively smaller values of 9 well below zero

as A increases. (The inflection temperature occurs at 9 1 + iJ,n(l - A/e).

The above criteria for 9^ are meaningless when the material is an adiabatic

condition, as the last column of table B-2 shows, because the material

always explodes.

B.4-8
Table B-2

^

X

dt f(r| ) versus 9; Exponential Approximation

0 0 1 2 e

-3 8.20 5.20 2.20 .05

-2 5.57 3.57 1.57 .13

-1 3.09 2.09 1.09 .37

0 1 1 1 1

1 0 1 2 1.72

2 1.95 3.95 5.95 7.39

3 11.9 14.9 17.9 20.1

4 43.7 47.7 51.7 54.6

^A = e - 1 / 5

'

B.4-9
When A is large, the simple model breaks down because, as Zinn and

Mader [33] and Merzhanov (see [14] p. 280), have shown; the material at

the container-sample interface reaches a critical condition even though

the spatial or volume average temperature of the material may be very

much less than T^. This phenomenon, transition to the ignition regime,

was mentioned in section B.l. Thus, it is to be expected that neither

the formula for t° or ri° will be valid when A is large. The effect of

f(ri ) on t° is different for order reactions where f(o) = f(ri ) and

m , m

autocatalytic reaction where f(o) « fCri ). t° will be much larger for

m . autocatalytic reactions when reactant consumption is included in the

model. Moreover, if the material is preheated and then again cooled, the

time to explosion on subsequent to exposure to a high value of T^ is

increased for decomposition by an order reaction but is decreased for an

autocatalytic mechanism.
B. 5 Non Stationary Analysis of the Isothermal Model, f(r|) Variable

From die previous two sections, which summarize the model for a zero

order reaction mechanism one can identify the following criteria for

distinguishing explosive from non-explosive states of the material:

(a) In non-explosive states, the temperature reaches a steady

state slightly above the ambient temperature while in explosive

states the temperature "takes off" after reaching this steady

state value.

(b) At the critical boundary between explosive states and nonexplosive states, the material is in a metastable situation

such that only a slight perturbation of initial conditions

(i.e. amount of material, ambient temperature, or heat exchange

with the environment) can shift the state of the material from

an explosive to a non-explosive state or vice versa.

(c) Explosive states of the material have an inflection point in

the 6 or T versus t curve before 0 reaches its maximum value

while non-explosive states do not.

When reactant consumption is now incorporated into the simple

model, criterion (a) disappears, mathematically. This loss seems to

have been a source of considerable concern to investigators in the field

of thermal explosions because it raises doubt about any analysis of

criticality that depends upon criterion (a) (all of the steady state

analyses). For example, one finds on page 126 of Gray and Lee'^s review

[13], the upsetting statement "...is one of the most important results

to have emerged from non-steady state theory since it rationalizes~^ the

problem of defining a suitable critical preexplosion temperature rise

when' reactant consumption is important".

Underline is ours; the quote is a comment on the analysis of Adler and

Enig (41) discussed later in this section.
Another group of prominent investigators, Merzhanov and coworkers, summarize

the problem more dramatically as follows. Based on the extensive analyses

of nonstationary analyses made by them and their coworkers, they state

(see [14], p. 283) "...there are no critical conditions in the mathematical

sense and likewise there is no heating-up preceding explosion. Merzhanov

et al. do qualify these remaks by adding (somewhat contradictorily) that

"...thermal explosion corresponds to a narrow region, roughly for values

of B between 100 and 1000, and the critical conditions have a clear

physical significance." Thomas [15] puts the loss of criterion (a) into

a more practical perspective by pointing out that "...assuming a finite

amount of reactant avoids the unreality of an infinite temperature rise"

(if an explosion occurs) but now "...there is no strict distinction

between supercritical and subcritical states which is not in some sense

arbitrary...". The ambiguity in this distinction is more important when

B is small (see comments concerning eq. B-45 in B.4.). However, if B is

very small, say 20 , the maximum dimensionless temperature rise is so

small that the practical (experimental) distinction is more of a qualitative than a quanitative problem anyway.

The main concerns of theoretical investigations can be viewed as -

attempts to answer the following: (a) What are the essential conditions

for any proposed definition of criticality? (b) What definition of

criticality duplicates the results of steady state theory when 3 is

large, independent of the form of f(ri)? (c) Can a definition of criticality

be devised that is derivable from instability considerations that also

fulfills the condition mentioned in item (b)?

"^Our survey of more recent literature shows this to be demonstrably incorrect.

“^For Q/R 400 Jg"^ c/p 1.5 Jg"^K”^, E/RT 'v 25 and T 600 K. p is the

density. ° °

B.5-2
We shall summarize the investigations in terms of these questions in detail

only for simple order reactions and will comment on a limited number of

the earlier investigations we have studied. A more complete review of

these earlier investigations are given by Gray and Lee (see [13] Chapter

5).

t From steady state theory it was inferred that if 6 is redefined,

as we shall do throughout this section, as

Bt

o *

, r then the effect of reactant consumption on 6 at criticality might be

« - Bk^ B-47

expected to be

, . (1 + RT /E ••)

6 '\j —

c e fCn ) c where n is some critical value of the fraction of initial material that

c has decomposed. This cannot*^ be derived from a Semenov diagram analysis

in which now varies with time (because f(ri) changes) by letting q^

=

q« and dq, /dT = dq«/dT at the critical value of n» h as was done before.

2 1 2 c Arguments developed in earlier work based on eq. B-48 prove to be correct

provided criticality is defined to be an inflection point in the 9,t

curve. In fact, B-48 is identical to the condition that in an explosive

state near the critical condition the inflection point in the 9,t curve

1/2

occurs at 9 =1 assuming [n/((l-ri )B)] is much less than one (r| is

c c c the value of r| at 9 = 1, n is the order of the reaction, assumed here to

' 2 2 be an integer). This can be derived from the condition that d 9/dt =0

for d9/dt > 0 which requires in the exponential approximation that:

1

6

»

|et£(n)eS] , 11-^^ iai fe®

B-48

dq]_/dT = dq 2 /dT implies d(f (n)exp (9) ) /d9 = 1/5 or d 9/dt _< 0 in the

dimensionless time variable while q = q implies d9/dt = 0. These conditions

are not simultaneously satisfied until all the material is decomposed and

9 = 0.
* _1

Noting that dri/d6 = {B[l - 9exp(-0) / (f (ri)S )]} one has letting n =

n , 9 = 9

c’ c Q

(X - i)(x - 9^) = nx^/[(i - n^)B]; x"^ ^

The only value of 9^ that satisfies this relation and yields eq. B-A8,

1/2 when RT^/E and [n/C(l - ri^)B)J are much less than one is 9^ equals 1.

An interpretation of this mathematics in terms of the Semenov diagram,

modified for the case where fCn) is not constant, is given later.

The argument developed by Frank-Kamenetskii (see [12] Chapter 7) to

derive eq. B-48 illustrates the main physical ideas and assumptions of

this earlier work (as well as their inconsistencies) and is outlined

below.

Using the exponential approximation, the rate equation eq. B-14

becomes

dri/dt = exp(9)(l - ri)\^ B-49

Assuming r| is small and exp (9) has an average value of e over the time

prior to an explosion

dn/dt e(l - nn)/x^;

where t^ is the time to explosion when n = 0 at 9 = 0. Rearranging eq.

I B-34 where 5 is now defined by eq. B-47 we can estimate t^ from

t T /

C 0° O

d9

9

[f (rij,)exp(9) - 9/6

= tJ d9

^

o [f (nj,)6' (exp(9) - 0]

B-51

In an explosive state near criticality, we have by analogy from steady

state theory

f(ri^)6 'V. (1 + y)/e B-52a

where y is positive but small. Assuming 9 = 1 4- X, one can approximate

exp(O) if X is small by

exp(9) e(l + X + X^/2—

)

Physically, the assumption that the material is an explosive state or nonexplosive state should not appreciably effect the value of t very close to

criticality.

B.5-4
Keeping terms to the first order of staall quantities QC or Xy) the expression

toi. t becomes

c ®e~^ 2 '-I t 'T/i [y + xy + X/2 ••] dx

c q -1 '

If (0^-1) /y is large one obtains

t IIt —

c q y

Substituting into the expression for t^ in eq. B-50 we have for small y:

4if - 5)C

n = eTT- c

The parameter y is identified by equating eq. B-48 with the right side

of eq. B-52a which gives for small ri^

^ —(1 + y) ; y nf) e(l - nn ) e

^ ^

'c c Substituting the value of y into B-53 one obtains

n - 2.7

nB^

Thus it is concluded from eq. B-48 that

B-52b

B-52c

B-53

B-54a

5 'V' c e[l - 2.7(|)^^^1

B-54b

The analysis by Thomas [34] makes the same type of assumptions as

those involved in obtaining eq. B-51 but departs in that the analogous

differential form of eq. B-51 is converted to Airy’s equation assuming y

is small, and solved analytically for the value of y that produces an

inflection point in the curve. Thomas derives an equation of the same form

Eq. B-43b yields the same value for except that 2 is replaced by e because the quadratic approximation should produce a bigger t^, than the exponential approximation. In either case eq. B-52b does not require that

9^-1 be large, only (9^-1) /y.

This is an added assumption and is not deducible from the previous equations,

in our opinion.

The quadratic rather than the exponential approximation is used.

B.5-5
as eq. B-54b but with the constant 2.7 replaced by 2.85' . The analysis of

Rice et al (see [35], also [36,37] differs from that of Thomas mainly in

that the analogous differential form of eq. B-52 is solved nximerically

(for a first order reaction). No formula analogous to eq. B-54 is derived.

Examination of their results shows that an inflection point occurs very

near 9=1 and not at 9 = 2 as stated by Gray and Lee ([13], p. 116).

The recent mathematical investigation of Gray and Sherrington (see

[38] p. 448, fig. 2,3) determines directly the conditions for an inflection

point in the 9,t curve. The previous work reported by Tyler and Wesley

[39] is essentially equivalent to determining an inflection point in the

9,t curve but is less direct.

It is useful to define the word "direct** because the associated

procedure has been used to determine critical conditions in more recent

non-s tat ionary analyses: (1) The locus of inflection points (or any other

2 2 criterion) is deduced by setting d 9/dt = 0. This yields a relation

between ri, 9, and the dimensionless parameters n (order of reaction), B,

t and 5 . (2) It is then required that the 9,ri curve just touch and be

tangential to this locus at the single point of contact. This requirement

is deduced by differentiating the locus of inflection points (or any other

condition) with respect to 9 and setting dri/d9 in this expression equal to

that for the 9,r) curve which is given by

dn/d9 = -{B[l - exp(-'i*)9/(5’f(n))] ^ B-55

"'"Our review of this work tentatively suggests that the analysis by Thomas could

be improved by eliminating what appears to be an unnecessary assumption.

Also, we do not see where 9^ = 2 at the inflection point is assumed to

derive the critical condition which we infer Gray and Lee to state.

B.5-6
where = 9/(1 + 66) and f(n) = (1 - n)^ (n > 0). (3) The values of

9 = 9 and ri = H at which the 9,ri curve first contacts and is also tanc c gential to the integral curve can be determined (usually numerically) for

t any given n, B, and 5 from eq. B-55 and the locus of inflection points.

(4) The integral of eq. B-55, the so called integral curve, is then computed

numerically for suitable initial conditions (usually n = 0 at 9 = 0) to

determine if the curve intersects ri at 9 , etc.

c c To see what the direct procedure means graphically for a 9,t inflection

point, a Semenov diagram appropriate to the case where f(ri) is not constant

(only a single decomposition is taking place) has been constructed in

figure B-2 for a first order reaction, q^ and q^ have been plotted in the

vertical direction or Z axis, 9 on the X axis, and r) (which varies from 0

to 1) in the Y axis. The rate of heat generation,

q^^

in terms of eq. B-15

and 15a can be described by

. ^dr\ B ,9

r Thus, q^ is a curved surface that intersects the X = 0 plane in the line

q^ = (B/t^)(1 - n)- ^2 rate of heat loss, is then defined by

9 _ B 9

^2 T

q T 5 r

1 q^ is a plane that has a slope B/ ("^^.^ ) in the Y = 0 plane and passes

through the Y axis. The q^ plane intersects the surface along the

solid line I whose projection on the Z = 0 plane or 9,n plane is the line

i. Along I or i, d9/dt = 0 and this defines the maximum temperature the

system can attain. Two integral curves (i.e. the integral of eq. B-55)

have been plotted in the Z = 0 or X-Y plane as and J^. Consider the

meaning of the inflection in terms of these two integral curves. Assume

is subcritical. Draw a vertical from any point on before its intersection
( 2 <xx IS ) jr

Figure B-2. Semenov Diagram; f(ri) = (1 - n)

I, intersection q]_ surface and q 2 plane; i, projection of I on Z = 0

plane; J 2 , integral curves; Z 2 , intersection of vertical with q plane;

Z^, intersection of vertical with q plane. ^

B.5-8
with line i. It will intersect the plane at the vertical height and

the surface at the height Z^. For the integral curve to be subcritical,

the distance must always decrease as point defining the state of the

system progresses along the curve from f) = 0 up to its intersection

with line i. For the integral curve to be supercritical, the distance

Z^-Z^ must decrease and then increase as the point defining the state of

the system moves along J

2 from n

= 0 to its intersection with the line i.

The direct procedure outlined above shortens the procedure by determining,

in step 1, a line analogous to line i in the X-Y plane which the integral

curve must cut in order for decrease and then increase. The

remainder of the procedure constitutes finding that particular Integral

curve that is just tangent to this Inflection locus line.

There are other criteria for an explosion from a 6,t inflection

point, however, and they are not less valid. As soon as one concedes that

f(ri) is not a constant, one must redefine what constitutes an explosion

either in terms of the character of the 9,t curve or the 6,n curve. Also,

one need not use the direct procedure outlined above to determine critical

conditions appropriate to any definition of an explosion. For example,

Tyler and Wesley's [39] procedure differs from the direct procedure in

that d0/dt and dri/dt are integrated numerically (initial conditions r| = 0

at 9 = t = 0) for a given value of n, B, 3, and a range of values 6 as a

function of time. Criticality for a particular set of n, B, and 3 is

I identified as the value of 6 which if increased by 1% doubles the maximum

value of 9 in the simple model or the maximum value of 0 at the center of

the material in the more complex models in which temperature gradients in

the sample are taken into account. The critical value of 9 is the value

of 0 (or 0 in the center of the material) at .99 6 (.99 6 ). c c

B.5-9
The equations for the simple model that were Integrated by Tyler and

Wesley are as follows:

S - I

<1 -

If

= 6(1 - n)"exp(j-f-gg-) - A6

= 6[(1 - n)^exp(

j^ ^ gg

) - a6] B-57

These are written down because they are the ones most commonly used except

in the Russian literature for comparison of the results of the simple

model with those of models in which the material has a simple symmetrical

geometry (sphere, infinite cylinder, or infinite slab), the boundary

conditions are uniform and the temperature gradient varies in only one

direction. The translation of the notation of eqs. B-56 and B-57 to ours

is as follows

Eqs. B-56, B-57

0

X

6

3

A

a(= A/6)

B

6/B

Our notation

t/(cr /A)

6/f(r) ); 6 is in eq. B-31 m

h r^/(VA);

1/6 ; see eq. B-47"*"*"

B cr\^ /A = cr^/(Ax^)

o The most important result of Tyler and Wesley is that the numerical

values of e/g^ for B ^ 25 and 3 < 0.1 agree within 3% of those computed

using the values of l/g2 cited previously in section B.3 for the more

complex models of the infinite slab, infinite cylinder, and sphere cited

in section B.3. In other words, the critical conditions can be evaluated

This may be derived from eq. B-30 and the subsequent discussion which compares

eq. B-30 with the steady state results of the same complex models and the simple

model

I ' I 6 of this section is 6 of B.3 and B.4 divided by f(n ).

_ _ in B.5-10
without taking into account the temperature gradients in the sample .

I I (If 6 » 6 one cannot, however, talk about when an explosion takes place

because the phenomena of the ignition regime occur as described in section

B.4. Also the simple model cannot be used if the environment temperature

is non uniform)'*"*'.

f Numerical values of 6 , or 6 for these more complex models, were

c c within 0.2% of the corresponding critical value for an inflection point in

the 0,t curve with B ^ 100 and 1.5% of the value at B = 25. No critical

condition (in their definition) occurs for B £ 14 at 6 = 0 and B ^ 25 at

larger 3, though in some of the cases inflections in the 9,t curve did

occur. They found their numerical values for (e5^) could be fitted to

' -1 2/3 an expression of the form eq. B-54 (i.e. (e6^) = a^ + a

2

(n/B) within

2% for B ^ 25 for n = 1 and B ^ 100 for n = 2 with values of a^ near 1

and a^ near 2.4. The value of 9 could be expressed to within 3% by an

2 c 2/3 expression of the form a^ + a^(n/B) where a^ is near one and a^ is

between 5 and 13. Numerical values of a^ through a^ given by Tyler and

Wesley are listed in table B-3 along with the other data relevant to this

review.' For example, 9^ for the various geometries do not differ greatly

from 9^ 'V' 1 of the simple model when n = 0 but they increase with increasing

n/B.

Induction times to explosion are continuous as 5 increases from

below to above 6 , in contrast to the behaviour of the discussion in

c section B.4., and reactant consumption becomes very large as B gets small,

in agreement with the qualitative argument of section B.4.

+.That is, correction for the effect of temperature gradients can be accomplished using the arguments leading to eq. B-30 and B-31 in steady

state theory.

A non uniform ambient temperature would require that temperature gradients

be taken into account explicitly rather than by the aposteriori approach

Leading to eq. B-30 and eq . B-31.

The only values of n considered.

B.5-11
Table B-3

Results of Tyler and Wesley [39]

(n - 1,2; B > 25; Bi - ®; n - 0 at 9 - 0)

' -1

(6 e) - a - (n/B)^^^; 9_ » a, + a,(n/B)^^^

*•

t

B ®2 *3

.00 1.000 2.28 0.880 5.58

.025 0.973 2.35 0.930 6.54

.050 0.944 2.41 0.985 7.93

.075 0.916 2.49 1.045 9.98

.100 0.885 2.56 1.110 13.1

Steady State Calculations

geometry gj^/Cg^e), see eg. B-30

• +

9 (0) c slab 0.8783 1.164

cylinder 2.000 1.376

sphere 3.3218 1.585

"*”9 (0) is critical

c. value at center of material of steady state analysis

nonstationary analysis it is calculated from 9 » 9 9 (0) c c c

•

Induction Times

(units: t/r ; conditions: 9 =« 29 , 0.99 5 < 5 <1.01 5 , any B < 0.1) ao» e c’ c — — C n » 1 — n » 2

B slab cylinder sphere slab cylinder sphere

1000 9.38 3.90 2.19 8.13 3.36 1.89

200 6.37 2.63 1.48 5.20 2.16 1.21

100 5.12 2.12 1.19 4.08 1.17 0.96

Percent Reactant Consumption in Sphere

(at maximum 9 at 6 » 0.,99 6^)

n(Z) , n - 1 n(2). n 2

B Center Wall Center Wall

1000 2.7 - 0.7 2.4 0.6

200 10.1 2.6 8.7 2.2 . 100 18.0 4.5 16.3 3.8

25 51.6 11.7 40.1 9.5

i

II

I

I

I

I

I

i

p

I

R

I

I

I

B.5-12
Gray and Sherrington, using Che "direct" procedure (see [40], p. 438)

showed that as y

= 1/B gets small for n = 1 and 3=0, that 9^ approaches

1 at the critical condition for an inflection in the 9,t curve prior to

the maximum value of 9 in the 9,t curve.

Adler and Enig [41] have deduced the critical conditions for an

inflection point in the 9,n cuirve by the direct procedure for 3=0 and n

** 1, 2, 3, 4, 5, and 10. They have shown that criticality in this sense

requires and necessarily implies that an inflection in the G,c curve has

already occurred if there is an inflection in Che 9,n curve. Physically,

t this means that the subcritical region in the a = 1/6 versus y plane is

larger for their criterion than for a 9,t inflection, as Gray and Sherrington

(see [40], p. 448) have pointed out.

f A comparison of a = 1/5 versus y results of Tyler and Wesley, Gray

and Sherrington, Adler and Enig, and Che earlier results of Thomas are

given in curves (a), (b) , (c) and (d) , respectively of figure B-3 for a

first order reaction. The curve (a) from Tyler and Wesley is for Che

range of validity of their formula (y _< 0.04) and the computed curve from

Thomas, (d) , is also valid only for small y. (Curves (e) and (f) are

discussed later. Below y

= 0.05 curves (a) through (c) are essentially

indistinguishable and their difference from curve (d) is remarkably small

considering the approximations Thomas made in his analysis"*". The agreement

between curves (b) and (c) in this range is to be expected because Tyler

t and Wesley noted that according to their criterion of criticality 5^

occurs approximately when the maximum value of 9 is 2 which is the requirement

on 9^ of Adler and Enig’s criteria for an inflection point in the integral

"*"We estimate Chat the net error in the approximations made by Thomas to

convert the right side of B-15a to a form that can be integrated analytically

is of the order of 20% or so. In this sense, reasonable agreement should

be expected.
a-

\

I

t Figure B-3. Sketch of 1/5 a versus y 1/B, n = 1, 3

2 2

(a) Tyler and Wesley [39]; (b) d 0/dt = 0, Gray and Sherrington [40]

(c) d^0/dn^ = 0, Adler and Enig [41]; (d) d"^0/dt^ = 0, Thomas [34];

(e) First trial function. Gray et al [38]; (f) Second trial function.

Gray et al [38]

m

B.5-14

I
curve when n = 1 (3 = O)"*". However, Tyler and Wesley noted that their

value of was within a few percent of that required to first produce a 0,t

inflection point. Hence the agreement is not coincidental.

Outside the range y £ 0.05 curves (b) and (c) diverge rapidly.

Further, curve (a) cannot be realistically extrapolated because Tyler and

Wesley's criterion for criticality is no longer valid. The reason for the

latter statement can be seen from a rough sketch, figure B-4 of at

f d9/dt = 0) as a function of 6 for various y taken from Merzhanov et al

[42] for n = 1, 3=0. For y

= 0.05 it is still possible to obtain a

large change in 0 for a small change in y identifying, thus, a critical m

I I value of 6 . Also, the magnitude of 0 for a large value of 6 is quite

c m

I large. However, for y ^ 0.1, <5^ by Tyler and Wesley's criteria becomes

I meaningless and the maximum value of 0 at large 6 gets progressively m

smaller. A representative value for the maximum temperature rise above

ambient, T -T , is calculated for an activation energy of 30 kcalmol ^ and

T^ = 600 K (327 C) to put the figure into a less abstract perspective. It

is clear from figure B-4 that for y ^ 0.1, Tyler and Wesley's criterion

breaks down. It is for this reason that Merzhanov et al [14], for example, state that there is no critical condition for large y.

I I The sketch in figure B-4 is too inaccurate to determine 0 corresponding '2 m

2 2 2 to 6^ for d 0/dt =0 and d 6/dr| = 0 but some rough estimates can be made

and they are shown by the dotted lines. It seems fairly clear (but this

must be checked) that both of these criteria would not consider a maximum.

The criteria of Tyler and Wesley and Adler and Enig are not exactly the

same. An inflection in the 0,ri curve (for n = 1) occurs whenever the

integral curve passes thru 9c = 1 and l-n,, = 2aexp(-2) + 2/B. 0,, and l-n^ are not the same as the maximum values of 0 and r| at the maximum 0 when

d0/dr| = d0/dt = 0.

Taken from a very small graph.

B.5-15
Figure B-4. Sketch of 6 (0 for d6/dt = 0) versus 6 (after Merzhanov et al

[42]) “

T -T for E/RT = 25, T = 600 K; n = 1, 3=0. mo o o

B.5-16
9 of the order of 5 to be an explosion at y = -05 and this would constitute

tn a temperature rise above ambient of 100 C if is 600 K (327 C) and 46 C

if T is 373 K (100 C) . Approximately 5% of the original material would

o be left by the time 9=9! It seems relevant to point out that if the

decomposition products were gases and the material is in a closed container

an explosion would occur in the real world at least in terms of container

rupture. Using the 9,ri inflection criterion one finds that even at the

time 9=2, roughly one half of the sample is decomposed. Thus it is

evident that an explosion criteria of the type being discussed is of

primary interest to the shipping agent only if gaseous products are not

liberated (i.e. this is an important part of the hazard rating that theory

cannot take into account explicitly. Implicity, it is done through the

statement n 0) •

Recently, Gray and Sherrington [40], have started an investigation

aimed at setting up an explosion criterion that has some physical foundation that is not purely intuitive. Their arguments, as we interpret them,

starts from the second observation noted at the beginning of this section

(see (b) of the first paragraph) ; 1) Because stability analysis alone can

be used to deduce all the results of the steady state theory, it should be

possible to deduce critical conditions when f(ri) is not constant by time

dependent stability analysis. 2) The time dependent stability analysis

should be selected so all the critical conditions of the steady state

'theory are generated as a particular case. Otherwise the limits of applicability of the results of the steady state results will not be- known. 3)

The preceding particular case or conditions are that in the limit when y

=

I 1/B goes to zero, 9^ = 1 and 6^ = 1/e. This corresponds to the experimental

fact that as Y 0 it is found that the steady state results apply with

f(ri) taken to be constant.

B.5-17
Gray [43] has recently pointed out that if the third argximent is

correct it should be true that our original time dependent equations must

have the same critical conditions as those given in section B.3 in the

limit when y

= 1/B ^ 0. Referring to eq. B-14 and eq. B-15a one can write

them as

1 d9 d0 \ /t-\ 0 I B dF' ^ 57" - p-: f' “

O

0

1 + 00

dt

f (n)

T

exp (ip) r The problem is then to prove that critical conditions for the solutions of

the limiting case, y

->• 0;

0 = f(n)exp(4;) ?

6

dn _ f (n)exp(il^)

dt ~ T

r

I are 0 =1 and 1/5 = e, for example, when 0=0, for a first order reaction

c c

I and l/<5^ = e/4 when 0 = 0 for a first order autocatalytic reaction. It is

surprising but nevertheless true that this was proved only recently [44]

We are in disagreement with the idea that the last set of equations

is the Semenov problem discussed in sections B.3 and B.4, however. (We

may be inferring something Gray did not mean.) The equations of B.3 and

B.4 are those solely for inexhaustible reactant (or starting material

which for large B make the change in f(ri) about some preselected value of

n = ri very small. We see no reasons for replacing one unreal situation

m

(inexhaustible reactant and preselected f(n )) with an equally unreal

m

situation of inexhaustible reactant with zero relaxation time"*".

, 0 = Bt /x = T / (yx ) for fixed 5 and x,x -^0asy-»-0.

q r q r r q
So far, results in regards to developing a stability analysis for

critical conditions consistent with the three arguments advanced by Gray

' and Sherrington must be regarded as preliminary. A stability analysis

using the direct procedure"*" of Liapunov with a Liapunov function that is

quadratically dependent on the perturbations in 0 and ri, but otherwise is

time independent, has been carried out [38] for a first order reaction for

two forms of the quadratic. The results are shown as curves (e) and (f)

of figure B-3; curve (f) is the better of the two estimates of the subciritical region using this type of Liapunov function. Recently Gray [45] has

I proposed a more restrictive stability criterion but no numerical results

have been presented yet. It is, as of now, pure conjecture but it seems

f likely on physical grounds that stability analysis will yield a a = 1/5

versus y curve lying close to the 9,t inflection point curve.

This is not the direct procedure referred to previously. It involves the

idea of, for example, guessing a function V(E, S, t) where E and S are the

perturbations in 0 and r|, respectively, such that V is positive definite

and dV/dt < 0 for ail t as the locus of stability in step 1 of our direct

procedure and then determining critical conditions.

B.5-19
II1ttII
B. 6 Temperature Gradients in Material, Various Geometries

' The heat balance equation assuming the material is a continuous

solid, heat transfer is by conduction only, and the heat generated power

unit volume under isothermal conditions is everywhere the same is

cf - og +

2 All the symbols have been defined in section B.3-5 except V , which is the

Laplacian operator, and now T is a function of position, x. For the

infinite slab, infinite cylinder, and sphere

.2 d‘ V T = —

dx

t n d

xdx

where a = 0, 1, and 2 respectively. In dimensionless notation the heat

balance equation becomes for these geometries

d0 - r 9 , . l,d^0 , n d9,

duTfj “

d(f^ - if(n)exp[3-^]

where all the symbols have been defined previously (e.g., 6 is defined by

eq. B-31) except £ which is x/r where r is the half width of the slab, or

radius of the cylinder or sphere. For these geometries, the boundary

conditions are:

e - 0; 30/9e = 0 (i.e. center of symmetry)

£ = 1; 36/3£ = -Bi0 (i.e. surface of material)

t/T = 0; 0. ; 0. = E(T. -T )/RT ^ < 0;

n

= 0

When Bi -> ® the boundary condition at £ = 1 is 0 = 0 and the material is

in perfect thermal contact with the container. However, if Bi -> 0 the

heat loss terra becomes

Limit

Bi ^ 0

1 , n30, _ 0

6 ^-2 £ 3

3 £

£^ T5

,-58

;-59

;-60

,-61a

:-61b

i-61c

B.6-1
where again 5 is defined In eq. B-31a. The simple model discussed in detail

in sections B3. -B5. is the exact model (see [13], p. 64). (This is true

for any geometry but does require that the rate of heat generation be the

same everywhere in the material under isothermal conditions . ) That is, as

the heat loss becomes small, the temperature gradient in the material also

becomes small. Physically, one can see if one has heterogeneous heat

generation in the form of reactions at or around surfaces of particles,

the same result will apply to the extent that the individual particle

volume is small in comparison to the total volume of the material.

Steady State Analysis (d9/dt = 0)

One can write the critical conditions in the form

where are the critical temperature and critical FK (Frank-Kamenetski)

parameter when Bi is infinite and 3 = 0. The,(])'s are the corrections for

finite Biot number and 3 = RT^/E, respectively. Strictly speaking these (]>

correction's are a function of the specific geometry. Merzhanov et al

[14], p. 281) have pointed out that to within 10%, (j)^(Bi) is the same for

the infinite slab, infinite cylinder, and sphere (see footnote, p. B-17)

They also indicate that, to a good approximation (no error limits stated),

(f>g(3) '^1 + 2 3 and <[>(5( 3 ) 1 + 3 in analogy with the simple model

of section B.2.

A detailed discussion of exact values for 6

° for the infinite slab,

c infinite cylinder, and sphere are given, respectively, in ref. [13] on

pages 39, 42, and 44 respectively and values summarized on page 45. The

corresponding exact values of t{)Q(Bi) and <{)^(Bi) are discussed in ref. [13]

64-69. Exact values of 6^“<})^(3 = .01) are known for the curbe and

regular circular cylinder - see table on page 63 of ref. [13].
Approximate methods of calculation of 5^* for more complicated geometries are discussed in ref. [13] on pages 51-63 and are sximmarized on

page 63; a brief sximmary is found in ref. [14], p. 282.

Methods of relating the critical parameters of the infinite slab, infinite cylinder, and sphere to those of the simple model are discussed in

detail in ref. [13], p. 47 and p. 69-74. (Our extension of the simple

model to these geometries is essentially the same as that of Barzykin and

Merzhanov [46] - summarized on p. 73-74 of ref. [13]).

Assymetric boundary conditions for the infinite slab and infinite

cylindrical annulus have been discussed in detail in ref. [13], p. 74-87.

A tabulation of more recent articles (i.e. since the publication of

ref. [13], [14]) is given below;

More (or less) Exact Methods Ref.

1. Mathematical properties of steady state equations (3 0) [47]

2. 0° slab (symmetric and asymmetric boundary conditions) [48,49]

3. “ slab (periodic variation of temperature with position). [50]

4. Sphere, solid and hollow cylinder (heat fltix prescribed [51]

at boundaries)

.

Approximate methods

5. Simple geometries (Bi = «) [52]

6. Slab, solid and hollow cylinder, solid and hollow sphere [53]

(Bi +

«)

7. Rectangular parallel piped, finite right cylinder, cone [54]

(Bi = «)

8. Arbitrary shape (arbitrary Bi) [55]

B.6-3
Other

9. Forced convection (references to other papers) 156]

10. Hot gas bubble in cool reacting liquid [57]

Non Steady State Solutions

A detailed discussion of the earlier work (analytical and numerical)

for the case where f(ri) is constant is given in ref. [13], p. 134-150.

The discussion in ref. [14], though published at the same time, updates

the discussion of ref. [13] in terms of the extensive advances made by

Russian investigators. It includes the case where f(n) is not constant,

autocatalytic decomposition, temperature- time variations of the environment

other than a step jump, convective heat transfer, etc.

B.6-4
B. 7 Autocatalytic, Multiple, and Heterogeneous Reactions

The rate expression given in eq. B-22 Csee section B.3.) is the

simplest phenomological representation of the catalysis of a reaction by a

reaction product (see [58]). The reactions is self-accelerating under

isothermal conditions but cannot explode as in the case of some branched

chain reactions (see [12], chapter 1). must be of the order of 0.1 or

less for the reaction be considered autocatalytic"*"; for chain reactions

-4 -6

described by this macroscopic decomposition expression 10 to 10

[59]. Both Gray and Lee (see [13], p. 151-167) and Merzhanov et al (see

[14], p. 284) have reviewed the earlier analysis of the heat balance and

rate expressions for this form of f(r))*

Mielentiev and Todes [60] carried out an analysis of the problem

using a method involving the integral curve (i.e. ri vs. 0) that yields

results identical to that of a Semenov diagram analysis (see section B.3).

1/ 5

' = e(l + n„^)/4; « 1

c 00

9=1 c

|-j^ [ The method assumes the temperature dependence of can be ignored,

which has been proved by Todes [61] (see [59], p. 1063). Merzhanov and

"*"lf Hq is 1, the reaction is effectively zero order from the point of view

of a themal explosion criterion.

Gray and Lee [13] discuss this in detail. It is simpler to understand

by using the method for a first order reaction. The locus of d0/dt = 0

is 1 - f) (Q/(S ’ )exp(-9) . A plot of this locus shows it intersects n = 0

at two values of 0 (one less than 1 and the other greater) if 6

’ < 1/e.

No explosion can occur. If 6’ > 1/e the locus cannot intersect ri = 0 and

explosion will occur. The locus is tangent to ri = 0 at 6 ' = 1/e, 9 = 1,

the critical conditions.

^ j- I Hq = (A^/A2 )exp[-(E]_ - E 2 )/RT] where A]^ and E]_ are the preexponential

factor and activation energy of the first order reaction forming the

catalytic product, respectively. A2 and E 2 apply to the bimolecular

reaction of this product with the original material.
Dubovitskii [59, 62] have developed a more detailed and physically more

useful interpretation, eq. B-14 and eq. B-15a, using the exponential approxi1 mation. They have shown that if 5 is in the range between l/(eri^) and

2

’ 4/[e(l + ri^ )] (i.e. 6 > but is not so large that the initial reaction

rate meets the explosion criteria) the system slowly self heats to 0 = 1

prior to explosion. This slow self-heating can be described to a good

approximation by setting d0/dt equal to zero in eq. B-15a. Acccording

to Merzhanov [14], the equations involving heat conduction in the material

for the infinite slab, infinite cylinder, and sphere have been integrated

by Barzykin et al [63] for autocatalytic kinetics for various values of

the Biot numbers. Ref. [64] also deals with the cylinder, Bi = “) . The

work of Barzykin et al [63] is cited by Merzhanov et al [14] to confirm

the fact that eq. B-14 and eq. B-15a may be used with a modification

developed by Kuydaev [65] to calculate induction times for these more

complex models.

The application of B-14 and B-15a with 3-22 to heterogeneous systems

(explosives that evolve gases) has been made by Dubovitskii et al. [66,

67] and Merzhanov et al [68]. In ref. [66, 67] the increase in decomposition rate due to a linear decrease in volume in the initial material (due

to gas evolution) and the autocatalytic acceleration of a reaction due to

the formation of a eutectic"*" of the condensed reaction products with the

initial material are examined. In ref [68] an attempt is made to take

f into account the increase in heat transfer coefficient, h , due to gas

evolution.

"*"The phenomena is associated with the idea that the decomposition rate is

accelerated by dissolution of the initial material in the product. Rates

of decomposition are higher in solutions than solids. Evidently in some

cases, if the gaseous products are allowed to escape, no autocatalysis

occurs (see [50], for example).

B.7-2
A general method of steady state analysis (i.e. f(n) in rate expressions

is held constant as in section B.3) has been used to determine critical

conditions for chain- thermal explosions, periodic cool flames, and two

stage ignitions [69-75]. While this work deals with reactions in gases,

the technique is generally applicable to multiple step reactions (the

general situation of interest here) though it requires the temperature in

the sample be uniform. Moreover, the technique is relatively simple

mathematically to apply (see [19], p. 277 and [76]).

Five relatively recent papers have been found that deal with the

eff.ect of reactant consumption on the rate expressions for multistep or

heterogeneous reactions. Bowes [77] has applied the 0,n inflection

criteria to two independent exothermic reactions, one zero order and the

other first order autocatalytic (i.e. eq. B-22) . The results seem to have

been applied with considerable success [78] to the prediction of ignition

of wood sawdust-vegetable oil mixtures for both small samples and large

samples. Walker et al [79] claim to have developed an equation to predict

ignition temperatures for porous solids undergoing slow heterogeneous

(gaseous) oxidation (in accordance with a specified rate law) that apparently

corrects for consumption of reactant without going through a non-stationary

analysis (i.e. as in section B.5). Two numerical non-stationary analy'ses

dealing with heterogeneous multistep reactions have been located [80, 81]

but have not been studied. '

B.7-3

B.8 Thermal Explosions in Linear Heating

The earlier work, on the subject, ref. [82-85] has been reviewed by

Merzhanov et al (see [14], p. 285-286). A review by Barzykin [88] includes some later work, ref. [86-87]. An assessment of the results in

terms of procedures and assumptions used in thermal analysis has been

made by Barzykin et al [89]. These three reviews are the basis for this

section.

Linear heating differs from the problems discussed in sections B.4.

to B.7. in that the uniform ambient temperature, T increases linearly

P

with time at a constant rate w from some initial value, T. , where k_

in T

.

in

is negligible. In sections B.4. to B.7., the time dependent problem is

for the case where T is constant and then is step-jumped (i.e. increased

P

at an infinite value of w) to a constant value T . In both cases, the

o temperature of the sample is assumed to be uniform and equal to T^^

before the time origin.

The description of the thermal explosion process proceeds in the

same manner as that described in section B.l except that now the sequence

of explosion temperature experiments with progressively larger values of

T^ is replaced by a sequence of scanning experiments in which w is

progressively increased (i.e. but other parameters are held constant).

For very small values of w it is found that T-T , where T is the mean

P

sample temperature, goes from an initial value of zero to a constant

negative value, AT , which does not change with time until T^ is near

the critical bath temperature, T^, in the explosion temperature test.

At T T. , T-T increases, from AT^ to a small positive value. After

P . P

the. sample has decomposed, , T-T^ returns to a negative value. The maximum

of T~Tp is the order of what is called the Semenov warmup in the step

B.8-1
2

jump problem (i.e. RT^ /E of eq. B-6 where is now T^) . It is assumed

that the sample container is closed throughout the experiment. For an

experiment with w above some critical value, w^, the value of T-T^

rises abruptly near to a maximum that can be much larger than the

Semenov warmup. A thermal explosion has occurred, w plays essentially

the same role in a scanning experiment that does in the explosion

temperature test.

There is, however, a basic difference (see [88], p. 32) between

development of a thermal explosion in linear and step-jump heating. In

step-jump heating, the explosion develops because of an inbalance between

heat generation and heat loss. The effect of reactant consumption is

merely to add a quantitative correction to the critical condition for

this inbalance. In the case of linear heating, thermal explosion occurs

when the reactant consumption cannot decrease f(r|) at a rate sufficient

to compensate for the increase in the rate of heat generation due to the

linear increase in the environment temperature . In this sense, heat

transfer plays a secondary role.

* An alternative explanation is as follows. If w is made small, a

*

P

P

large amount of material is decomposed prior to the time T reaches T.

P

*

f Thus, 6 of eq. B-48 is less than 1/e because f(ri ) is very much less - c c than its maximum value, f(n ), which is used to calculate T.. Thus, no

ni X ’ thermal explosion occurs at T^. However, a thermal explosion could

occur at some higher value dictated by equations B-47 and B-48. If

sample decomposition is always sufficiently large enough so that the

calculated thermal explosion temperature exceeds the actual material

temperature, then no thermal explosion occurs.

I

p|

II

II

II
Numerical solutions in the tefetences reviewed here are for the

isothermal equations, eqs. B-14 and B-15a, and the unidimensional conduction

equations, eqs. B-59 through B-61c. The reaction mechanisms are for a

first order and simple autocatalytic form of f(r|) (l.e. eqs. B-21 for

n = 1 and B-22) . 0 in eqs. B-14, B-15a, and B-59 to B-61c is defined in

terms of T. which is the critical value of T of sections B.4. through B.7. “ o That is

0 =

RT. T

(t-t*)

Time is replaced by the variable 0^ defined by

RT. ^(Ip-T.): Tp - + wt

This yields for eqs. B-14, B-15a, for example

ten) -^(6-6p)

P

*

B-62

B-63

where

n = Cw cw

Q Aexp(-E/RT.) QAexp(-E/RT.

)

RT*

E

1 + 3

ef (ti^ exp(-E/RT.)

h RT

RT

*

2

* C = ® q

' "‘‘‘q

I The same variables appear in eqs. B-58 to B-61c except that C, Q , and

t 6* are replaced there by c, Q, and 6*. 6* is given by

X _ 1 *1 „ w B1 , QY^ A E ,-E . ^ - If(v <1 + V(BrT7[> - X

B-64

B-65

B-66

B-67

B.8-3
Shortly after Tp starts Increasing, the material goes into what is

called a quasistationary state (known in heat transfer literature as the

quasi-steady state) in which the rate of change of temperature everywhere

in the material is equal to w. In this state, the lag of the surface

temperature (i.e. for the infinite slab, infinite cylinder, and sphere)

behind the surface temperature, are given by eqs. B-68. These

equations apply only when dfj/dt is negligible.

In eqs. B-68, K is the thermal diffusivlty. It is evident that since

equals 2/Bi that if Bi is 20, 90% of the total temperature

lag occurs within the material while if Bi is 0.2, 90% of the total

temperature lag is between the surface of the material and its surroundings.

Stic dimension r while AT ^ increases linearly with r. AT ^ is independent of X or, equivalently, the heat transfer coefficients inside

the material, independent of a^. Finally, it is useful to note

that the time, Tqg» required for the temperature lags to reach 99% of the

values given by eq. B-68 is independent of w and is given by

AT, ^ ^ wyV ^

^2

ext a S BiKS BiK

B-68a

o

AT

wC wrV

2KS

2 wr g 2 B-68b

int XS/ (r/2) 2K

It is useful to note that increases as the square of the characteriT qs

B-68c

B.8-4
As soon as heat generation becomes appreciable, the material still

remains in a approximately quasistationary state except during a thermal

explosion (if it occurs) . That is for most of the scanning experiment

d9/d0p is very close to one. Numerical solution of eqs. B-62 and B-63,

or the equations corresponding to eqs. B-58 to B-61c show that the maximum

9-0 , A0 , depends primarily on ^ and y. and is insensitive to the

p max

magnitude of Bi"*". Curves of A0 versus have the same S shape and

? variation with increasing y. shown by 0 vs 5 in fig. 3 of section

* m

Using a criterion for a thermal explosion analogous to that used by

Tyler and Wesley [39], discussed in section B.5., one finds that Q must

exceed a critical value that is insensitive to Bi and to a first

c approximation is linearly proportional to y^. The proportionality constant,

g^, depends upon the type of mechanism and varies with (as well y^ and

for an autocatalytic reaction) . For example, the numerical results

for a first order reaction can be represented by

- gjY*; gj = 5.23 [1 - 23* + 139 (B*)^][l + lOy*]

For y less than 0.01, g_ is constant to within 10%. This value of Q can

J c be predicted within 50% or better by solving eqs. B-62 and B-63 with

d9/d0^ and 6^ set equal to one and zero, respectively. The resulting

value of ri from eq. B-63 is put into eq. B-62 and the value of 9 that

makes d(0-9p)/d0p a maximum is determined . is determined from the

condition that the corresponding value of 9 be a maximum (i.e. d0 / d^O) . P P

Apparently, by contrast Qdri/d0 , whether a local or a volume average

value, is sensitive to Bi. ^

In other words, thermal explosions become degenerate or difficult to

detect when y^ is large (e.g. y^ ^ 0.1).

At present, we are unable to justify this step adequately on physical

or analytical grounds. It gives good semiquanitative results.

B.8-5
Assuming is strictly proportional to one can derive the form

of as follows. From eqs. B-64, B-66, and B-68 one has

0, = wC/h

5*RT* /E

(AT. ^ + AT ) int ext

RT*^/E

B-69

Replacing by g^Y*, one has «

,2

"c

= ^

To eliminate T^, one notes that varies slowly with T^^ so:

”c

' ^*«3 ^

1/B* - 2n

h «*

)

B-70

B-71

B-72

The dependence of h on Bi can be written as

f h = a^S/[Bi/g^ + 1] B-73

Because the material remains to a first approximation in the quasistationary

state throughout the scanning experiment if w ^ w^, it seems reasonable

to propose that the effect of a thin metallic container similar in shape

to that of the material will be to alter w^ by changing c to the heat

f capacity of the material plus container and chainging h to

I h = a S/[Bi/2 + a S/a S +1] B-74

o o e e

is the external heat transfer coefficient per unit area of a container

of total area S^. (Abstracts of the articles indicate that the effect

of the sample container has been studied by Grigor’ev et al [90,91] at

least in the step jump case of appendices B.l. through B.7.)
B. 9 Experiments on Gaseous Reactions

The most common experimental method is to admit reactant gases to a

thermostatted pyrex sphere fitted with a pressure transducer and containing

one (usually movable) or more fine (y .040 mm diam. ) Pt-PtRh thermocouples.

The couples are coated, if required by the chemistry, with silica and

arranged to measure temperatures at positions along a diameter of the

sphere (to verify thermal explosion theory; at one position to detect the

onset of a thermal explosion) . Gas at varying pressures is admitted to

the vessel at a fixed temperature to determine the critical pressure, P^,

for- explosion. Data are analyzed using eq. B-3l"^ for simple order reactions.

Induction times to explosion are viewed as being unsuited to exact analysis

[94c]. Precise control to maximize reproduceability of experiments is of

key concern. The use of diluents, particularly gases having small heat

capacities, to determine the order of reaction from the variation of the

critical pressure with thermal conductivity is unreliable. This has been

shown [94a,c] to be due to the compressional heating of the inert gas which

can lead to a transient temperature rise above that of the reaction

vessel of 200®Cl Reliable difference calibration of thermocouples can

be carried out in place from the temperature change caused by expansion

of nitrogen gas (i.e. assuming the expansion is adiabatic and reversible)

[94a]. Thermal conductivities of reactant gases can be calculated,

apparently with reasonable accuracy, from thermal diffusivities measured

during the warmup of the gas after an adiabatic expansion [95].

Noting that A Z(P*/RT*)^, a plot of Zn(P^^/T^ ) vs 1/T^ has a slope

of -(E/R)n. Also, (P^^)^ at constant T^ is proportional to A/r^.

The dependence on A has problems (see above) . The course of the reaction

is either followed optically or with the pressure transducer.

The heating effect decreases as the gas heat capacity increases, increases

as the pressure increases up to some maximum and then stays constant,

and in the case of some gases is accompanied by an initial cooling effect.

The transient rise decays in about one second or less for vessels of five

cm diameter.

B.9-1
The various predictions of thermal explosion theory involved in the

method have been verified in detail for simple order reactions when

convective heat transfer is excluded. See ref. [94c], in particular. An

anomalous critical temperature distribution observed in the early stages

of H^/Cl^/NOCl decomposition is attributed to a shift from a heterogeneous

to homogenous initiation step in the reaction mechanism [92] . The supercritical temperature distribution corresponding to the ignition regime in

methyl nitrate [93] appears to be anomalous, however, since a temperature

distribution corresponding to the normal explosion regime is not observed.

Raleigh numbers less than 600 are required for convective heat transfer

to have a negligible effect on pre-explosion temperature distributions

[92].

Recent work in technique development [96, 97] has shifted to development of a stirred gas vessel to take advantage of the simplifications in

analysis when there is a uniform temperature distribution in the bulk of

the gas. The aim of this work is indirect measurement of heat evolution

(i.e. product of temperature difference between gas and container times

an empirical heat transfer coefficient)

B.9-2
B.IO Experimental Work on Explosi-<res in the U.S.S.R., to 1970

The kinetics of decomposition of Tetryl (N-methyl-N, 2,4,6 tetranitroaniline, m.p. 130C) and DINA (dinitroxy diethyl nitramine, m.p. 52.5 C)

were determined by measuring heat evolution, weight loss, and for DINA

also by pressure increase. The experiments were carried out at various

constant temperatures in the ranges 130-155 C for Tetryl [98] and 150-170

C for DINA [99]. The mass to volume (available to product gas) ratio, m/V,

was the same in both methods for Tetryl because of the known variation of

the rate constants due to product catalysis"^ (see [102, 103] and later

discussion) . The measurements were made under conditions as nearly

isothermal as possible; heat evolution measurements were evidently carried

out in a differential calorimeter . The results of the various methods

agreed within their combined experimental errors.

The validity of the applicability of thermal explosion theory to

DINA [100, 101] was verified by the agreement of the measured and the

calculated critical ambient temperatures, T^, critical temperature increments

and induction times to explosion. Thermocouple measurements of the

The corresponding phenomenon for DINA (see [110]) was evidently unknown

then; it is probable that m/V was small for all methods.

++

The reference, identical for both DINA and Tetryl, was unavailable.

From ref. [98], the measured temperature difference between thermostat

and substance was kept less than 0.5 K by, apparently, keeping, sample

weights small enough (e.g. 200 mg) that heat evolution was less than 23

mW. It seems probable the instrument was not a DTA apparatus since the

electrical current strength was altered in steps to give the experimental

curve in the form of separate points. This procedure alone would certainly have been insufficient for DTA measurements on DINA.

Induction times to explosion for DINA were calculated by numerical

integration.

B.10-1
temperature onset of explosion were made oh 70 to 260 grams of sample

contained in cylindrical glass vessels whose outer surfaces were thermostatted by flowing liquid. The samples were kept isothermal by stirring

the liquid with a propeller (T^'^>150*C) . Measurements were made using five

different inner diameters, d, of vessels ranging from 3 to 6 cm with the

sample depths, il, equal to or double the diameter. The procedure evidently

follows the explosion temperature test described in ref. [8]. The calculated values are for zero order"*" kinetics and the isothermal model

parameters are expressed on a unit volume basis. Heat and kinetic^ data

* are from ref. [91]. The value of a was determined by electrical calibration below T^ and agrees within 10% with the value calculated from the

thermal conductivity and thickness of the glass vessel wall. Similar

agreement was obtained between measurements made on Tetryl [101] using a

ring stirrer (T^ 'V' 150 C) and calculations using quasistationary theory

with autocatalytic decomposition parameters from ref. [98].

Quanitative methods for studying thermal explosions in small amounts

of material, whether they are solid or liquid, should, according to ref.

[104] , avoid designs that aim at making heat transfer between the material

and its environment as large as possible. Small volumes under this condition mean high critical temperatures which can lead to foaming, swelling,

etc., that can restrict the applicability of the results. Larger sample

Ref. [91] indicates simple first order kinetics. However, the volume of

initial reactant decreases as l-pri where y is 0.95. Thus, in a cylinder,

h varies as 1-0.84 ri for S,/d equal to 2 while h /V varies much more

slowly as (1-0.84 f])/ (1-0. 95 n) . Division of q’ dri/dt by V yields Qk^ (1-n)/

(1-yri) • Thus, if n were 0.1 at explosion, then h and f(n) have decreased by 8.4% and 10%, respectively, while h /V and f(ri)/V have increased

by 1% and 0.6%, respectively.

++ ' ' a = h /S or the net heat transfer coefficient per unit area of sample.

B.10-2
volumes reduce the critical temperature but increase errors caused by the

increased (generally unknown) temperature gradients in the material. Both

critical temperatures and temperature gradients in the sample are reduced

by the opposite procedure -reducing heat transfer by introduction of an

insulating layer"*" between the sample and a thermostat. Biot numbers of

one or less for solids apparently can be achieved for one to five gram

samples with cylindrical glass Dewar vessels whose inner diameters (i.e.

of the reaction space) are 1 to 2 cm, and outer diameters (i.e. inner

-4

face of thermostat), are twice as large as the inner (ct 1. 3-2.1 x 10

-1 -2

WK cm ). Sample depths are of the order of four inner diameters.

Validation of the approach was carried out by demonstrating agreement between measurements and calculations for DINA and Tetryl [104] as

in ref. [101] as well as measurements and calculations of a ntimber of

induction times under supercritical conditions for Tetryl. Thermocouple

temperature difference measurements showed a maximum internal difference

in the material of 1 K for a self-heating increment (i.e. T -T.) of 10 K.

c *

t h is determined for each sample depth since heat exchange parallel to

the vessel walls is not negligible.

A standard size reaction vessel for one gram samples was developed

for critical temperature (and induction period) measurements for absolute

ranking of thermal instability [104]. Absolute in this context means

- I that h of eq. (B-9) is kept at the same known value for all materials

This is apparently not common practice in thermography; it corresponds

to the analysis of the possible errors in our own experimental work.

Of necessity, however, the layer must not have a long thermal time constant.

+4

*Values of E/R can be derived from critical temperature measurements if

they are made for different values of h -differing a or S/V with due

allowance for end effects. From eq. B-9, E/R is the slope of a plot of

£nT^h /V vs 1/T^. QA can be determined from the intercept of the plot if

is known (i.e. order, mechanism etc.),

c

B.10-3
and abnormal swelling or foaming of the material has been eliminated.

Consequently, the results can be used as a part of calculations for

different conditions. Values of and critical induction times for

seven explosives are cited and compared with flash point temperature'*’

rankings.

This type of reaction vessel was shown to be suitable for use in

single calorimeter scanning (i.e. linear heating or cooling with the

thermostat) as well as static (i.e. constant temperature) experiments.

This was demonstrated by the agreement between kinetic parameters obtained

in the two types of experiment for the decomposition of Tetryl [106] and

the polymerization of styrene [106, 108]. Maximum temperature differences

in the materials during scanning experiments were 0.2 K for Tetryl and

0.3 K for styrene. The measurement procedure is based on eq. B-3 written

in the forms

q'n = C[T(t)-T(t^)] + /^h’(T-T^)dt B-76

In these equations, C is the sun of the heat capacity of the material,

which varies with n, and that of the adjacent wall of the reaction container.

Besides temperature measurements of T-T^ and T^, auxiliary measurements

are required to determine C and h and their dependence on n and/or T.

This appears to be some variant of the explosion temperature test discussed in ref. [8].

4+

' ' h is ordinarily measured as described in ref. [104]. Variations of h

with n and T occur though S and a, respectively. In ref. [108], a

for a glass Dewar. For cylindrical vessels h /V will be nearly independent

of ri if the decrease in sample volume, u, is small (e.g. for Tetryl, y is

0.24) and, S,/d is large. If y is large, a metal reaction vessel is introduced to make h’ constant and keep the time constant of the ’’insulator"

small throughout the reaction. This procedure works to the extent that

heat exchange parallel to (i.e. within) the walls of the Dewar flask is

negligible and to the extent that a a << S/S ,where a is the heat

transfer coefficient between the thermostat and a unit area of the metal

tube having a surface S . e

B.10-4
The variation in C with n is neglected when its contribution to eq. B-7

I is small. The variation in h can be eliminated if the reaction vessel

wall is made of metal (i.e. large thermal conductivity) rather than

glass. Measurements at various scanning rates, w, or constant temperatures are analyzed by setting up plots of dri/dt versus r| for various

+

temperatures .

It is of interest to note that linear heating is recommended for

decompositions with large initial rates while a program of linear cooling

followed by constant temperature is recommended for strong self-accelerating

reactions. Dilution of the sample by an inert solvent, such as alumina

powder as a diluent for pyroxylin [112], can be used to increase y (and

thus degenerate the thermal explosion) to extend the upper temperature

limit of static experiments. The same apparent effect is produced by a

sample container of large heat capacity as mentioned in ref. [109].

This does not achieve an identical result since y for a unit volume of

sample remains unchanged (unless heat transfer between container and

sample is much greater than between container and thermostat).

Apparently an appreciable number of explosives decompose by an

autocatalytic mechanism over fairly large ranges of n and the degree of

autocatalysis increases as the fractions of gaseous products in the

reactant is increased. Examples are tetryl [102, 103], pyroxylin [105],

and flour [109].

is determined from plots of dr|/dt versus 1/T at constant r).

B.10-5
DINA changes kinetics from first order. when no gaseous products

remain in the reactant to an autocatalytic type of mechanism when product

gases are not allowed to escape [110] . A 50 percent increase in Q and a

100 fold increase in decomposition rate also occur. Measurements [111]

first order reaction, give values of A and E identical to earlier work

[99], apparently when autocatalysis is minimal, and a value of E at

Confusing these observations is the fact that an analysis of critical

temperature data at 1 atm in references [110], [104], and [100] by a

used in calculations of in refs. [104] and [100]. One can show that

calculated from the first pair of E, A is only 3.7 K larger than that

from the second pair. Thus it may be that the only significant problem

is the disagreement between ref. [110] and the original data from ref.

[99]. An additional complexity in the case of DINA is the decrease in

the Semenov warmup as T is progressively raised above T. when external

pressures are 0.4 atm or less! This behaviour and the experimental

Semenov plot are consistent with the heat absorption effect of vaporization of reactant in product gas bubbles [113].

"^By virtue of y

= 0.95, measurements of T-T^ times aS/VQ yield k^.

Apparently T-T^ were maximal values corresponding to small n*

The isothermal experiments of ref. [110] were analyzed in terms of

first order reaction with E and A varying as a function of r).

DINA decomposition reaction at small ri*^, when treated as

I [ pressures above 1 atm identical to that in ref. [110] ' for small n*

2 ' -1

ilnT^ h /V vs 1/T^ plot yield an E and A of 35 kcal mol and 2.5 x

13 —1 *“1 *18 *1

10 sec instead of 45 kcal mol and 3.8 x 10 sec , respectively.
The experimental work on flour [109] demonstrates the need for

studying heat evolution kinetics for various types of mass exchange to

establish the gross features of decomposition mechanisms.

The dependence of 5^ on the Rayleigh number, Ra, needed to account

for the effect of free convection in liquids on critical conditions, has

been established experimentally for plane parallel and cylindrical geometries [114]. This was done by comparing calculated and measured values

and benzyl alcohol at large Bi. It is found, for reactions of integral

order and these geometries, that 6^ is given by eq. B-77 when Bi is large

or equivalently, and that 6 in eq. B-31a should be divided by 4)(Ra)

The predicted effect of the addition an insulating layer of heat transfer

Hexamethylene diisocyanate. The decomposition proceeds without formation

of gas bubbles by a second order mechanism with E and A varying with the

solvent; the parameters were determined thermographically . In the formula

for Ra, AT is set equal to the measured critical Semenov warmup and the

viscosity at criticality is the measured value.

++

An alternative form of the terms on the right side preceding A is

^^^2^a(j)(Ra)

where a is defined in eq. B-25.

of critical temperatures for various known Ra with HDl"^ in octyl, heptyl.

after g^/Bi is set equal to zero.

6 c B-77

B-78

f(n^) = 1 - 2.7(ny) 2/3 B-79

coefficient per unit area, a^, outside the material on eq. B-31a would then

, -H- be ;

5 B-80

o

+,

o

B.10-7
The agreement of based on the Semenov model with measured values of

when Bi is small (i.e. 2.4-6) but Ra is large (10^<Ra<5 x 10^) is

consistent with eq. B-80 [114]. The applicability of eqs. B-77 to B-80

depends upon the induction time to thermal explosion being much less

than a characteristic time required to set up the free convection temperature

distribution. The applicability of the equations is also limited to

high Rayleigh numbers when decomposition products form gas bubbles

[115]. At small values of the Rayleigh number, free convection caused

by bubble formation dominates for DINA which results in 5^ passing

through a minimum as Ra increases.

B.10-8
B.ll Hazard Evaluation Experimental Work (Ref. 116-140)

Hazard tests are generally classified as sensitivity tests or damage

potential tests (see [123], section 2.; [124]; [126]; section 1, [128]

par. 2, 17; [129], section 2-2.2)'*’. The former assess how easily some

external stimulus may provoke an accident or, for self-reaction hazards,

initiatedecomposition. The latter assess the consequences or damage

potential of the accident.

The applied stimuli or initiating impulse in sensitivity tests can

I \ be classified as being mechanical, thermal , or an electric discharge.

Mechanical stimuli are impact, friction, and shock. Thermal stimuli can

be, for example, different constant ambient temperatures, ambient temperature that increases with time, exposure to a constant radiant heat flux,

exposure to flame, or exposure to a hot surface. Associated with each

classification of stimuli are the ideas of variations in strength or

magnitude of energy input degree to which the energy input is localized

spatially, and the time duration of energy input. Boyars [128], for

example, recommends usage of "susceptibility" in place of sensitivity and

use of the term "sensitivity tests" to refer to stimuli of short term

duration and "stability test" for stimuli of long tera duration.

The structure of the content of the cited references of this section

is consistent with the explosion research concept that, whatever the

type of stimulus, initiation of decomposition is ultimately thermal in

"*"The distinction is termed initiation and results of initiation in ref.

[129].

Radiant flux is included because in transport accidents it is thermal

in origin.

B.11-1
origin. (This is explicitly identified in ref. [130], for example, which

is concerned with fire and explosion hazard evaluation of materials in

general, not just materials designed to function as explosives.) In

other words, the result of any initiating impulse is to produce an elevated

temperature in a material. The magnitude and spatial distribution of

this elevated temperature and the length of time the material is exposed

to this temperature distribution would, in principle, be sufficient to

characterize the effect of the stimulus [17]. However, the detailed,

quantitative mechanisms of the degradation of the energy of the various

mechanical initiating stimuli into heat is, in general, xinknown.

Damage potential tests assess either directly or indirectly the

amount and rate of liberation of energy and gas per imit of mass of

decomposing material ([128], paragraph 2) and the possible dependence of

these parameters on the total mass of material. They are power tests.

Apparently, current practice aims at distinguishing between what are

called the three types of chemical explosions. In order of increasing

violence, these are usually called homogenous explosions, deflagrations,

and detonations. The phenomenology of the three types is discussed

clearly in ref. [122]. The distinction betwen the first two types'*", as

described, is less important than between the first two and the last type

from a practical hazards point of view.

A homogenous explosion corresponds to the spatially uniform thermal

spiral and associated decomposition events when the isothermal model of

thermal explosion theory applies to the material throughout the course of

"*"The distinction is not artificial in the design of thermal tests. For

example dilution thermography aims, by limiting the adiabatic temperature

rise, at producing a degenerate homogenous explosion.
the decomposition reaction (i.e. when conditions are supercritical).

This type of explosion would evidently apply, to a first approximation,

to exothermic homogeneous decompositions of liquids and gases that are

initiated by a uniform elevated ambient temperature and that are present

in bulk size (large Rayleigh numbers).

Deflagration is characterized by a distinct reaction zone which

propagates through the unreacted material during the course of the

decomposition. Propagation is by heat and mass transport; thus, the

propagation rate is less than the speed of sound (at most, tens of meter per

second) . It is evident that this type of explosion can be associated with the

thermal wave that ensures under supercritical conditions in the thermal explosion

theory model involving heat transfer by conduction when Bi and y are large

(see section B.2.). It would apply to this model for any value of Bi if a

supercirtical condition is produced by local heating rather than a uniform ambient

temperature (i.e. ignition regime). Deflagration initiated by an elevated

uniform ambient temperature would be associated with solids and heterogeneous

kinetics, in particular. In practice, hazard tests are primarily oriented towards

detecting pressure sensitive deflagrations and conditions, such as degree of

confinement, that might cause a transition to detonation or direct initiation

of detonation.

Detonation differs from deflagration in that the propagation rate is

supersonic (1-9 Jm/s). A shock wave leads the reaction front; the reaction

products are gaseous . Detonations can be initiated by the shock or fragments

produced by detonating neighbor charges or by spark, friction, etc.

Lack of a quantitative description of the decay of non-thermal stimuli

into heat requires that there be a sensitivity test for each stimulus.

In transporatlon accidents, these stimuli are usually considered [116] to be

collision, fire, and direct ignition of material spilled from broken

containers. From the preceding discussion, the sensitivity tests

must assess the ease with which these stimuli

B.11-3

can
initiate an explosion, deflagration, or detonation and the power tests

must assess the damage potential of these events once they are initiated.

A variety of thfe mechanical stimulus test in refs [116, 124, and 128] are

listed in Table B-4. Although not all of them were designed for testing for

transportation hazards, all can be explained in texrms of this scheme of accident

stimulus and type (i.e., sensitivity or power). The reader is cautioned that

the tests listed in Table B-4 are examples. Many of them are not included among

those permitted in the U.S.A. for testing for transportation hazards. Nor

do they match the United Nation Recommendations. Table B-4 shows that this

scheme converts an otherwise bewildering array of tests into a more

rational assembly. It should be noted that positive results (i.e.

material can detonate) in tests 4 to 7 are of significance even if

sympathetic detonation is not an issue (i.e. no multiple stores).

Positive results mean the possibility of detonation by transition from

deflagration must be investigated since this is the more likely course

of events in an accident ([122], section 6.2).

Table B-5 lists some of the thermal tests in the cited references

of the section in the same way used in table B-4. The list, as is the

case with table B-4, is not Intended to be complete. It does include,

however, many of the thermal tests used in combination with mechanical

tests in table B-4 to evaluate the self-reaction hazard of a material. For example,

tests 8 through 11, tests analogous to 12 and 13 (see later discussion),

and test 16 are used in combination with tests 1, 2, 5, 6, and 7 by RARDE**”

[124] to construct a laboratory scale picture of the hazard of the material.

Tests, 8, 10, 11, 14, 15 and a burning test (unlisted) in combination with

tests 1, 3, and 4 are recommended by Boyars [128] for evaluating the selfreaction hazard profile of organic peroxides. The results of these tests

are organized for purposes of regulation of transport, etc. so as to

indicate the degree of hazard of material and both if and how it is to be

shipped. One illustration of how such a categorization can be developed is

given in ref. [128] (see pp. 26-33 in particular). +

Royal Armament Research and Development Establishment; the official

organization concerned with hazard evaluation, in Great Britain.

B.11-4
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a o o o U 9 00 X 44 H a Z B. z a z aX •o fH 44 iH 00 44 44 44 a a a ^ 9 X a •H a • a a ‘H a o a cj a u z cd a oox •“4 <H fH O fH o o o H i4 iH fH >v VI o Vi 00 Vi 00 Vi 00 u «0 44 a a VI a a fH *44 44 *44 B *44 B *44 B Bi

ac

a > 3 o a

a

XB i4 •H •H o a X a X 44 X- 44 X 44 00

44

a 1-1 •B

X

a i4a B.V

auaua

fd

a

O Vi >4 44

a Vi

O o a o a

O

a *n

u

a Vi

a

a ^ aaX V4X iHX fHX fH • •He H •B > .2 i a > a o a X a X a X cn 0X B B fk X u O o 4 a fH 44 U V4

d)

B i4

B 14

a 0 a o a

X

Oa

O 44

B Vi

X

a

X

a

CM l4 o • i4

OO *44

X 44

a iH

a

a fH

aa iH 44 fH • o a. > O VI a Vi u a >%iH > fH K 0 iH <44 •B a 00 a X X 44 1 H s* a fH P< 44 Vi 44 9 Vi to >< ^ X K -B •X Vi a Vi 44 B a

•>

aa

a

B CH o OB

Q «n

O *44 0 a a CO 00 a CA O a a a fH iH 1 rH a o B m *x B a a 14 m X a a CO 4J • i4 CO 44 - a o • o u a a a o (d B Vi CJ B U u a iH Q «H iH 44 B z a H 9 (0 0 B . a a ^ 44 44 44 a a o sra

44 Bf P. •H

a 44 Bf

UX

a f>

a

a

a 44

X Vi

a

u e 44 fH

u

B

T iH

O

B •H

a o a z 3 a 44 cd ^ U 00 a -H *B B fH n o iH a fi •B 44 H a 9* ^

i4

a Vi

O fH 44 iH Vi

K

a 8 >% Vi

a >1

a iH

a

X

a 5 * o r

O lH • H

fH

M ^ 0*1

X

M

XO 44n S -Bn

Q *n CA *44

a

O • • • z CM cn in

X ri

•o

CO

CO

a o

iH 9 4J •H 44 44 cn U U U U X 9 Cd O a iH u 8 9* <1-1 a Vi o

•H 8 a *44

X

4J t/i •H M-l •H

a eV

•o

•HU

U

B

•tH

O

C

B

B 44 iH

4) o O

Oa 44 •H •H •H X a CL (0 a a 44 B >v <H i4 iH a o H rH f-H @f 44 fH rH F a 0 O O ^ *B a U u a

O

T3O

•o o

•o oT3 o

•o o•o

NOTES:

1,

No.,

number;

2,

S,

means

susceptibility

to

Initiation; D

means

damage

potential;

3,

hydrodynamic

or

detonatlve

strength

shock;

4,

see

references,

comments.

Table

B-4a.
Table B-4a References, Comments, Table B-4

References and Comments

[8], p. 50, [116]; [123], critique; [124] RARDE description;

[128], p. B5-10, explosion detection by noise; [134], explosion

detection by gas evolution; [129], force gauge on anvil.

[8], p. 57; [116]; [123] critique; [124] RARDE description.

[8], p. 57; [123]; [128], par. 48, p. B-3.

[8], p. 45; [128], par. 48 indicates D, par. 36, 37 indicates

S. As distinct from blasting caps, this test involves only pure

hydrodynamic shock.

[8], p. 60, test evaluation misleading-see Appendix A, this report,

and [116] (sect. 6.1.4), [124], [128]; see [123] for critique.

[116]; [123]; [124]. This text evaluates effect of containment in

determining response to shock.

[124]; see [123], section 3, for comments; [122], section 5.4.1

describes a similar test for ammonium nitrate solutions.
o a

a e a

a

u QD

u 9 0 a 44

aa 00

G JS Vd e 4J

u

a

aa ft 44 e oo 1 1 9 u 9 a o c a a o V a M CO ft c a a

•H

U

ha

h •H ft

M 4JW 44O ^s.

u

u ft 9 a

u

u 9 9 c o O..S

G o e CO

a «w <ua s VM

M

a 9 0 a

a t*t

U

a 44U o 44 e O

ua hi Oi 9 (0 o e > V a > CO CO ft 9 X 9 a 9 w 00 o. o a h u e a u «o4 0) a a CO .O tu

01

a CO c X ft

u

a

a A4

O ft 01 9 u 44 4J CO

a

a

O

u 9 •H

a

a 9 9 tj CO

a 9 44

V 9 a 9 9 Ita

a • A

a o O

a

a JZa 09

a 0) 44 44 Q>

a

M

M

O

Q

aG

Oo CD s e ft

a ft

a CO

h

a

W

a CO 44 9 ft O. ft 01

aa

O

a

09 e a CO o a cu Q.

Xa o e 0)X CO

U 9 uX tJ

u 44 c u ftb

0» 0) Q. ft

XX

a ft 0 o 4)

a 9 j9

a

C <44 CQ (0 'Ha 9 H a 44 9 <fat a a a •o u a *0 M CO 44 a a o > ft u u a

aoe Gou

a.e«

ca 3

O’

•o9

O’

41O, 4J . a w u a a

a

a «M ft «

”0 i-l QOf^ ft a w 4J o a a a ^ a w ft 10 9 0

tn > ^ > 9

O’

"O

ft

O ft o oa I w o f « a b

4J

UBV

a 00

C

a

a f>. c 0) U ft ^ b o e o S w irf ft 9 >. a

O.WS .a tJ 0 9 c

ft • • a

a

o.

a

a e a

a

a ^ a ^ a

u

a 9 fta ft

a

w *o <o

a 9 a •^9 o a a a G b a •o b •a b 1 a a

ft

aaa

a • e UM 3

t«t hi f ab

a

b 3 a

a o. o e a u a a j: a a O ft ft ft a ft

>N e a

a

a

abba ft

a .9

a o b

a

a fta tb JS

b

a >.

b

a

b

u

aa

a > 0..0 ft b S b ft e b a ft a •a b > b a o a lb 9 >« a a ft a a ft > b a a 9 9 a a ft > a •a > a a b o b b M ft o o ft a a a a b u b ft e a 9 u a b b •a a ft a a ft a a a e ft

4J

a 9 O

a 9 •a

b 9 *o

a ft

a 00 ft 09

a fta o a js

a

O

a o > >saa ft >» esi 0)

a

b

b ft

b

b ft

a fta ft Vf o ft Z Wl

H

a 9 b 9 - a

b

u ft

b

u - • a ft ft

a ft 9 a a • a a ft a a ft ft •a a ft a • CO "V 4J o > a. O b a > lb a > «b a a a b Q. u a m b a flu a a a .o ft a ft a a b a s ft . 0) a a X hi a X o. 9 b b a b b a b a a a b ft u N H a a hi a a b a O > a o a a b a b w cu 1 1 e a a a 01 0) 44 b A ft ft ft a a ft • e c O a 3 m e f a b u b a ft o 09 a 1 c ex 00 a a a e a b a ft a <44 c a

01

a ft ft

b

aa

a

b

a ft

au o 09 <44 ft •a

X o. <b cu

aa ft 3 b

b o 0)

U

U 09 ft

H >t f o a

a 3 o a Uta ft 44 o. 9 Ut

H CO

a

b utb ft - o b Q. 09 G0<^ e O

aa

O > <b

aX -o 09 CO « o

lA1 • a • s _ >s ft o o a

a

a

a e 09

aa

aa baa

b e a •«

u o o U

a

b 1 a o c ^ o ft o a e . • G Ut a 44 44 ^ CL • b a e CQ a a. ft <b b ft ft b a cu b a o 3 1 9 44 3 b o z b O <b ft b a a a ft O A 0) O 44 00 a •H

09 3' O. >. ^ ft

a o a

a o . e o w ^ C 09

a 00 44

b O.

A

b

a e e b

U J3

b ft cu

a 9 O. CQ 44a c CQ

a Ji 3 b 00

aa

a 3 > a 00 * CO 44

b ft

w 9 a b a c o e b •a b «n o o 44 CO 09 9 ^ cu e 00 H a a o cu b e a H "O u a GO 09 c a b 9 a ^4 ^ CO ft 00 • o f ut e > C a o 9 U <44 0 o ut 3 • H AJ > a M cn 0^ a O ft o« a •o s 44 O. 0 O .9 <44 • d # • en

00 9t ft ft

cu .X

O 1 1 z a 0 b a ut b b •a a e 1 00 a b a u 0 a a T3 o X O ft b cu a b a a ft

a 01

b ^ a

a ut

aa £ a 3 f b 3 b b b b e • • b a a ft ft 44 a • e 3 • 3 3 ft >» o o 3 a b a 3 u b a a a a a a a •a ft iH a ft o a 09 3 a ft o a o b a a a Ut Q. ft u a a XI a. a b cu e a 00 a 09 a a a b •H O ft a X ft o X o b c a 09 b ft >. 0

CO cu Ut

a

a Ut tsa

a

a fta 9> cu

b ^ u 1G 1 a 9 o *o a u a a U b

09

b

a f a 44 o a a

Q. 2 > o

>.

a

a

b ft

H

ab o a

ft

a

09 ft c a ft

a

a 09

a

W ftb ftb

b

Wb

o.

a "bO • ft •H ft

<44

a

b Ut

U ft ut Ut <44 ut

B.11-7

NOTE:

If

No.,

number, *

isa

test

that

determines

fundamental

parameters;

2,

S,

means

susceptibility

to

Initiation, D

means

damage

potential;

3,

Pressure-Volume-Temperature;

A,

see

reference

comments,

table

B5a,

and

text;

5,

self

accelerating

decomposition

test.
Table

B-5

(continued). .

Thermal

Hazard

Test

Survey

a

tu 3 1 o> o o 1 a •3 u u u o u 'O 4S

4J 01 OS

V ca •• 3 3 14

u to 0) (U

O > 43 00 i4 3 a C o a u T3 >

00

u o u •H « T3 CO 1—1 *

0) (Qu 4J 0) (U 0) cQ •r4 SO • • Wl 0) t4 0) to > a c Q. m 0)' o e >> CO a M <0 1-t a V r-4 4J 0.1-1 IS O. (U U « to O o

00 s o. o iH i.1 toC 00 to o u 4)

VVX U-> 0) •• 3 43 <u c (0 0) o 01

H 4J lU CM •o CO O.

O t4 6010 00

M 1 a 3

cQH

C % Vi 3 3 a ^Hf •• <d 3 CO 14 Wl o OO 44 iH* 00 3 3 c a .4 CO H •3 3 to a CO u (U C •H o a

44 •H

O c *H 14 Oi 3 •a •H O. 3 44 •H

•H rH j3 o M • D. vn iH 3 O. 3 U o O 44 •H 3 cu in iH 1—1 c 3 ir4 c 3 ca e to T^ 44 44 3 O. •H O 43 *rl 43 •H o ^ 3

t4 40 43

a

0>* 3 .H- 3 u

u to s CO

U 3 T3

O

a o 1—

3 0) ta 00 O 3 cd u a t4 <44 c 3 3 u

Oi •3

C o. 1^ VJ

C >4 6 o •H

a o •H •H

O o

iH •H tH

a 3 44 -n

O

a Vd CO Wl

a CO O. 3 44 o 3 .H 3 u & 0) 14 c 4^ 3 4<S u a 3*r4 u 14 44 s CO 3 4J 3 o c 14 CO <0 iH 3 0) u C C 3 0> 4= a 43 •rl 0 3 <44 U3 fH a •3 3 irl O Wl 4J 44 S 44 44 o a o o y -T3 3 3 3 W4 3 3 3 3

00 o 3 Wl 3 Wl c 44 o 3 O 3

<44 -H

H •H 44 44

U

O Wl

U Wl 3 >4 3 3 3 Wl 3 3 44

flO eo 3 O. > 3 >. > 3 a s 3 O. 3 O.

iH •r4

C 3 Oa 3 o a

a

U 44 3 44 3 a 3 >. 3 14 B 14 •H 3 1-4 .a eo 3 3 3

iJ 43 3 Wl W4 14 > • Wl 44

ca —4 3 3 3 B 3 3 B 3 Wl iH 43 3 3 3 iH 3 3 u 3 3 -o 3 O •H O -H

ca 1-4 44 3 3 0.43

M ' 0) 0.43 o 3 3 3 -H X a O O X a H o. a B -3 3 a <44 CO 3 3

01

0> <U i-t

m «k Vi Vi o.

M ' u 1 o.

a «k

a

a

01 ca

X

a eo 4J CO ij

a

a &

(U 3 3 A o B

a

a

aa

ij 3 u

u o 3 3 Vi

m Vi o H 3 iH * a o 3 • Wl a XJ a •c iH a > 3 CO 0) CM . 3 CO 3 C. a Ou Xi

Ok 3 eo -H O. eo <u 00 s a

a •H

M 3 3 u 4J a •k 0} s U4 'O CO a u a a a O a O >i o u a U •H O M u XJ a 3 O. ca 3 43 M ca a 3 44 X a OJ C a .n z Wl )-• a <u M 3 3 o a fM o o 3 B o H a a 3 O u M CO a •M a •H M u O. a ca O CO a 4= B CO a a o XJ O 3 a (U -4 tn a 1 3 CL <u •k •M c •H a 3 14 u •a 3 u H •H O. VM 1—1 s u CJ U a H 3 0) O a Q c a rH O u c CA •H S Vl N 3 JZ * 1—1 0) < •H Oi 0) •14 o a 0) •H Vl a •H “ H H V3 0.4= CO a u CO > «M :s u a CJ > a o • <r m M3 z a 1

iM rH rH o a O 1 a

Xi XJ

a

aa Vl

a iH fM *H

a

a

aa M-la -a •a Xi

a

M 1 a

u

a • • fM a a 3 u c u m a a a CJ ^ a eo a a a f-H rH a Q a Vl M <J u > •H o.

CL #-» 3 a *H 3 a Xi

a

a

a

Xi

X 3 O

a Vl 4= fM

B

a

aa

14 CO 3 O.

aa Xi 3 a

a

u

a

a e

01u<

0»a. ouuO (So

0» -H

Wi

a

<4.1 f-t o u C (0 ua oi (Q JS

m fH r4 a 3 a 3 a 3

B.11-8
Table B-5a References and Coimnents for Table B-5

Test No.

8 .

9.

10 .

11 .

12 .

13.

14.

15.

16.

17.

References and Conments

[116]; [117]; [124]; [128], par. 21, 23 ASTM Tests D92-66,

D93-71, and D1310-67

[116]; [124]; [128], par. 48.

[116]; [116] [124]; [128], par. 17, 48 [133].

; [117]; [124]; [128] par. 17, 24, 48; [133].

[8], p. 35; [122].

[ 122 ].

[8], p. 25; [128] par. 35, 48, [129], sect. 2-6; [135]; [136].

[117]

; [128], par. 21, 48.

[116]; [124].

[8] p. 37; [119]; [120]; [126].

B.11-9
Because of the importance of thermal tests in establishing and

developing thermal instability, the following additional commentary on

table B-5 seems relevant. Tests 10 and 11 illustrate the attempt to

duplicate actual accident conditions in a laboratory test. Evidently, the

tests assess the possibility that a violent deflagration or detonation can

be set up by confinement and destroy the sample container. In other words,

it seems clear that neither of these tests are a simple damage potential

test, as Noller et al [117] uses the PVT test. Rather they are a combination of a power and sensitivity test as Boyars (see [128], par. 19)

suggests.

Tests 12 and 13 may be viewed as the Dutch (RVO/TNO) version of the

second generation of tests 10 and 11. Tests 12 and 13 attempt to distinguish between types of chemical explosions, which test 10 or 11 cannot

do, and test 13, in particular, attempts to quantify conditions that may

lead to violent deflagration and possibly detonation. Tests 12 and 13 have

aspects that are characteristic of a third type of hazard test (which can

be either a sensitivity test, damage potential test or both) which is

called a fundamental parameter test [130]. The distinction is that the

physical properties measured in this type of test are used to calculate,

for example, sensitivity or power for any hazard situation. It assumes one

knows the engineering formula for sensitivity or power. In this particular

case, then, the results can be used to evaluate the possibility that

decomposition will burst any vessel rather than a 'specific container as in

test 10 or 11.

"*'The U.K. (RARDE) versions of test 12 and 13 seem to be the Time/Pressure

[116, 124] and Time/Temperature/Pressure [124] tests. The test stimulus in

the Time/Pressure test is ignition of the material via a match head which

ref. [124] equates with an accident stimulus of hot spot ignition rather

than the equivalent of fire engulfment as in test 12. However, both the

Time/Pressure test and test 12 are damage potential tests so the difference

will be unimportant for many cases.

B. 11-10
Test 14, the thermal surge test, can be viewed as a pulse calorimetric

(see [137]) version of the Koenen Steel Tube Test with the vent in the

latter omitted. Heating of the sample container (syringe tubing with an

O.D. and I.D. of the order of 0.36 mm and 0.18 mm, respectively) by an

electric current pulse produces high temperatures in times that are apparently assumed to be short enough to neglect heat loss by the container to

its external environment. However, heating of the material is not adiabatic.

For large B, the relation between the time to explosion, and' the

container temperature, T^, should be approximately that of appendix B.4. or

the numerical results of Zinn and Mader [33]. This necessarily means that

the slope of a plot of iln versus 1/T^ is not equal to E/R. Wenograd,

in his original work on the thermal surge test [138], incorrectly states

the opposite (i.e. the slope is E/R) but observes that the values of E so

obtained are smaller by half or more than the values of E obtained in other

methods. Zinn and Rodgers [139] point out that the relation between t

and T^ in either the thermal surge test or the explosion temperature test

(see [8], p. 27 or [140]) is dictated by the condition for rupture of the

sample container. Zinn and Rodgers also show that the experimental results

of the explosion temperature test (i.e. results of ref. [33] and [139])

and the thermal surge test (i.e. results of ref. [138] on TNT, PETN, and

tetryl) are consistent with the relation between t and T predicted by

Zinn and Mader. The values of E are in reasonable agreement with those

obtained by other techniques. Neither Kendall and Rosen [135] nor Janswoude

and Pasman [136] mention this second result. Janswoude and Pasman cite

(i.e. no derivation given) the following expression for t :

Strictly, in the adiabatic case a plot of £n (t /T ) versus 1/T would

have a slope of E/R. ex o o
t = -T iln(l - T /t ) ex q

00 q

This result is not consistent with the analysis given in section B.4 or

the numerical results of Zinn and Mader and, in our opinion, seems to be

in error. Various other possible problems associated with the thermal

surge test are discussed by Janswoude and Pasman"*" [136]. It seems evident

that the potential of the thermal surge test to be a fundamental parameter

test has yet to be confirmed in the sense of verifying and/or extending

Zinn and Rodgers second conclusion.

The SADT test, number 15, (developed . for organic peroxides) detects

self-heating by onset of an increase in the temperature of the

material (i.e. up to 140 kg) in its actual shipping container relative to

ambient temperature. As such, the SADT temperature, which may be less

than the critical temperature for thermal explosion for the system , has

the undesirable attribute that it is directly dependent on the signal to

noise ratio of the output of the thermocouple-amplifier detection system.

This is not true for a critical temperature test. The SADT test illustrates the following points of general signficance. Since the test is made

directly on the material in its container, no scale-up of the results

from laboratory sizes is required. The effect of the container is also

automatically evaluated. As Clancey [123] points out, the role of the

container should, in general, prove to be as decisive for borderline

hazardous materials as for explosives. (Test results for the effect on

the hazard potential of a material due to a difference in the type of

commercial container is given in the applications section of ref. [124]

and also in ref. [117], p. 25).

Effects associated abnormally high critical temperatures of the material

(or model distortions) in this test are mentioned in ref. [136]. See

also section B.IO. The test was originally deliberately devised to

produce high temperatures (see [138]) to examine the hot spot explanation

of initiation of explosion by impact.

B. 11-12
Elimination and/or evaluation of the scaling problem does not

eliminate the other major obstacle in relating laboratory tests to

accidents. This bottleneck is the characterization of the mechanical or

thermal stresses (and other factors) that may be encountered in an accident.

Evidently there is no quantitative description of stress available [123]

suitable for transportation conditions.

Test 16 was listed, despite lack of information on details, to

serve as link between test 15 and test 17, which can be viewed as a

laboratory scale, fundamental parameter version of test 15. Test 17

illustrates an experimental approach to evaluation of the effect of size

on heat evolution kinetics. The test rationale is the extension of the

thermal instability test developed by Russian investigators to replace

"flash point" measurements discussed in the text and a footnote of

section B.IO.

In test 17, developed by Bowes, the procedure is to measure the

critical ambient temperature, T^, as a function of various amounts of

material in vessels of different size but the same geometry. The critical temperatures are correlated using eqs. B-30 and B-31 written in the

form

5^ is calculated for the particular geometry, a circular cylinder of

radius r in refs. [119, 120], using measured values of the sample dimensions and heat transfer characteristics of the material in the initial

stages of decomposition and of the container. E is an apparent activation

energy since it will include the r and T dependence of 5 through the

Rayleigh number (i.e. (()CRa.) of eq. B-77) and mixing of the material by
gas bubbles (see end of section B.IO). is a constant which would have

the form indicated in the absence of free convection or mixing by gas

bubbles. is the correction for reactant consumption prior to

explosion. Separate measurements of E and A, when free convection and

the effects of mixing are small, are apparently recommended to obtain

some feel for possible limitations in using the extrapolation equation

for other material sizes, geometries, or heat transfer conditions.

There are a number points about Bowes' work on test 17 that deserve

clarification because of its importance.

(1) Because measurements were made on benzoyl peroxide, the value

of Bi, as defined in eq. B-31a, is effectively infinite. Only succinic

peroxide, of the organic peroxides, (see [117]) shows appreciable decomposition in the solid phase before melting. Under these conditions the

dependence of 5^ on Biot number would still be given by eq. B-30 provided

Bi is redefined with replaced by a^, the heat transfer coefficient per

unit area of the sample container with respect to the external world"^.

Values of comparable to those encountered in real packaging are produced by keeping small. This avoids so called "model distortions"

mentioned in section B.IO and emphasizes the fact that avoiding these

distortions is of practical, as well as experimental importance.

(2) The value of 6^ for a circular cylinder can be evaluated with

sufficient accuracy for the purposes of the extrapolation equation by

writing eq. B-30 in terms of defined in eq. B-17 using equations B-23

to B-25 as: „

5 c Kr

ex

q

"*"This can be derived by multiplying numerator and denominator of 5^, in the

footnote at the end of section B.3. by and assuming and (t^Xg^)/

(rX^g^) are small in comparison to one.

° ®

B. 11-14
K is the thermal diffusivity of the material and is the correction

factor for the error in asstaming that is the reciprocal of the smallest

eigenvalue of eq. B-58 with the boundary conditions appropriate to the

situation indicated in comment (1) . In this case one deduces from heat

transfer texts that is given by

r^K/x^ = + (yD/L)^

where y and y are the smallest values that satisfy

I f yJ^(y) = Bi Jq(u); Bi = a^r/\

f y tan y

= Bi D/L

D is the diameter of the cylinder and L is the height of material in the

cylinder and is a Bessel function of the first kind, g^ can be estimated

as the ratio of the known value of 5 for infinite Bi and zero D/L divided

c by the predicted value of 5^ (i.e. 2.00/2.13). An alternative would be

2 2

to use eq. B-30 directly with replaced by g^[y + (Ilr/L) ] where y

is the smallest root of jQ(y). g^ can be evaluated from g^t g^ §2

which equals [2(1 + r/L)

]

(3)' In the case of experiments of benzoyl peroxide paste the value

t of gj^/Bi was determined directly from thermocouple measurements of the

f temperature at the center of the sample, T^, the container, T , and the

surrounding constant oven temperature, T^, at criticality. Using eq. B25, one has for the steady state heat balance equation per unit area of

container:

f Si t t (T -T ) = —, (T -T ); Bi = a r/X

o „ . s e

B. 11-15
In the event that the material did not melt one could still determine

g /Bi

I directly by replacing T -T

t with a measurement of T -T

It

where T

® 3 S S S

I I I is the temperature of the surface of the material. -T divided by T -T^

would yield a /a and 6 would be estimated from ^ e o c

. „ 1 ®1

0 'V C e g^ §1

[1 + —r (1 + ct Bi

This extension of the method would apply to homogenous reactions. It might

apply to heterogenoeous reactions but no experimental or theoretical work

dealing explicitly with this question has been found.

(4) The induction time to explosion of benzoyl peroxide paste (35%

dimethylphthalate) is roughly a factor of 100 shorter than that of dry

benzoyl peroxide under essentially indentical conditions of 5^ and r. T.^

is however, identical (i.e. within 7 K) . Both the damage potential indicator of the SADT test (see [117]) and Bowes’ observations on the nature

of the explosion, once initiated, (see [120], p. 293) indicate that the

damage potential of the dry peroxide is much greater than for the diluted

peroxide. These facts indicate that induction times to explosion are only

indirectly related to either susceptibility to thermal explosion, the

damage potential of the explosion, or the degree of self-reaction hazard

based on a composite of the preceding. Induction times are more relevent

to an emergency response system to prevent an explosion by cooling the

material once the container has been exposed to ambient temperatures near

the critical value.

B. 11-16
(5) The results of Bowes' work in combination with the Russian work

[ 113-115 ] on mixing by free convection, gas bubbles, and evaporation

suggests that Groothuizen et al's comment (see [122], p. 240) that analytical methods are not applicable to these complex situations should be

restricted to the qualifications given by the authors. That is, the

statement applies to the question of analytical versus numerical calculation of the hazard implications of a fundamental parameter test.

A number of fundamental parameter thermal tests in the cited references for this section are not listed in tables B-4 and B-5. These are

analyzed along with similar types of methods in Appendix C. It should be

noted in this regard that the autoignition or self-ignition temperature

test discussed by Gerlicki [130], Hilado and Clark [131], and Setchkin

[132] could be converted into a fundamental parameter test for initiation

of combustion of vapours in air. In view of current work on gaseous reactants (see section B.9), it seems probable that results obtained by Bowes'

method would be more reliable if stirred reactors of various sizes were

used"*". There are some suggestions (see [116]) that hazards posed by the

thermal explosion of vapours from condensed phases with boiling points well

above normal ambient temperatures are not important (in comparison to

explosions in the condensed phase)

.

"*"Extrapolation to practical situations involving free convection would then

be estimated using eqs. B-77, 78, 79 for the dependence of 5 on Ra.

c

B. 11-17
IS

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c. Fundamental Parameter Tests

C.l. Adiabatic Calorimetry

Three existing tests, the adiabatic self-heating test developed at

NBS (see [8], p. 21 and [142-146]), the adiabatic storage test (see

[8], p. 29, [122], [133]), and the ARC test [147] involve adiabatic

calorimetry. The first two tests have been evaluated in the second NBS

study (see [8]) and the following comments are to correct, where needed

j

and amplify our first evaluation of these tests.

The purpose of the adiabatic self-heating test is to determine

t values for Q A/C and E, assuming it is found that the Arrhenius form of

is found to be valid for the initial stages (i.e. r| 0) of a decomposition reaction. This test is carried out in an adiabatic calorimeter in

which air or another gas is blown past the surfaces of the sample which is

suspended below a fan. Fan and sample are mounted in a glass dewar.

Depending upon its integrity during the initial stages of decomposition,

the sample is suspended unenclosed in a wire basket [144], or in an open

beaker [145] . During adiabatic operation, the circulating gas is heated

to be equal to the temperature at the center of the specimen. Heat is

supplied to a guard cylinder around the dewar to minimize temperature

gradients outside the vicinity of the sample. The remaining parts of the

test procedure is to determine C, the total heat capacity of the sample,

by the method of substitution using a glass dewar, stirred water, isoperibol calorimeter. The calorimeter can be calibrated electrically

[144, 145] and is operated at room temperature. The thermal conductivity,

X, of the material is determined for example, by a heat flow meter type of

thermal conductivity apparatus (see [144]) that is calibrated with materials

of known X. Relatively large quantities of material are involved in all

C.1-1
three tests (i.e. 50-100 gm or more). Thermal instability is evaluated

from plots of calculated critical temperatures as a function of the

radius of a sphere of the material. Bi is taken equal to zero.

The adiabatic storage test, in practice, seems to have the same

purpose as the adiabatic heating test (i.e. see [122], p. 126, first part

of section 5.1.4) and not primarily that given in our first appraisal of

the test (i.e. [8], p. 29). As indicated in the latter, the quantity of

sample is larger than in the adiabatic self-heating test. Some provision

for heat capacity measurement is evidently included (see [122], the

ordinate of figure 7 requires one to know heat capacities)

A commercial adiabatic calorimeter, called the ARC, has recently

become available that is designed to study exothermic reactions. The

instrument is marketed by Columbia Scientific and is based on the design

of a unit by D. Townsend, H. Kohlbrand, and coworkers at Dow Chemical

[147]. In the commercial instrument [148], samples, nominally of five

gram size, are contained in a high pressure vessel. The temperature of

the outer surface of the bomb and a surrounding adiabatic shield are

measured by thermocouples. Heat is supplied to the shield to keep the

two temperatures equal. An additional feature is the ability to monitor

the pressure in the vessel.

The main idea behind the adiabatic method is that upon eliminating

external heat losses, the temperature of the material undergoing a homogenous exothermic reaction becomes uniform. Thus, from eq. B-58 and the

discussion following eq. B-61, one has for a single reaction oh a unit

volume basis eq. C-1

C.1-2
C-1

or for the entire sample eq. C-2

og - Qf(n)k^

q'g = Q'f(n)k^ C-2

Equation C-2 will apply to a heterogeneous decomposition reaction to the

extent that the rate of heat generation is approximately spatially uniform.

If the material, as is the usual case, is placed in a container,

neither eq. C-1 or C-2 applies exactly if the adiabatic condition is to

minimize heat loss from the container. Heat is supplied to the container

by the sample and this introduces temperature gradients in the sample

unless it is a well stirred liquid. When the temperature gradient introduced into the material is small, one can replace eq. C-2 by eq. B-3

where T in the latter is now interpreted as an average temperature of the

material. For the case of heat transfer inside the container solely by

t conduction, h can be approximated by eqs. B-23 to B-25. Assuming the

temperature of the container and its external environment are uniform and

equal to and T^, respectively, one has for the heat balance equation

for the container

dT

, c 'odt = h (T-T ) + h^(T -T ) o e e o C-3

In eq. C"3 C^ is the heat capacity of the container and h^ is the heat

transfer coefficient between the environment and the container. Using

eq. C-3 and eq. B-3 one has in place of eq. C-2: '

of-

= q'f(n)k^ c-4

where f is given by

f = 0 , no container. T

e T C-5a

II 1 , container. T

e m

1 o C-5b

II X/(l + h /h ) e

, container

,

T

e T C-5c

C.1-3
Equation C-5a applies, for example, to an adiabatic self-heating test

where there is either no container or a wire mesh support whose heat

capacity is negligible in comparison to that of the sample, f in eq. C1 5c is close to one if, as usual, the value of h^/h is much less than

one. is not necessarily small in comparison to C. For example, from

the adiabatic storage test resxilts cited in ref. [22] one estimates C is

o of the order of 0.25 C. Clearly, two. heat capacity determinations are

required -one for the C of each sample and one for C^.

Data analysis is greatly simplified if one can assxime that dT/dt

equals dT^/dt or, equivalently, that material and container are in a

slowly varying quasi-steady state. Intuitively, one can see this assumption will hold during the initial stages of the decomposition reaction in

terms of time in an adiabatic self-heating experiment. An approximate

error analysis confirms this and shows that effect of an error in the

asusmption that dT/dt equals dT^/dt can be expressed as follows. Writing

eq. C-4 in the form

O

where

D = d/dt; P = Q Dn C-6b

one has for £ when C is constant

e = +

dP "^1 dT (C + C ) o where

T CC

1 o h (C + C ) o

C-7a

C-7b

h

h

h , T = T

e o

h + h , T

e e = T

C-7c

C-7d

C.1-4
This follows from the solution of eqs. C-3 and C-4 which can be expressed,

when C is constant, as

DT-DT^ * dX

•^1

'\j — DP, DP constant

P dP

'''

. - r TSr » DT DT and DP constant

C (C + C^) dT * o

e in eq. C-7a is proportional to P since dP/dT is given by

dP d£nf(n) . d£nk„T ^ o dT

^

One can see from eq. C-8 that when is of the Arrhenius form and f(ri)

is for an reaction of integral order, eq. B-21, that dP/dT is small

compared to P. e is small when P is small which occurs during the

initial stages of a decomposition reaction.

In the adiabatic self -heating test, data analysis is carried out by

plotting £,n(dT/dt) versus 1/T for the initial stages of the decomposition

f reaction. The slope is equated to -E/R and the intercept to £n[Q Af(0)/

(C + ^q) 3• f(0) is f(n) at the beginning of the decomposition. It is

evident that for this particular type of analysis to be valid, the

followirg conditions must be met:

a) only a single reaction must be occuring

b) k^ is of the Arrhenius form

t c) h must be much larger than h^ (i.e. T equals T)

I d) (C + C^)AT/Q , where AT is the temperature range of the plot,

must be much less than one.

Condition d) is the requirement that both C and f(ri) be constant for the

data plot. In both the adiabatic self-heating and adiabatic storage

tests, the material is not enclosed in a pressure tight vessel so C is

C.1-5
not necessarily constant. That is, C is of the form

c c (1-n) + c n

r p

where C and C are the total heat capacity of the initial material and

r p

the portion of the products that are condensed, respectively.

In the work of Townsend et al [147], or in the commerical ARC calorimeter mentioned above, the material is enclosed in a pressure container.

In this case C is nearly constant because in the preceding relation is

now the heat capacity of all the products and because now

much less than one. In data analysis condition b) is replaced with the

2 assumption that has the form AT exp(-E/RT). This makes little difference

since AT is small"^ and has the advantage that eq. C-6a, with £ set equal to

zero, becomes simply integrable. Townsend et al extend the data analysis

to larger values of AT when f(ri) is not constant by finding a satisfactory

fit of the temperature-time data with an assimed form of f (n) . is

t evaluated as (C + C^)(T-T^)/Q where T^ is the initial temperature in a

t self-heating experiment when n is zero. (C + C^)/Q can be evaluated as

the total temperature rise in the calorimeter caused by complete decomposition of the material.

It is evident that the extension of the data analysis to temperatures

where f(r|) is not constant is practical only when C is constant. Also it

seems clear that the extension requires not only that condition a) , given

above, be satisfied but also that DT remains nearly equal to QT^ and that

the assumptions implicit in the use of eq. B-3 remain satisfied. It

"**The maximum deviation of a 2.n(dT/dt) ver^sus 1/T plot from a straight line

occurs near the mean 1/T of the plot, 1/T, and is of the order of 1/4

(AT/T)2 in magnitude.

This seems, intuitively, to limit the extension of the analysis to some

value of n less than that when DT is a maximum.
seems condition a) is, at least in part, a required condition because

there is no direct way to test the assxnaption that P is separable into a

product involving n, f(ri), and T, k^. A satisfactory fit of the data is

a necessary but not a sufficient condition for this condition be be

met. The condition that DT equal DT^ is desirable both for simplicity

I and because h will change during the course of the reaction. It should

be noted that because the assximption is made that T equals in eq. C6a, (as well as DT equals DT^) in the work of Townsend et al, the value

ofe in eq. C-7a must be replaced by

^ ' dT

o h

The adiabatic method clearly mimics the hazard situation involved

in, for example, loss of cooling of a chemical reactor. It does not,

however, exactly mimic the hazard transportation situations mentioned in

section B.ll (nor need it do so).

C.1-7
ectiXtmoo

tui ' 97md:i

' > . S$^ Hi

'-o\^

,ar.6i^....^^ ,i-r * jfc# ^--Vjt^ltii ^l^l^iiP^. O' ""

^ " '' ^ ‘ e^' i*

Nsco**^« Ut«^tiij^'f*'V

,V ...t »i?"i«^^^v-4!i«t*'^'l^#*^*-'^^

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jiM* iit jdbl •Ki.tftJLAt.

OT :« tji *

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it

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fclj* 'tsjf'-a

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f !«:

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C.2 Conduction Calorimetry Thermal Instability Tests

Table C-1 lists some hazard tests that involve heat conduction

calorimeters. Tests 1 to 4 involve large sample sizes while test 5

involves (with one exception, the last entry of table C-2) small sample

sizes and sensitivities corresponding to commercial DTA apparati that are

given in the fourth and fifth columns of table C-1, respectively. The

term DTA is used generically in column 1. It is used in a more specific

way in column 2 (see entry 5) . Nomenclature for DTA, QDTA, and DSC in

the second column are based on ref. [154] in the sense that DTA refers to

classical DTA, QDTA to Boersema DTA, and DSC to compensated instriiments"^.

The definition of the term QDTA given in ref. [154] has not been extended

to include twin calorimeters in which a thermopile is interposed between

each cell and the common block. The calorimeter in test 1 is essentially

that of test 2 but with reference cell omitted. Table C-2 is a summary

of the purposes of the tests listed in table C-1 and can be viewed as

supplement to and/or revision (where applicable) to the discussion on pp.

16-20, 31-34, 39-40 of ref. [8]. The following comments are made in

explanation of the third column of table 7.

Exact details of the various purposes of tests 1 and 2 are unavailable.

The method of data analysis in test 3 seems to be to assume the isothermal

model applies and to use eq. B-75 divided by C:

f + |-(T-T^) =

, f^Af(n)exp(-E/RT-)

I h /C is determined from observed values of dT/dt and T-T when dri/dt is

o

I negligible. Values of Q dri/Cdt are computed from observed values of

dT/ dt and T—

T^

between the exotherm onset and peak temperatures and

This nomenclature is adopted for clarity and is not an endorsement.

C.2-1
Table

C-1.

Conduction

Calorlnetry

Tests

60 «n >« lb e • a • 4S b

eo

b o lbbab

CMb

b 60

ubb

a > 60 • e a a o a a a

6< :s oo"^a

b

ab 60-bi

u

y >s

B

B

u

—6^ 3 aa JS

a Jd 3 CD

b

bb

•H :s 3 b

b b. a.

u o a

b

a S o 3 60

b 3 0 b

B •9 e o a o Cl b a >n,o B a a

a

aO

b ul o 1 a

b Aim <0b

B

a

CO «n

b

w

mba

a

b

b

a

•9 9 <44

a

«6 60 • b c a

a

aa

bs •9 •9

ab o MU

b •9b

b

m

aa

b

a *9 1 a

y

"O 1 *o • b • b 3 y

a

B <u

b JS CJ • B 1 a e a O b b O b 4J a b 3 C9 a CO y b C b 9 b b a b 00 a e 6 3 9 Q b a c c • b b b » b u a b a • o a b a • a b a 9 b b a 0 b > *9 CD 9. 0 0 • > b > b i9 a b C eu b a > •< b « b w b a a y a 3 a • b 41 b H a l—V U b a • y b a y 41 •9 y a B lb a s a b b a <T> u *a Cb b a * U c 3 00 a -9 •n y 0 O' a e o u a a b a 9 b b a 3 a •9 a > a u b U a a b b b a a a a b 00 a B b CD b 0 • o b

o.

b

b

a

ba

a

y eu GOy

BO 3 b

N

b < a

a e a b b <n a a b a 41 a a CM b <Q y Mm H C us a b O b • b b b • a U b b V b b JZ a Q a b

CO CMa in

a

b

a o.*o

ba CU-9

b o ab •a >b JB o a b a

(Q «h • 41b < b a >b

U •9

H

ay • •H a O b B b a b a a B y o lb %4 a •* a b a b b b o a a b a b b o 1 1 s 3 b b a y

bb oo * B 00a e OaBa

b

b

b •o 9.

bb

a o b e • 9 * * c a a a b y eu o 60 fl 60 60 CQ o 3 y

•rl O-tb Cb Bb

b

B 6^y

b • •9

b

U

b 1 ab 1 b 1 a

b

U

a 60

a

b

B

a c e JS e e B

a

X s <r « * B

ba

a

b ctJ

b

C

b c b S O JZ «ub eu JSb

a

a .a o a

a

aa o a

u CO

B 3 b

eua

y , a

y vj

Ma

4)

H •<

C o a 3 o CO bi b CO CO ss a y b u y o b 1 « •• a a a B 1 b b b M 0 b b b e fi b 1 b b 1 o O c b a 3 C a jC b a a a a a a • z 00 o a JS b b a o. U y o y o >t T3 *9 < « H c mH b b a iB a a 9 a b y o y y u fi b H < i-\ 1 4) 4J b b 60 » 0 aba a e c b o C B CQ a a o H u o CO 0 S a a b b U a < .9 b 9 b b 3 S b era CO > a jQ 94 a a e » a 9 JQ o b 3 a OS 4) > b a » A 9 b e -a a a « a a • b O 'w' t-i 0 a s a a o a j= ^ S b 1^ a 9d ^ 3 u a b a a • >o 1 b y 9. jC y b O.JS CD y B y a 9 • oO <b y b

CMb

a

a 3 b

O 3 bO

O

b

a

C CQ

b jt

a

a

b

Oa o aa o a b* o a 00 JS be

b 0 U <! 9 * y

a

eu eu o b o c b

y .9 .9y .9y COa

aa 4)

b o a

O

b

b

a

aO

b

b o ^ a

HB

a 4J a. • O

ba

a .9 Q.-a cub

a

b fl

a

a -9a

B <QB • CJ

a •9

b ocaa

b .jj

a .9b Jd 4> • BO.

a

U 4> 0 CO s 9 b a b a a a 9. a y y a y a B lb a a cu 4J u o a O a a X < b u a j: o •9 .s o b m: o a G a a b b b a b e a b b « c b b A H b y o M4 0 •* <W a •o .C b 9 JS •9 a .9 •o CO a b B y 4.J o 2 a a 1—

b e , • c b a c * a 4> H w e a c B b b B y b a a y « b • a e b a a B u y * 41 c O a 3 O a b b • ji b b a b b b CD Xt o u >> b b eu o eob e iC y b y 60 a eu 60 00 B y .B ^ < H b a b a

lb e bb

y o b 0 c *o c a c -9 e b o y cj

H

U

w

a

b

y

a

b

a > « ba

bb

Ca

a

b

C (U 3 ba COO fH

m 0 B

a

a

b

u COu

H

a .9

y i9 CO

a 9.

a CO

a o.

H j9

a

a O' <Q fj JZ

a 9.

aa

C o a

b a. o O b a a b y a m •k a b b o • CQ 4J b * a a

•« z o N CB o a CO

ub

b

H e c j2H

a

a JL c b usB

b

O

b CQ

w • 41

b

O

a > a

b fiO

U o b l»^ o b

a

a

O

u

a

b

X

b

b o b

Xa 3

bC e • * •9 0 o 44<'

a

b e o U >* b •o u B b b O o b ca b b c 0 0 a o >

'rib 0)

b

O (Q c c b

B <b

a

u

bu

ba 00ab

a

a

b 4J 0 >b • b

u o 41 ja CO o b •9a • a

b eu 00

a «n 41 e b

C • a

b

a C9b r*

a o CO •9

a

b 00 us

O. o Ua 1 b

aaH Sb e

Ouu 0 a SO

a

b

a e a CM

a < a 4J b b b y s a a •H 0 > •X c 3

•a CDa e b

a > b

b

B

y e a

a

au

aO

b

b

a T3 Q. 0 ba o 0 b

a

b

bb

a z bay 4) 6 U

a 3 b

O

C i 9 (0 a b b JS a H a o b 00 0 b b a 0 b a ib a X a a y b 1 B.b a 3 0 4J b o a z b o C b 9. a a o o 9. a 3 C y <b a z 4) o H •9 a > b X iC O U4 O a o 0 •9

JS

H JZ Du

bb

b 00 04' w

uO 3 * a

a

b 1 4J < eu (Q aM (T

m 60

b

b

O

O o z H cr < w c 4ia

>! 5 (0 < •• CO

b

b 4 • • 4 0 3 fls >s

a

M QO

M

n v9 ><

b

aa .9y •9 <u .3

C < b

b

u E H

b 0 b

CQ 3 O

a c 0 'e • • • • • Z 4i O' a 35 2 u

tn •«»

m CM <n •«»

C.2-2
Table

C-2.

Details

on

Conduction

Calorimetry

Tests

.

•

'

• u

Wl a a

u a u 0 o B ^ V 1 a X X •rt 4J W e e IWI a a 1n 44 a —— a o a a • •o CO Wl .o w u a Wi a a B a*3 c cn c a

a a u o a IWI 'H c a a B •o CO • •H Wl ^ a <w X WI W a O wl a a Wl a o CM «D a

*3 M o o IWI IWI e X a Wl X Xt3 a a T9 a a Wi a a a a Wl • u a wl '•T o a o a a Wl Wl O a o Wl Wl c a 1 u o a u A a O X • Wl a o o a • CM 44 o o u Wl Wl a a -H a a G

a 00 • a a X a Wl Wl • AJ •H

• ^ a CU>r4 a a •3 a « a a 3 a 00 o. •G a m e u C X c • IWI Wl a B X • 0) •3 u a *M a a a a o a a O "H O B CM a

G a 0 u . . X a a a O Wl u • a

« w a a ax Wl a a Wl a a a 4J 3

a a a U 01 a o a *o Wl a a Wl a CO 44 CM § 00 •w e a a a X a 3 Wl Wl a H 0) « cn o

o a e a 4J WI <H a Wl IWI o X a O 3 IWI 01 1 . u a wt CO a 0 a a ^ «wi a a ea in 44

. a V . WI IWI a a a X o B >1 O' CO * a e U a X o Wl a o a a . Q. a X 00 • 4j « G • > o 4J > o 3 a a - S a X u CO c a* w 44 a •HH u • a Wl a a a a w X X

o flu a CO eo o a a 3 •3 Wl a lU A4 • o. a a w a eJ S WI a a a a ^ U 00 CO CL w e a a 3 <0 a u a IWl X . c a < 0) < a • 3 u.* W h O •o WI a Wl a a O wl a 4J O a 44 > a-T

eu a a Wl a

a o a 4) fH X IWI O IWI wu a

C wi e Wi A a a Wl a X a CO • o 44 O iH a o o a >v JS X a 3 • UJ a w C ^ •G CO H B tn

uH WJ a a a 3 O C o V B 0 0) B O 1 a uj 4J u <wi O B H a a * fi C9 o a O o Cff 4) O tU X

a a a u G • i-t X O X a X • o X -H u C wU a

a Wi a a a a a a a -H CO CB Wl Wl f"4 ««4 Wl o • o w a Wi 00 a a a a a a Wl C Wl a <0 s a iH 3* e a o '•M c a B a CD O o a B 4i a u B a a

a a c a.^ <WI O * Wl m Wl B •H Hw w •H 14 0) X

>u 00 00 o cu— 0 a 3 a X 3 a 0) 44 a a a

a a 'H < u Wl • a a Wl Wl wU • ^ • a Wu 4) Wl > Wi 4J a a a a O. a > a a o o. -o a o a m a X >

Wi o a Wi IW Wl a s a Vu wl a Wl 6 C s a 4J o 0 wl Wl 3 w a a . 0) a s a 0) u 3 a B 3 o> o a X a CO a O a wH «

to a X > M Wi X 0 a a U Q o a U Wl a a 3 44 a 3 H

-

«n«n

s • a a a

a X B X mi a X X mm u ww • • as a . a •a

Hw - • W a a mm w ^X| X «X es • ex as A

«x «N X ex CM esi <n X ww wi es

.«

. eses X cn s .w 1 1 o Wi >s a CM CM a a Wi Wl o M

o UJ a H B a CD Wi H a a 4» a O ' Wl a wH ^ • W m4 M a M a o a X ^ tn a < Wi o a a X a a f'* a H a a a Wl a 1 ^ Q a a w o Wl X o X a O' Wi Wi a a a o b a S B U a a a Wi ^ a -T X j. a a X > < a X S L « o .b ^ a

a •• X z .b wi a • wH U X X H a a • o a S a a Wi wi 1 a u a < wl a a a X • a 3 o O X O H a a Wl O Wl o H O a 3* > u a Wl Q X z 2 S a z Z H wl X X a O' X X H X H X w4 w 1 Wi 1 << Wl z Ok • A « w o o 0 O z a H X < < H -< < < X > * 2 < a Ui s H H Q H H M X X i-i X n H < a Q Q O' a a Q

es

a • .

. « s . s a X .CD a fH C*4 <n m m m m

C.2-3

5d.l

DTA,

Chemicals

[130]

Determine

maximum

ambient

temp,

for

which

ex(

ASTM

E-487-74,

DTA

therm

occurs

within

2

hours

after

exposure

t(

^

temp,

(exotherm

onset

temp,

at

zero

scanning

rate)

.
Table

C-2

continued.

Details

on

Conduction

Calorimetry

Testa «ooa.b3

CkOua«

)H <W

•I« X a q

a*

aM

U 4> I ^ .C q 00 a • u C h O u n 'H 0) *a e a X c

oik quo • o o

Q.okU

X JS V

aaa ,o •

•• ik a « a

•H 41 iH J O. j a 4> a Ik • asa

>» Ik * 4din

aa

a •H

H

a

a

iH

a > c ^ u & o JmQa

a

ok

a « < a

u hi Ik

a « u • « kS o ok 00 o.

ik

CH QB

Ma

H • » e a

u Ik

a e > a o •H

a e C JO a o a Ik U ok hi a a a e U OB a «k a •a Ik hi e a § a Ik ik Ik • a c S a § a B XJ C9 a Q. a 0 ok a • m

<3

a • c •H «H hi e u •H o a

a OI

jSs; O roI m

M

4) Ik JB to Ik N

hi

a < j:

V

u

a Mk .s

a c a o u a O 4> ik

j: -H Ik .X » u c a

u

00

hifk

O OI e <hi

au

m

04

<*> in

mmm

aMa

o.

4)

JOaa

Oia

J3a

Ika£>

e.a Ik

u 0)y Ik > O okU

a

a Ik

a 0)

a

a 3 a

u

a • • h a a e 4J o •3 fk 3 a a

X S hi a ik i

Ik • o Ik

B o a Ik • •vk

a J4 hi 09

M > a o U Ika

O •3

a hi 3 > a

a ss o a Ik Ik - B

ua « a

a oka k= gokaa o a in 3 3 0 4) a a 00 N o.

hi c O' iH

O > a o s a ok s Bi ChO B "O a CO a 00 B CD k= a -H •3 a

fi.

a

a

a >1

a 3 Ba * o a

ok 00

a

BB okB ok ik ikB

B ^ 1 ik ik

a iTa ik • Ik ok 0 O

O ea

UH Ik

a

g

g > >i3 ik ik »

ik hi

a

a hi Ik hi Ik ok

a fHaM 4)

H

a

UU kS ok 00 hk 4J <n Ik a a ok N a a hi o a 3 cn a 0) 1 a j: 4) a ik a a a Ik O hi s < a Q A Q hi Q ooko j: Ik u > • 3 a

Iks

•aauo

hio. •aa h u a a jsuoa

o.a

4Jaa

hio§o5 . -se a o

.a Ik

OI

00

>•

•a

•aaas

ck

a. 04a

Ikoc u bk

CO 3 S a

O o a 4J a

Ba

aX ik

a

O * o hi

a

a Ik hi ik :< o. u 3 > hi ik 3

Ik < H o cn hi Ik 3 a H a Ik a < a > u B a u Q cr a. H M a (J ik • <a a 6 X 3 o

ok •• * • •< a. ol ik O' Ik Cb a a a H «* a 3 o. Ch ok ok oi a ok a Mk C >1 < a a a or a Ik o a u a a u a a Cb a a • .H ik •• ik o Ik a. ik > a a a a hi a . o a ok ok a a a a a a o. ok ik O a j: k= JB a a Cb a X > z u CJ U hi CJ a a 3 B

Ik >< CO < M • • * iB • ok Pk cn «» m < -< •< P < H H H CO. H O a a < Q .. CO

Ck <n <T

m

M • . • • H •3 3 3 3 O m m m m Z

C.2-4 II1IH
plotted versus 1/T. Near the onset temperature (since n 0) , the plot

should be linear with a slope of E/R. While test 4 is based on test 3 it

does not seem to incorporate the preceding method of data analysis. This

is assumed to be because test 4 is also applied to solids and has no

design procedures attempting to insure that the isothermal model applies

to the sample. (Test 3 does so although demonstration of the efficacy

of the procedure is not given. Test 3 would be relevant only to liquid

samples.) Test 4 is signfiicant in identifying gas evolution as being

part of a measure of damage potential. Also, in specifying that "safe"

sample temperatures are less than the onset temperature, test 4 indirectly

takes into account that a small sample (M).! g) under good heat transfer

conditions has a much higher explosion temperature than that appropriate

to the material in bulk. This is confirmed by the statement in test 4 to

the effect that measurements on small samples would have to be made under

conditions"* nearly adiabatic *” to obtain T^ for bulk conditions.

•4^ In test 5b. 1, the definition of the term auto- ignition temperature

is inferred. The maximum process temperature is stated to be either of

those given minus an unspecified safety margin. This is consistent with

the preceding comments on test 4. Test 5b. 2 determines the values of A,

E, and Q from measurements on small samples (_< 20 mg of propellant). The

cook off and isothermal self-heating tests are large sample explosion

I I [ temperature tests. The former is evidently similar to the type of

"*”This is consistent with section B.IO and from refs. [88] and [89]. One

would expect the peak rather than onset temperature to be near T^.

The term is used to denote an "explosion” temperature in test 2—5 of

• [129], and in ref. [150], The same usage is used in autoignition

tests mentioned in App. B.ll.

) I I Reference containing details was unavailable.

C.2-5
explosion temperature test used by Bowes (varying propellant sample size

under constant heat transfer environment) while part of the latter seems

to involve measurement of the Semenov warmup (i.e. under critical conditions

2

T-T RT /E) . Test 2-4 of ref. [129] clearly identifies that there is

o o a scaling problem and that the three parts of the test procedure apply to

propellants that decompose by a single reaction and are both homogenous

and without voids. (The latter insures that eq. B-31a applies).

Test 5c is significant in that ref. [23] demonstrates that for a

number of explosives the measured T^ for 40 mg of sample confined in an

aluminum blasting cap is in good agreement with that predicted from eq.

B-31a (with gj^/Bi equal to zero"^) and kinetic constants (E,A) determined

under unconfined conditions by DSC. Values of Q (normally 500 cal gm

at times 300 or 600 cal gm ^) were estimates based on DSC measurements at

various degrees of confinement. Values of X (between 0.8 and 5 mW cm ^)

were l estimated or taken from the literature. The critical value of 5 c (see eq. B-30) is 0.88 (i.e. the infinite slab approximates this test

arrangement). Test 5c is consistent with test 2-4 of ref. [129] for propellants in recommending that a separate explosion temperature test be

made. However, test 5c seems to suggest that it will turn out that for

+4

* many explosives the scaling problem in regards to T^ is not serious

Tests 5d.l and 5d.2 are similar in that both are in effect DTA

sample size and heat transfer conditions versions of a time to. explosion

and/or explosion temperature test. The ASTM Test E-487-74 correctly

"*’With the exception of TATB, the explosives cited in ref. [23] decomposed

at their maximum rate in the liquid phase.

This would not apply to explosion time measurements. Values of T^ are

quite high (200-300 C) so it is not unexpected that E and A determined

from DSC measurements are consistent with these T^ measurements. However,

we are unaware of T^ measurements on larger samples or small sample

measurements under poor heat transfer conditions that can be compared

with predicted values to further clarify the T scaling problem.
identifies the fact that the value of T^, so determined, may not apply to

bulk size or heat trasnfer conditions by stating that is neither the

safe storage or processing temperature. The ASTM test uses the values of

to order materials with respect to their thermal stability.

Tests 5d.3 and 5d.4 represent the current status of the state of the

art of thermal instability evaluation using commercial small sample

apparati. These tests are discussed in detail below and in section V.

The last test listed under 5d differs from the other DTA tests in ’

that it involves relatively large sample sizes and happens to be designed

so that temperature gradients in the sample due to self-heating are

reduced"^ (i.e. since the cell is in poor thermal contact with the block).

Identification of gas evolution as an important factor in hazard evaluation is consistent with the aims of test 4.

An analysis of the third column of table C-2 clearly indicates

that a defensible common test purpose for the tests would be to determine

those heat evolution kinetic parameters that can predict T^ for materials

in the bulk under their storage or processing conditions and to measure

gas evolution as means to assessing damage potential. Thermal stability

is greater as T^ is larger. Explosion temperature tests under well

defined heat transfer conditions can serve as an independent check of -the

heat evolution measurements if these conditions are the same as the

latter. They can be used to evaluate scaling problems if conditions are

closer to those in bulk.

+

From what they would be if the common practice of good thermal contact

between block and cell were used. The particular design was adopted in

an effort to increase the DTA signal.

C.2-7

C.3. Simplified Measurement Theory of Quantitative Differential

Analysis (QDTA) and Differential Scanning Calorimetry (DSC)

The nomenclature in regards to DTA, QDTA, and DSC is that specified

for column 2 of table C-1 in Appendix C.2. Only QDTA and DSC types of

apparati are discussed and of these one specific example of each: the

QDTA apparatus described by Baxter [173] and the commercial DSC apparatus

used in our experimental work, to be referred to as the DSC-2, that is

described in refs. [154, 174-178]. This presentation supplements the

comparison of the operation of the two types of instruments worked out

by Gray [179]. It extends the comparison to provide a complete set of

equations for analyzing the experimental data. In particular, our

purpose here is to show how dri/dt and n are determined from the measured

output signals, assuming only a single decomposition reaction is taking

place . The QDTA measurement equations are summarized first in Sections

C.3.1. and C.3.2. The corresponding DSC equations are discussed in

section C.3.3. Calibration of the energy and temperature scales are

summarized in sections C.3.4. and C.3.5.

C.3.1. Measurement Equations for QDTA

A simplified cross sectional diagram of the QDTA apparatus to be

analyzed is given in figure C-1. The hollow cylindrical block, whose

temperature can be programmed to be constant, increase, or decrease, is

in good thermal contact with the constantan sheet, CS. The chromel

wires, Cr , are attached directly to the sheet and ^form a difference

couple with the constantan for determining the difference in temperature,

Tgj of the area of the sheet immediately below the sample container, 1,

and the temperature, T , of the area of the sheet beneath the reference
(Uu

C.3-2

Vi

<u 0)

OJ « "3 3

1—t DS iH •H c. o 3 3"- VI o <u 3

CJ Vi (U 0 0 3 0 0 i VI c S CO 0) iH

0) Vi V4 3

x: (u (U •H

VI 0» *vi Vi s <u 3

fH (U VI VI

Qj VI 3 E , E 0^0 VI u 3 X o a; 0

CJ iH

M 3

Xi 3 3

VI VI Vi CJ 00 CO 3 C VI 4H • « -H O 3

•3 iH CU Vi t—

<a <H 6

cn iH

O <U rH VI VI <r»

r—l

O VI

Vi

C -

VI

O ViO

VI

C

O

H iH

(UO 3.

Vi

O Vi

g

O ^ 3

cdm <vi o 3

ca -rlO

Vi r-l

—

(U (U ^ CM

Xi VI VI 3 cO U-i • A

'T3 VI <U

C VI

<U CO x: 3

Xi Vi <u 3

Ti 01 r-l Vi •H

VI 3. CO 3 3 o g U VJ VI

00 <U CO 3

(U VI V4 Vi 0 Q VI O 0

(0

a cu

C CO (U

g 3 H O tH (U <H a x: di u a

o* 5 g3

4-1 ^ Si -oco O O H Q •« — c fH 4J fH o ^ 3 Vi •H <U O • •

Vi •>

g T3 VI

c» pq -H 3

lU

OO TO

cn •“ Vi j: iH

0) -H 0

00 Vi > <U x:

CO •H 3 fH o 5 0 3. 3

V4

g rH o (U r CO 3. tH W CO g • 3. 3 fH o ^ • A 0k 3 3 o 4J cn •H x: O (U CH Vi

CJ o OJ •« 0 3

VI

g Vi VI

(U Vi 00 0) 3 3

0) TO Vi

g

on 3: 3 rH 3

VI CO 0 VI 3 • VI xi 3 a

iH -H 3 Vi 3 ! (U CO (U 3 3 U g Vi 0 0. Vi 3 CO 3 g 3

lU rH 3 <U 3 U4

Vi 3 0 Vi VI 3 3 0 0) Vi 00 » <VI 0k •H o?cn 01 3 « u* •< CJ Vi H •<r t
material container, 3. The single alumel-chromel pair forms one junction

of the two junctions of a thermocouple to determine the absolute value

of T^. The symbols S and R of the diagram refer to portions of the

constantan sheet that are assumed to have uniform temperature, T^ and

T , respectively, for the purpose of a simple lumped constant or finite

element analysis (i.e. see O’Neill [176] or David [180]). The corresponding

electrical diagram for this analysis is given in figure C-2. The node

potentials are temperatures, capacitors are heat capacities, and .the

ideal current source of magnitude is the total rate of evolution of

heat by the material. The ideal voltage source of potential T^ is the

temperature of the block. The conductance from nodes j to i is a heat

transfer coefficient h, . and figure 15 assumes that h. . = h..**". Sample

ij iJ J

1

and reference materials, their containers, and sample and reference

holders are all assumed to have uniform but different temperatures. In

effect, we automatically assume there is no calorimetric error due to

non-equivalent sources (i.e. see [181]).

The energy balance equations corresponding to figure C-2 are summarized

in table C-3. D is the ordinary differential operator in time, d/dt,

and the remaining symbols and subscripts for entries 1 through 5 of

table C-3 are defined in the key to figure C-2. Entries (i.e. row) 1-,

2, and 3 are ^he energy balance equations for sample, sample container,

and sample holder, respectively. Entry 4 is the sum of entrie's 1 through

3 (and assumes heat transfer coefficients are symmetric) . Entry 5

corresponds to entry 4 for the reference material, container and holder.

Entry 6 is entry 5 minus entry 4. Entry 6 contains the approximation

This will not be the case where convective transport is concerned

unless turbulene dominates laminar flow in the gas flushing the outside

of the sample and reference containers and in the gaseous products of

decomposition of the sample.

C.3-3
co

•H

4J3

(U I O U CQ G 3

lU "G >. M c ua

iH (U o O <4-1 CJ *3 > (U <u

(UM i-t • G

«« • ^ O O fo O T-l O *3 3 •* >H 6

4J 3 ^ -H 3 <“) M o. •* a a .o 3

rH 3 O 3 3 G

4J 'i-l 3 0 « 3 U

4J *3 3

tH 3 “ O '3

J3

G

01 3 O G

•« iH 3 3 3 CJ a 3

Vi3

iJ3

VI3 3 G3

Vi3 O <4-4 *H 3 G 3 Vi -H

CU Ou •>G 3 a a Vi *H 3 3 3 U 3 3 ^ VI ^ • » 3 O

-rliH -3

O 4J

E-l fH

U • • O "3

•• 'f’l ”3 3 3 - 3 8

4J <H 3 a 3 G iH 3 3 3 CU 3 3

•H /-s

a 3 3 U VI 3 3

I—

•H

O 3 3 1 4-I 4-1 3 3 u <4-1 3 0k Vl 3 3 VI 3 3 3 O 3 3

VI 3 G 3 3 3 . 3 1—4 3 3

00 Vi 00 3 •3 3

•H 3 •HO JS

PC4 <4^ 4->

U

G 4-1 3 G 3 G

Vi

G 3 3 G 3 O 3 Vi 3 3 U

4^ V4 Vl

a 3 3

4.4 3 G.

a

O 3 4.4 ^ 3 U 3 3 3

VI 3 3 G Xi

00 3 -H 3 3 .G U -3 4.4 •H CM 3 3 t4 a P-» <4-1 1—4 5

T-1 3 3 '3 ' *H • Vl X ”3 o j:: 0) 3 3 G • -S' F—

T—

3 • M 3

4J QO • 0k 4.4 -Q O 0) 3 O 3 3 •H 0) 3 G I—

3J 4J rH

G

G •H G.-H 3 -3 O 3 a 3 Vl G u 3 3 4J 3 3

C. 3 G • 3 O

U 3

cs 3 >% 3 3 G 1 jQ JS -H o 4J 1—4 u <—

3 4J 3 3 3 3 t-l 3 "3

VI -G 3 U 3 3 3

-- 3 3 4^ 00 4-1 G 4J •H Hu-l 3 3 o O O a 4J "3

C.3-4 /
Table

C-3.

Measurement

Equations

for

QDTA

JS

CM

01ao

•H

4J IoO

0>ua

(d iHa

00

IM

0)0M

0) uao

(U

ro H'w'«n

rM X+

iH HI H

JQ1—( IIII HI CO H

CO

iH + a iH CO

HHH I I I CO ^ CO H H H

CO

(0 rH

JC ^ rC

CO

•H +

00

a

H

HI CO V4 I—( E-* CM j3

PM

U + ^I CM CM CO H H H I r" H

CM ,a

fH

H

rH Xi

CM CO ^ II x:

CM

Pm+HI

JS+HI

II rH II HQ CM rH CO H O H QCM O ••

010 +

CO II

f—I *H CM H HQ

Q rH *H II CO CO U u x: rH W -H

V

rHIC/1 0

4J0OO 0

•O

rHos oH

x:

II

X3

II

CO HI

CO

>M0

MJIpo

rH0U0U00

V40

>H0

>Mo

<4M0O00

rH0

XI

>r

0000M +

CO »M H H

XI rH H H

ja rH

CO

M

x: x: HI 0o

•H

MJ000

00

XI Hs-r x:<

COT)

MM

S3ICM

PM

XI0

X3

<?CO0

X3

4-

X3

CO

rH

X3

CM+

JQrH

X3+0M X

CM+X0 XnS3

CO

rH XCM

CO0 X

H

X <H

V—r 0 H 0 0 P

Ml

W -H

X CJ

•H

w + X

/-“S CO

Ml

H

MM 1 •H X H H P

rH

X P

Ml •H

XW

X

II

XX ft •H CO X

HP +

•H II

X

CJ

H

•H P

tH *H

X

U • • 1 rH 0 0 H X P o 0 H CJ • in X

CO

tM

CO

CM / « CO

rH rH X

II CM

•H +X+ o

<?

XCO XI XI—I X+

Wl xf“I CO XJ“

II X<HI 0 H

II CO•0

MM 0O00X1000

4-1 O0

•0O

IMCM XX00000O0O

i*o

CJ0

*0

IMo00

cy0§000000

?.0000M 0

•H0§0

•00§0000

Ml0

cn

0S Ml0

II

”00

•00HP

CM HP

•>

H

•H P HP

•>

M •H C

CD 0

rH

•>

X

XH r00

•r 0

Ml

H 0O

•N *H X X < -H0

•>

O

CM P X BO “ o

rH 0 X r0

COI CO

'--r i4I

PmI&w CM

CJ+

iH

CJ 0O

•HX0

•0

X0PX0OP0OaQ0XoX

CMX0 X

•H

HP

33

•H II CJ CM

•H W i4

Ol 10

C.3-5
that temperatures in the terms involving small heat transfer coefficients

can be estimated by values derived from the heat balance equations when

all the small heat transfer coefficients of figure C-2 are assumed to be

zero. The values of f^, f^, H, and Ah of entry 6 are given in

entries 7 and 8. The first value for each is for the aforementioned

approximation and the second value is for the case where small heat

transfer coefficients are assumed to be zero at the outset of the analysis.

Entry 6 neglects the time constant of the block temperature regulation

system (i.e. detector-electronics-electrical heaters plus the heat

balance equation for heaters and block) and thus will not apply until

transients in starting a scanning or step jump program have decayed out.

In a sprogram, T^ is increased or decreased with time at a constant rate

w. In a step jump program, T^ is changed as rapidly as possible to a

new lower or higher value.

S or (T^ - of entry 6 is the Instantaneous ordinate output

signal of the instrument, f^ is the product of the thermoelectric

coefficient of the chromel-constantan difference couple of fig. C-1 in

K*V ^ divided by the gain of the difference signal amplifier. The

observed ordinate output signal, 0, lags S because of the thermal inertia

of the difference couple and the finite response time of the difference

amplifier. In the next section. C.3.2., we shall assume the relation

between S and 0, the ordinate transfer function, is known. Discussion

of this transfer function is postponed to section C.3.4.

C.3-6
C.3.2. QDTA and DSC Equations for Kinetic Analysis

The energy measurement equation for our DSC, which is discussed in

section G.3.3., is similar in form to entry 6 of table C-3. Also the

temperature equations are similar in form to those to be derived below.

Thus the following summary of determining kinetic parameters applies to

both instruments.

In kinetics one is interested in obtaining P^, since this is equal to

Qdri/dt, and the temperature of the sample, T^* To obtain P^, it is first

necessary to subtract out the effects of the sample holder and the reference calorimeter and its contents. It is convenient to generate the

blank signal, S^, by rerunning the temperature program for T^ used in the

decomposition run with everything the same except now only the sample

products, having a heat capacity are in the sample container. The

measurement equation for this blank run will be entry 6 of table C-3 with

C

2

, S, and P^ replaced by S^, and zero, respectively.

The products of the reaction are assumed to undergo no transition or

phase change during the blank run. Also, we assume DT^ = DT^ = w and that

both T^ and T^ increase with time as wt. This automatically requires DT^

= w for i = 1, 2, 3, 4 so long as the heat capacities and heat transfer

coefficients of fig. C-2 change slowly with temperature. Hence, the

measurement equation for the blank run is

[Cg - ^2^ " ^3^^3 = "^^d^o

Subtracting eq. C-10 from entry 6 and rearranging with the assumptions

that f^, f^. Ah, T^, T^, DT^, and C^(i = s, r, 3, 4) are the same for both

blank and decomposition runs yields entry 9 of table C-3. K = Hf ,/f, is

d' 1

the calibration constant of the instrument and will vary with temperature.
The second step is to evaluate + CDT^. Three approaches are

presented in Table C-4. Following Brennan et al [182], the Method #1 is to

assume T,12s = T_ = T . This method asstunes the sample, sample container,

and sample holder are in perfect thermal contact. C

2

is assumed to vary

linearly with n as given by entry lb (i.e. row b of colximn #1) of table C-4.

C

2

is the heat capacity of the sample before the decomposition starts and

C

2

is the heat capacity of the products left in the sample container

after the decomposition has ceased.

Entry Ic is derived by inserting entries la and lb of table C-4 into

entry 9 of table C-3 to obtain

(C

2 - C2)(i - n)w - ^2 “ "

^o^

Denoting K, S, and before the decomposition has started by K^, S^,

S^, one has

o

(cj - ci^)w = -K^(S^ - S^) C-11

2 2 O

Multiplying eq. C-11 by (l-n) end subtracting it from the preceding equation gives entry Ic of table C-4.

The reltion between dr|/dt and is given in entry Id of table C-4.

-<5(T ) is the enthalpy of decomposition referred to some arbitrary reK

ference temperature T^^. C^ is the heat capacity of all the products of

the reaction whether they remain in the sample container or not. When

the sample container is closed, C^ = C^, and C^ are constant volume

heat capacities, and -Q(T ) is the energy of decomposition referred to

R. T^. If one integrates -?2 + ^^^^2 ^ time t^ before the decomposition

starts to a time t^ after the decomposition is over, one obtains for the
Table

C-4.

Baseline

Treatments

for

Kinetics

C.3-9
so

Q(V

Table C-4a Definitions and Comments, Table C-4

heat capacity of sample before decomposition

heat capacity of sample products left in sample container

calibration constant (see entries 9, 10, table C-3)

instantaneous ordinate signal for decomposition run

instantaneous ordinate sigjial for blank run with products left

in sample in container (see eq. C-10)

S before decomposition starts (see eq. C-11)

at temperature of (see eq. C-11)

K at temperature of (see eq. C-11)

Enthalpy or energy of decomposition at reference temperature

\

heat capacity of all the products of decomposition

instantaneous ordinate signal for blank run with no sample or

sample container present. (See eq. C-10)

apparent sample holder temperature (see eq. C-13c)

total heat transfer coefficient between sample container and

its environment (see eq. C-13b)

heat transfer coefficient between sample and sample container

C.3-10
total change in energy of the sample, AE

2

, as expected

AE^ “ -Q(Tj^) + C?(T^, - T,) + CjCTj^ - T,) C-12

and are the temperatures of the sample at times t^ and t^, respectively. If the sample container is open and products are lost quasistatically, < C?; C?, C^, and C, are constant pressure heat capacities;

and -Q(T ) is the enthalpy of decomposition referred to T . Integrating

"^2 ^2^^2 before gives the enthalpy of decomposition, AH^. "The

equation has the same form as eq. C-12 for AE^ plus “ C^) (T^ “

added on the right to account for product loss.

As Speros [183] has shown, the integral of from t^ to t^ is

equal to Q(T ) entry le, if T is defined by entry If of table C-4. The

value f) is given by entry If. Ordinarily, the heat capacity term is

small in comparison to Q(T ) so that one is required to solve Ic, Id, and

R

If iteratively.

General practice aims (or should aim) at insuring that the sample

and sample container are in much better thermal contact than the sample

container and sample holder. Thus, a somewhat better approximation to

entry la is to assume that T^ = T

2

, as do Heuvel and Lind [184]. The

corresponding entries for this method, 7’?2, are listed in entries 2a to 2h

of table C-4. To derive entry 2c one must derive entry 2h. Adding

entries 1 and 2 of table C-3 one obtains

C-13a

where

One uses: /L ndi^

T

2

T

2 - /QT2dn.

C.3-11
f C-13b

f

h. = + b- , + h,

Is Is lb 13

C-13c

T is the apparent temperature of the environment of the sample container.

Eliminating E.C.DT. + P, from eq. C-13a and using entry 9 of table C-3

gives the first line of entry 2h of table C-4. To obtain the second line

of entry 2h, suppose one carries out a new blank run in which neither

sample and sample container are present. Replacing C^, C

2

* ?£ entry 6,

table C-3, with zero and assuming DT = DT = w, etc., as in deriving eq.

s r C-10, one obtains

[Cg - - f

2

(C

3 + C^)]w = H f^S° + Ah(T^ - T^) C-14

where

^2 " ^2 ^^13^\3

H H - hib - 2h^3 -

Subtracting eq. C-14 from eq. C-10 and assuming f^ - f^ and H - H are

small yields

(Cl + C2)w = “K(S^ - S°) C-15

Substituting eq. C-15 into the first line of entry 2h yields the second

line of 2h. Differentiating the second line of entry 2h, assuming DT^ =

w, and substituting inco entry 9 of table C-3 gives entry 2c of table C-4.

•It is useful to note that a third blank riin in which no sample is

present and for which the instantaneous ordinate signal is S^ provides a

check on the assumption that C^ is independent of temperature if the sample container is closed. One can show

-C^w = -KfS - S ).

2 o e
Since no corresponding measurement of is possible, the assumption

that both and are constant is still necessary in table C-4.

The second method requires that one not only determine K over the

temperature range of interest but also determine h^^ for the sample

f container. Usually, h is sufficiently large and the experiment is run

Xw

I so that the correction to T is small. In this case, h, , for disposable

s Is

containers of nearly uniform type, can be determined from the melting

point-enthalpy of fusion calibration runs of metals used to determine

and to calibrate K as a function of T . The method is summarized later.

s To obtain some feeling from the nature of the errors in method 2, it

is useful to tabulate the measurement equations when the temperature of

the sample materials, container, and holder are each assumed to be uniform

but unequal. These are given in table C-4 under method 3. The addition

to entry 3b is C^DCT^-T^) where l2”^i given in entry 3h. It is evident

I r that for a given (C.DT./h^ + T^-T ) that T„-T, is made smaller as h, /K

i 1 is 1 S L 1 is IZ

decreases. Another way of examining entry 3h is to note that

given by

T -T

2 ^1

?2 - C2DT2 V(Qdn/dt - C^dT^/dt)

Aa

12 o

Q and c^ are the heat of reaction and heat capacity, respectively, per

unit volume of sample. V is the volume of sample and A is the area of

contact of sample and sample container having a heat transfer . coefficient

per unit area of a^. Assuming Q and c^ are constant and that the sample

is a thin layer of thickness 2r and area A, T^-T^ can be minimized by

reducing 2r/a^. This is done in the proposed ASTM E-27, test [153] by

sandwiching a thin layer of sample between the container bottom and lid

C.3-13
so V/Aa equals r/a . Since T^-T is given by

o o is

T,-T

1 s Is

C^DT^ + C2DT2

Is

» reducing h, increases T--T which reduces temperature gradients in the

Is is

sample as discussed in Appendices B.6., B.IO., and C.l. The main problem

in adopting the procedure of reducing h^^ is that the inaccuracy with

I which h, is known must correspondingly decrease. Preferably a procedure

Is

f

,

' must be devised to determine h, of each sample container with the sample

Is

in place prior to the decomposition run.

Method 3 of table C-4 clearly cannot be used to evaluate T

2

unless

h

^2

known during the decomposition. There is no method for determining

h

^2

S under these conditions.

C . 3 . 3

Measurement Equations for DSC

The instrument used in our experimental work is a commercial unit

made by Perkin-Elmer Corporation, Model DSC-2. It consists of two separate

calorimeters'^ which are mounted on supporting tubes in separate cylindrical cavities in an aluminum block, E of figure 4 (reproduced here for

convenience). The instrument differs from the Boersema type (heated

block) of twin scanning calorimeter of figure 3 in that the block is

thermostatted at or near room temperature while the calorimeters are

operated at the temperature or in the temperature range of interest. In

other words, if the temperature of the instrument is set at 500 K, the

temperature difference between either calorimeter and its immediate

environment is of the order of 500-300 = 200 K. This temperature difference

"^The sample calorimeter is sample, sample container, and sample holder.
produces larger temperature gradients in both calorimeters than would be

present if the block temperature was nominally equal to that of the calorimeters. However, these temperature gradients can automatically be taken

into account to a good first approximation by calibrating the energy and

temperature scales of the instrviment using the same thermal geometry inside the calorimeter (i.e. same sample containers, same holder lids, etc)

for both the calibration and "unknown" experiments'*" . This first design

feature permits the calorimeters to be cooled, other things being equal,

more rapidly than in the Boersema design (i.e. the latter requires cooling

of the block also.)

The second difference in the design of this instrument from the typical commercial unit based on the Boersema design is that the difference

between the sample holder temperature, T , and reference holder temperature,

3

T^, which are sensed by the platinum resistors and respectively,

is not the ordinate output signal, S, of the calorimeter. Instead,

electrical power is supplied to the sample and reference calorimeter

heaters, and R^, respectively, to keep T^ and T^ nominally equal. S is

proportional to the difference between the power supplied to the calorimeters to maintain T^ and T^ nominally equal. An additional increment

of electrical power is supplied to both calorimeters to keep T^ and T^ .

nominally equal to the program temperature, T^, of the instrument.

To accomplish this type of operation when an exothermic reaction

takes place in the sample calorimeter requires some way of withdrawing

heat rapidly from this calorimeter. The large temperature difference

takes care of this"*"*".

"**This is an intuitive observation based on the study cited in Appendix C.4.

Rapid cooling is achieved in some heated block (Boersema type) calorimeters by using a Peletier cooler.

C.3-15
The particular features of the design of the instriiment that are relevant to how an experiment is carried out are as follows. Referring to

figure 4, each calorimeter holder consists of a cup, or R^, with a

platinum lid, or R^, that can be removed to insert samples, 2 and 4,

enclosed by their containers, 1 and 3, into the sample and reference

holders. The heater and temperature sensor of each holder are sandwiched

between metal plates that form the bottom of the cup of each holder.

According to the patent description [174], these plates, the wall of the

cup, and the tube supporting the calorimeter are made of Pt-20% Ir which

are electron-beam welded into a single iinit. The heater and sensor consist

of identical resistors made by winding (nonbifilarly) 0.005 cm diameter

platinum wire on a thin rectangular chip of alumina. The wire is insulated

from the adjacent metal plates by thin alumina cloth. Thermal contact of

the resistor wires with the cup occurs through the cloth. Possible air

gaps between the cloth and wire or cloth and plates, which would reduce

thermal contact of the wires with the cup, is avoided by welding the

plates in position after the sensor and heater have been compressed between

the plates, thus, the construction reduces the temperature gradients in

the cup and insures that the temperature of the sensor and heater are, near

that of the cup. The use of platinum, platinum iridium alloy, and alumina

produce an oxidation resistant, physically stable (i.e. nearly uniform

coefficient of expansion) calorimeter.

In the usual experiment, the sample container is in good thermal

contact with the sample, and, also, it is in much better thermal contact

with the calorimeter cup than the calorimeter lid. Thus, to a first

approximation, sample, sample container, calorimeter cup, calorimeter

heater, and calorimeter temperature sensor all are near the same temperature.

C.3-16
Figure C-3. Simplified Sketch of DSC-2 Output Circuitry

a, heaters connected to average amplifier circuit; d, heaters connected to

difference amplifier circuit; f 3^, proportionality factor of average temperature detection circuit; fp, adjustable parameter of program temperature

source; fd, proportionality factor of difference temperature detection

circuit; f^v» Proportionality factor of energy (ordinate) output circuit;

G^, gain of average amplifier circuit; G^, gain of difference amplifier

circuit, 2 Iq, magnitude of constant current source; R^, heater resistance;

(R, ) , reference holder heater; S^

^ gauged heater switches; (S^) sample

holder heater; T , program temperature; T , adjustable parameter of program temperature^source; T , sample holder temperature; T , reference

holder temperature; AV, dixference cirucit voltage sourcef V , average

circuit voltage source. ^

C.3-17
An understanding of how the compensation circuitry works can be developed by first considering the operation of the output power circuitry

and neglecting details of the input or temperature detection circuitry.

Combining the appropriate parts of refs. [154, 175, 178], one can view the

operation of the output circuitry as the continuous, simultaneous switching

of the heaters and of figure C-3 alternately to the power output of

an averaging circuit (e.g. ganged switches and up as shown) and to

the power output of a difference circuit. 21^ is a constant D.C. current

source while V and AV are D.C. voltage sources whose magnitudes are

A

represented by the expressions in the appropriate "input” rectangles.

f^^V is the ordinate output signal of the instrument.

The averaging circuit detects a signal proportional to the average of

T and T,f(T +T)/2, and subtracts it from a signal proportional to

S 1C ^ S IT the program temperature T . The net signal is linearly amplified by a

P

factor G and supplied as voltage across both heaters to minimize the

A

difference between program and average holder temperatures. The corresponding time average power, P^, suppplied to each holder by the average

circuit, assuming V is an ideal source and the resistance of each heater

A

equals R^, is given by entry 1 of table C-5. The parameters f^ and T^ of

the input signal are adjusted in calibrating the temperature scale while

the parameter f is, in effect, controlled to be independent of temperature.

The difference circit detects a signal proportional to T^-T^ that is

linearly amplified by a factor and appears as a voltage in series with

the holder heaters to keep the difference in holder temperatures to a minimum.

C.3-18
Table C-5. Equations or Simplified DSC Measurement Analysis

Average Amplifier Power

2

IV. T + T

“

' A

-T -f^( =- A p p o a Z -)1

Difference Amplifier Power

Sample holder power; Pg -

?jj/2

Reference holder power; Pg + h W* h

Energy balance for sample container

a. sample; C

2DT2

*'21 ^^1*^

2

^ ^ ^2

b. container; C^DT^ - \2

^’^ 2

"’’^ 1

^ ^Is^’^s"^!^

c. holder lid; C^DT^^ -

d. holder cup; C^DT. - ^ " f ^

s s si s 1 B 2 D

Energy balance; sample calorimeter

=l“l - ^

-

’>'s* h*h-l ’di - 1 . 2 ,s,Jl

^ergy balance; reference calorimeter

£. C^DT, - Q; * P4 * Pb * i Pd= ^

1 - 3,4,r,m

Energy measurement equation

C,DT, - =l”l - ^

*

'’a-’r - S - K - ^

O^AV

i-l,2,s,Z 3,4,r,m

Energy equation for blank run with decomposition products in sample container

t^lCi + 4

- + Q3 -

1 “ 3(1(1 r(ffl(3(4

Energy measurement equation for decomposition run minus blank with products

Ei C^DT^ - + C^)v -

?2

“ -^CS-S^); DT^ (decomp.) = DT^ (blank) = w; 1(2 1 8(r(3(4,l,m

Q^-<5^ same in decomp, and blank

C.3-19
The corresponding time average power, (P + P_)/2, supplied to each holder

by the difference circuit is given in entry 2 of table C-5. It is assumed

both AV and 21 are ideal sources and again that the resistances of the

o heaters are equal to R^. The parameter f^ of the input signal is temperature

dependent

The basic measurement equations are obtained as in the QDTA instrument

analysis. The energy balance equations for the sample, sample container,

sample holder lid, and sample holder cup are given in entries 3a, 3b, 3c,

and 3d, respectively. and T^ are the heat capacity and temperature

respectively, of the sample holder lid. h^^^ and h^^ are the heat transfer

coefficients between the lid and sample holder and the lid and the sample

holder cup, respectively. is the rate of heat loss to the block from

the lid. The subscript s is used to denote the sample holder cup; -Q

s is the rate of heat loss to the block from the cup. Definitions of the

other heat capacities and heat transfer coefficients are the same as in

table C-3 for the QDTA instrument.

The total energy balance for the entire sample holder plus contents

is given in entry 4. The corresponding energy balance for the reference

holder is given in entry 5. The subscripts m and r refer to the sample

holder lid and cup, respectively.

Subtracting entry 5 from entry 4 yields entry 6. Entry 6 of table

C-5 is formally identical with entry 6 of table C-3 for the QDTA instrument.

Cssrr DT , C DT , f T , f _ , and Hf, of the QDTA equation are replaced by C DT +

l 3 d ss

CgDTo, C DT + C DT , 1, 1, and K, respectively. In the expression for

x> X. r r mm

the calibration constant, K, of the DSC, f^^ represents the gain of the

electrical circuit. Since both I and f.„ are constant, K, in this

o AV
instrument is independent of the program temperature T^. Subtracting the

energy equation for a blank run with sample products in the sample container, entry 7, from entry 6 yields entry 8 which formally corresponds to

entry 9 in table C-3.

• I *1 The assumption that DT. and Q -Q are the same for the decomposition

Jv S IT

• f • • • and blank runs in entry 9 are interrelated since Qg ~ Qg

depends upon T^, the temperature of the sample holder lid. Qualitatively,

the assumption that DT,^ and are the same in both runs requires that the

rate of heat transfer between ,the holder lid and sample container be much

less than the total rate of heat transfer between the holder lid and the

holder cup and the holder lid and the block.

The equations in table C-4 apply equally well to the DSC instrument.

f » The only essential differences are that now T^, and the energy equation

for S° (see columns 2 and 3) are;

h, +

Is ‘IJI

' IJl T = T + -4^ (Tg-T ) s s ,

' 2, s

h.

Is

[Z.C. - Z.C. ]w = -K(S-S°) +

q' -

q’ 11 11 s r s,£ r,m, 3,4

C-16

C-16b

C-16c

As for the QDTA instrument, eq. 6 of table C—5 and the equations of table

C-4 apply only after starting transients have decayed out.

C.3.4. Transfer Functions and Temperature Calibration

The relation between S (instantaneous signal) and 0 (observed signal)

for the DSC instrument was first investigated by Flynn [190] who measured

the time response of 0 caused by exposing the empty sample holder cups of

an earlier model to infrared heat pulses. It was found that

C.3-21
s (1 + T,D)0

d

C-17a

is the longest time constant of the two holder cups and lids, holder

heaters and difference power circuit to the extent that the rise time of

caused by the pulses is small in comparison to T^. (An assumption of

this type or appropriate modeling is always required to relate the known

power input to P

2 O Another method is to determine the time behaviour of

0 caused by the very rapid rate of liberation of heat caused by the

freezing of a supercooled liquid metal [191, p. 33-39]. The assumption is

that S due to freezing has a rise time much less than T,. The preferred

d

direct method of determining the ordinate transfer function is to determine the time response of 0 due to electrical heating.

For calorimeters having considerably greater stability and precision

than our instriiment, an expression of the form

S = (1 + E^T^D^)0 C-17b

is used. The more detailed experimental procedures and analyses required

for eq. C-17b are discussed in references [192-196]. Eqxiation C-17b does

not seem to be warranted for the relatively slow changes in 0 encountered

in our decompositions run in the scanning mode and the present uncertainties in realizing assumptions of procedures or commercial instruments

discussed here. In addition, the methods seem inappropriate because

•electrical calibration experiments would be very difficult for the commercial apparati considered here and also modeling would be required since

the thermal time constant of the electrical calibration heater system

could not be neglected in comparison to x,.

d
The instantaneous abscissa signal of the DSC and QDTA instruments

described here are proportional to T and T , respectively. For the DSC,

p r

I following Flynn [189], we assume is related to in a scanning experiment by

T - T

s p

AT

STAT

T W

a C-18

where AT„„.„ is the static difference between T - T when the calorimeter

STAT s p

is operated in the isothermal mode (w = 0) . T is the longest time constant

of the holder cups and lids, holder heaters, and the averaging power

circuit.

AT can be determined directly [185] by determining the smallest

^ XaI. nominal temperature, T (t,), for which a melting or transition temperature

P 1 standard begins to absorb heat if the temperature is stepped to T (t ) P ^ where T^Ct^) - Tp(t^^) is the resolution of the programmer of the instrument (0.1 K for the DSC- 2).

AT„ T + T, , and h^ (needed to evaluate entry 2h of table C-4)

can be determined from variation of the apparent melting or transition

point, the so-called extrapolated onset temperature, of the material as a

function of scanning rate. The extrapolated onset temperature, T^^(w), is

constructed from a sketch of the observed output signal, 0, from an Indium

melting point standard in figure C-4. The argument used to interpret the

variation of is as follows. Suppose first that the sample both

before and during the transition is in good thermal contact with its

container so that T

2 = T^^. Then entry 2h of table C-4 will apply provided

we replace S by S^ and C^ with C^:

T^ = T = T +

2 1s

Is

i i i ^1 ^ ^2

ts-s^l - t‘w;

Is

C-19

C.3-23
o g

0) UHOGO

CM 4J < a

•H

4J

O 0) O. TJ

C

W

O 0) •H (0

(d

(d 0) n o

43

•H c <u (d

CM c u o 4J M O cw

•H -H 0) T3 o • U 4-t iH (U 3 -H e* M

Oi (fl 0 •• cd 3 a (3 O (U • ** o 4J •H cd O C 0) Cd •rl rH M o u (U iH rH 0)

4-1 >-l (U (• O CD • <u * o 0 cd 00 u d) 43 •H >4-1 4-4 iH 60

•H

a

)-l O »• 44 cd 0 M -H pC4 M 0) o Cd

cd 4J 0)

u 0 -H • CO 60

•H 0) /-stH cm

O

M 6 S o o l-i P. c &

CO

C 1 3

•H

O

M

M rH

<u H*4-icM 4-4 3 3 O 60 U 3 S O 3 3 tH

Vw/ 4J -J- •H 4-1 0

iH c 3 M

O 3 O >s-H O 3 3 3 H H CJ 3 rH

60

O <! « •H 60 cd CO CO

C CO C 0) .H 3 H

•H

C cd 3

(U 0 -H 3 O •«

4J 3 rH cr <4-1 «4HiH

cd (0 (1) o 3 O C CO *3 43 ^ •H cd 3 r CO TJ 4J<4H O 3 O -3 M C 3 CJ V4 3 O O rH 3 r Cd

4-1 •rH O 3 4J Pi iH 3 O ' *3 •H O 3 C.) V4 „ ^ 3 3 V4 -d- 3 U 3 4J •> P rH <4H M 4-4 X CO 0 3 O 3 3 m 3 •H •> 3 3 O 4H 3 w a 4-4 60 3

•H 3 o •H

O

4J > 3 V4

M rHM •H

•rH 3 3 4-4 X 3 3 + 3 4J 3 3 u 3 3 •H 3 rH 3 •H MtH cm <4H P O

.H

OU

0) 43

(U

(0

cd

a

(U P3 taiS 42

O

cd cdC

(3, CdM

4JX

(U

rio? 4J

_ 0) H 43 3 44 .« •3 W r •H •3 44 3 O 0 3 3 3 P "3 C 3 •H O <4H 3 3 3 O

<4H 3 43 CMH 3 •3 O 44 3 3 3 3

>4H

O 3 3 > o

43O ’3

M cr 4 -H

O 3 3 3 44 3 44

44 3 3 44 3 3 U 3 60 <4H - 42 42 3

4<i •3 <4H 3tH

O

O 3

CO 3 O

u 3 1<4H 3 •a (J) • 60 3 q

•« •rH

O 44

•d- 3 > rH 3 1 •H M •r iH

CJ Td 3 vO 44 44 3 U •• 00 43 3 •> 3 3 <U tH 60 3

*H >*» iH •—

•rH 44 > 3 cd p 3 3 '•^ 60 »» CJ 0 3 3 3

•rH P de 3 3 3 3

443 O

O

O

H

C.3-24
Inserting eqs. C-17a and C-18 into eq. C-19 gives

T, = T + AT... _ -T w + -T- a + T ,D) (O-o’-) - T^w C-20a

2 p STAX a , a s

ti.Is

Just after melting starts K(O-O^) is observed to decrease linearly with

time at a rate a so

K(O-O^) = -aCt -T (w)]; T > T (w)

^

- p on p on

Inserting the last relation with eq. C-20a gives

‘^^STAT - '"a

- tl - T„^(w)l - T >

‘'is ‘'is

Assuming is constant and equal to the transition or fusion temperature,

T , for T > T (w) , one has from the condition that DT» = 0 that a equals

tr’ p — on 2

^

f h^g. Inserting this value for a in eq. C-20b and rearranging we have the

results

a = h

Is

= ^tr

- ^^STAT

T = T + T, + T

o ads

C-21a

C-21b

C-21c

Thus, h^^w is the slope of the leading edge of 0-0 divided by K, AT

STAT

equals T^ -T (0) , and the slope of a plot of T (w) versus w should be

tr on ’

- on

constant and equal to T ads + i, + T . These results were first cited by^ Flynn [189], Experimentally, it is found that for very pure metal melting point standards (e.g. see [187, 188, 189]), the slope of does

vary linearly with w and the value of seems to agree within a few

STAT

tenths of a Kelvin with the direct method cited above.

C.3-25

C-20b

i
The value of can be eliminated by determining for a cooling

run (see [189]) or by determining the decay constant, of the observed

s decay curve of the endotherm. Since for = T

2

is

.f

<=1 + =2

Is

C-21d

Tg equals x^ if the usual situation in practice.

Another method for determining x is given in the next section.

To evaluate the temperature in a decomposition run one uses eq. 20a

with x^ replaced by x^ but now is the heat capacity of the decomposition

products and 0^ replaced by 0

T2 - T, = Tp + ATg^^^ - (X^ + X^)w + V C-22a

(1 + X^D) (0-0^)

^Is

C-22b

V is called the correction for self heating. In our decomposition runs.

To is calculated for the value of T when 0-0 is a maximum or when D(0-

2 p o

0^) =* 0. Inserting this condition into eq. C-?2b yields eq. 16 of the

text since in our decomposition runs « C^.

It is pertinent to note that if sample containers of the same type

and weight are used in both the decomposition runs and the dynamic

t temperature calibration procedure discussed above (so that and h^^

are the same in both types of run) , some economy in determining the

corrections on the right side of eq. C-22a can be achieved. If

« in the temperature runs, from eq. C-21b can be inserted in

eq. C-22a to obtain

T

2 - T^ - T + [T^^ - T^^(w)] + X^w + V C-22c

C.3-26
Thus, if T^w is small in comparison to the other corrections, T^-T^-V

is the value of T -T (w) at T . This value is taken from a plot of

tr on p

T -T (w) versus the nominal temperature constructed from the temperature

tr on calibration runs for the w corresponding to that of the decomposition

run. This procedure is essentially that of ASTM method E 474 which is a

part of the proposed ASTM Kinetic Method.

C.3.5. Determination of K

Calibration of both instruments is essentially the same. The details

will be different since K for the DSC-2 does not vary more than a percent

or so with temperature while it does for the QDTA instrument.

As discussed by Flynn [185], the value of K can be determined independent of time constant considerations by determining the time integral of

0 from one constant nominal temperature, T^, to another temperature T^

of the sample holder for a material whose enthalpy increment, 112(12 ) -

HzCT^) is known. If one subtracts the corresponding time integral of a

blank signal, 0^, when only the container is in the sample holder, one

obtains 'from entry 6 of table C-5 and eq. C-17a

tf . . t

/ [C2DT2“P2]dt = H2(T2) - H2(T2) = -K/^ (0-6 )dt C-23a

T2 “

'^p ^"^STAT’ ^ - C-23b

It is assumed that the baseline has been adjusted so 0^ - 0^ + 0^)

e e is zero. Also, T should be varied between T^ and T^ in the same wav for

p P P

^ both runs so that Q -Q for blank and sample runs cancel . Operation of

It is also assumed that the integral of the starting and stopping tran—

si®^ts not shown in entry 6 of table G—5 cancel between measurement and

blank runs.

C.3-27
the instrument in this way is called the step jump procedure. Both heat

capacity standards (e.g. sapphire) or materials whose heat of fusion or

transition, used to determine K. For a heat capacity

calibration

AH, = - H,(Tb + A C-23c

2 2 p 2 p

while for a fusion or transition run

AH., = AH + cj(r -T^) + C^(T^-T^ ) + A C-23d

2 tr 2 tr p 2 p tr

where

A =

^2^'^STAT ”^2^'^STAT

C-23e

ATgijAT known when A is small in comparison to the nominal enthalpy

increment. Neither or need be known for fusion or transition

Ic £ i. measurements if C

2 (Tp-Tp), k = i,f, is small in comparison to

K can also be determined independent of time constant considerations

and the temperature scale calibration by using fusion or transition

standards and running the container with sample and empty container in

the scanning mode through the transition. As outlined by Flynn [185]

and Gutmann and Flynn [186], the times t^ and t^ of eq. C-23a refer,

respectively, to times before and after the transition in which the material

has reached a quasi-steady state. The added assumption is that the temperature corrections due to running in the scanning mode of the sample

container and the sample in both runs at times t^ and t^ (and only these

times) contribute negligible error in evaluating eq. C-23b. This occurs if i

if is arranged to be much greater than or C^, or if nearly equals

C^. The alternative procedure is to evaluate the cross hatched area of

figure C-4. As Gutmann and Flynn [186] have shown, this area equals, AH^.^

if the onset of the transition occurs at T (w)"^.

on

T

This can be derived using eqs. C-19, C-17a.

C.3-28
An alternative method for determining that does not depend on

I the dynamics of a transition or fusion has been developed by Richardson

j et al [187, 188]. They determined the temperature lag of different

T weights of sapphire (heat capacity standard) from the area under the

total area of a 0-0 versus T curve and the area under the stopping

e p

I. transient when heat capacities are measured by the step jump procedure

* outlined previously. No container is used, material is in the form of

I rods, and it is arranged that h^^ is as large and constant as possible

for each sample weight. T^w is obtained by extrapolating the temperature

I lag of the various sample weights at the same w to zero sample weight.

This method gives results that are consistent with the dynamic method

j cited in section C.3.4.

I

I

I

i

I

C.3-29
V;-<s' ”'i»o bndrgAb

®Vf-.

'

'

' ', I

, ^•:

, S! ,,i . i \‘'ff*3

[Aj»" c*M!»t{>'>'.stNuii^|ift lo

W'i j

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.

•, . rik

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,i

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' “’*

' "SI

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' ^T^laawaBi: "t?' Kii?:

..~;4li . .. m Vig^®vTM^ #JP wS<r’^ . ..'Zfc > 1- N iipjTf- pi

io.'ino'i '»tiCa ,' '^xln^Oktya

g

kX ^ ':<!it9Til'«m «1 4i &fl* /aSq^j

. j^sidv., 03j»4s v 9d4f" . »w<vHav.- #rfa ' If |•X,

LodasJ'ffl ;©Jfcw.i^^ is
C.4. Instrument Calibration

C.4.1. Main Experimental Results

The temperature scale of the DSC-2 instrument was calibrated with

solid holder covers in place using disposable unsealed crimped containers

(Perkin-Elmer type 0041) containing melting point standards and using

either the normal block lid (see Ef, of Fig. 4) or a new type of vented

block lid (supplied by the manufacturer) which reduces the effect of deposition of decomposition products on the lid upon the operation of the

calorimeter. The vented block was obtained after the polyacrylonitrile

studies were completed. A summary of the temperature scale calibrations

and time constant studies using the dynamic method discussed in appendix

C.3.4. is given in table C-6 and the accompanying key. The vented block

was involved in experiments having condition B or C. onset

temperature of melting, was determined as specified in fig. C-4 from a

chart record with correction for the lag between the pens indicating 0,

the signal output, and T^, the programmed temperature. A typical plot

of T^j^(w) versus w is given in fig. C-5; the estimated imprecision in

T^^(O) is about 0.02 K. Graphs of given in fig. C-6. The Pb

and Zn corrections, expt. 8b and expt. 48, are unrealiable and were not

used. A better calibration curve than fig. C-6 is clearly desirable.

The difference in T^j^(O) between experiments 11a and 11b and between

12a and 12b are due to removing deposits of decomposition products of

polyacrylonitrile on the normal block lid after experiment a of each

pair of experiments. It is evident that T (0) seems to be stable to on + 0.02 K for condition A so that the difference between condition A and

B of 0.1 K is real. The difference in due to the output filter of

about 0.016 min (i.e. compare experiments 1, 2a with 11a, 11b and experiment 24 with 25) agrees with the longest time constant of the two pole

C.4-1

C.4-3

«L 5*-

il$

Cl* 4«

0-0,

-t^ ol

O CVJ ±J

£ ^

3*

Figure

C-6.

Plot

of

Static

Temperature

Corrections,

AT

,

versus

Program

Temperature,

u1

A1
Table

C-6i

Temperature

Calibration

and

Time

Constant

Study

C.4-4

Table C-6a. Key to Table C-6

Conditions

A; Normal block lid, 20 psi (input) N

2 flushing gas, solid sample

holder cover and specific settings of the temperature calibration controls (temperature zero, temperature range, and AT

balance) determined procedures specified in instrument manual

(supplied January (1977)).

o: Ordinate signal output filter bypassed (i.e. + 50 mV signal of

section 9H of manual)

f: Ordinate signal filter in (i.e. + 10 mV signal of section 9H

of manual)

B: Same as A except vented block lid, 30 psi (input) flushing

gas.

®

2

’ ^

3

*

Same as B except the following are placed between melting

point standard (in its container) and sample holder: B- pyrex

covers lip, B^ inverted empty container bottom (type 004l)

,

and B^ preceding plus coverslip.

C: Same as B but new temperature calibration controls settings.

D; Same as C except Indium sample in stated container (type 0062)

as described in proposed ASTM test [155]; see also [194] fig.

2b or [195], fig. 3.

Sample formula (No . )

No. indicates various preparations of standards supplied by

instrviment manufacturer except for Sn(2) and SiO_ which are RM

758 supplied by NBS and ICTA and Sn(4) which is 0.999999 pure

tin supplied by OSRM at NBS.

T

on

( 0):

T^^(w) extrapolated to w = 0.

T : slope of a plot of T (w) versus w.

H: H is slope of leading edge of melting isotherm, see fig. C-4,

section C.3.4. H is nominally h^^ of appendix C.3.4.

Tg(calc. )

(C. + C

2

)/H; is a measured value and C- is taken from

[196]; see later summary of measurements of
Tg(neas.)

;

Slope of iln(O-O^) versus time of decay curve (i.e. A

2

, fig*

C-4, section C.3.4.)

T + T,: T -T (meas)

;

a d o s

T w is the lag of the apparent sample holder temperature

behind the nominal program temperature, T . is the longest

time constant of instrument's holders, ho£der heaters, and

difference power system (see Appendices B.3.3. and C.3.4.)

AT

STAT* T(transition or fusion) - T (0)

on

The temperature correction added to T to obtain the temperature

of the sample container in the isothermal mode of operation of

the instrument, see Appendix C.3.4.

C.4-7
-I-

' output filter . The assumption that H is- equal to h^^ or, equivalently,

that the sample and sample container temperatures are equal during the

initial stages of melting was checked by comparing the calculated value

of T , T (cald.) (see key) with the time constant of the final part of

s s decay curve of the melting endotherm, T^(meas.)* These two time constants

t should be equal if H = h^^ as is discussed later in appendix C.4.2.

With the exception of experiment 28 (where T (meas) decreases rather

s than increases as 0 approaches 0^ and experiment 14b, the agreement is

reasonable. The value of T + T, seem roughly constant for a given

condition (A, B, C, or D) and nearly independent of T^.

T, was measured by the analysis of a cooling curve produced by a d

partially supercooled exotherm of indium assuming the initial part of

the exotherm is given by the dotted line of the insert in figure C-7 (see

[191], section e) . The initial part of observed exotherm does not fit

the model as can be seen from figure C-7 but it seems reasonable to

conclude that T, is 0.5 sec or less. The resulting value of T of about

d a 0.06 min, obtained by subtracting T, from T + T,, is in agreement with

d ad

that determined by Richardson, et al (see [188], fig. 1, the slope of

their line is 0.06 min) using the heat capacity lag method summarized

in Appendix C . 3 . 5

Some preliminary measurements aimed at determining the extent to

which the sample holder cup and lid depart from temperature uniformity

were made in experiments 20, 21a, and 21b (see key, table C-6a) . The

inverted uncrimpled sample container raises the Indium melting standard

'^’'pie filtered output signal, 0, is related to the filter input signal,

0 , by (0.35 sec^D^ +1.4 secD +1) 0=0*.
:l 9

cO 4J Xi (P i + COCO /^S 9

00 f—. c •o * H V N c

(U 4J

<U 1 > a M- 9.X 9*

r— (U 9

CO

•P- 1 9

-C\ •n iH C c

-H < 4J

II 9

<u 9 e > >% • 4J 9 >-l c C

•H (U CO o 9

TJ CO -H *H 9. C u X M 0 «% c.) 9 9

T3 o S-i CO

0) 4J • 4J iH <H CD O <u CO 9 9 O > 9 O H O •H 00 CJ 4J •H 9

(U CO CO <U 9

o. iH 9 VI 9 CU 4J -H 4-1 CO h 9 iH

O. 9

<4-1 (U 4-1 O S 9 (U

H •H O 4J 00 4-1 4-1 c 0) O • •H CO 4J "O 9 N •H CO H

<U 9 a

(U 4J •H

O iH

u •9 >4

<UM <4-1 CO V4 O •H

<4-1 (U •9 o JS 9 9 9 9 4J 9 s 1^ cO O

4J 9 9 I—(

<a eaU rH cn JS <—1 cO 4J + •H > o <4-1 9 • X •> 9 4J M < 9 9

>-i CO e CO <4-1 CO 9 o •H O CO CP Vi

<4-1 ' 4J 9 9 9 CO -9 >

•9

a 9 9

,9

O CO r—l •H 9. < <4-1 CQ -43 • o 9 -—\

4J

<U Vi 4J o

(U • CQ r—(

4J o T3 ' 9

9.

nj 19. 9 h* 1 S a 3 9

•H T3 9 >N ^ <U CO 1 4J

CQ

a CO 1 9 <5 - <4-1 CO <3 o • CO 9

<3 • CO 00 9 1 a *H O H

CJ 9

•• •H cO

a

<u 4J 9 9 M )-l 9 9 «• a 9 (U •H Vi 4J •H 00 CO 9 o 9

•H 9 <4-1 J3 r—l •H

PC4

M

O 5 CU

a ii

C.4-9 t
in its container to within about 0.03 cm of the sample holder lid. From

the discussion of Appendix C.3.4., one should expect to decrease

or T (0) to increase in proceeding from condition B or B, to conditions

B„, B« if the holder lid temperature, T. , is less than the holder cup

2 3 ^ temperature, T . These changes are observed. T -T^ ^ 1 K. In spite of

s s ^ this the value of the calibration constant, K, for expts. 21a and 21b

discussed below do not differ significantly from the other values of K

for conditions A, B, or C. Thus, for normal thermal geometry (i.e. conditions A, B, C) it seems reasonable to conclude that effects of temperature

gradients can certainly be taken in account by normal calibration procedures as was suggested in Appendix C.3.3.

A summary of energy scale calibrations using the single sample plus

container run and base line analysis of ref [186] mentioned in Appendix

C.3.5. for indium and tin fusions is given in table C-7. To confirm

that K should be independent of temperature, a heat capacity calibration

using SRM 720 Al^O^ was carried out using the step jump procedure outlined

in Appendix C.3.5. Initial and final temperatures are listed under

"temperature range". No corrections were applied to these temperatures

(i.e. corrections are assumed to cancel"^). It is evident that

the variation in K with T is 1% or less. (Areas were calculated

P

numerically from the chart record using the trapezoidal rule with an .

uncertainty estimated to be of the order of 0.5%).

"**The uncertainty caused by this assumption is estimated to be of the

order of 0.5%.

C.4-10
Table C-7 Energy Calibration

Conditions

Expt . No. Conditions^ Sample mW/unit"*"*” K/min or 1

1 A.f In(l) .0726 10

.0731 .625

11b A,o In( 2) .0717 .3125

11b A,o In( 2 ) .0719 .3125

24 A.f Sn(2) .0725 10

26 A.o Sn(3) .0728 2.5

28 A.o Sn(4) .0725 1.25

29

30 A.o Al^^CD .0721 475.1

.0727 821.1

.0729 871.1

.0721 921.1

14a B.o In(2 ) .0709 5 .0718 .625

14b B.o Shd) .0717 5

16 B.o In .0719 5 .0722 0.625

46 C.o In(3) .0713 10

.0713 5

50 C.o Sn(3) .0710 5 .0724 1.25

54 D.o In(3) .0707 5 .0694 2.5

.0704 1.25

20 Bj^.o Ifi( 2 ) 0.0716 .625

21a B2.0 ln(2 ) 0.0737 .625

21b B3.0 ln( 2 ) 0.0730 .625

"*"See key to table 24.

++-.1 unit on chart recorder when setting of Instrument is ”1 mcal/sec". chart

0.1 mV full scale.

C.4-11

Range

to 525.1

to 871.1

to 921.1

to 971.1
The heat capacity of the aluminum containers was measured as a part

of two experiments on the decomposition of nitrocellulose using the

dynamic method described in ref. [199], p. 17, with the exception that a

heat capacity standard run was omitted. (w was assumed to be exact and

an average K from table C-6 was used) . Although experimental conditions

in experiments 14 and 16 were far from optimal, reasonably good agreement

of both experiments with the data for pure aluminum was obtained as

shown in fig. C-8.

C.4.2. Analysis of the Calibration Procedures

An analysis of the performance of the DSC apparatus was undertaken

to provide a basis for interpreting the calibrations and decomposition

experimental results. This is an extension of the treatment given in

Appendix C . 3

The operation of the input temperature detection circuit was analyzed.

Calculations indicated that, if the instrument is adjusted at the In and

K

2 CrO^ melting points as specified in the instrument manual,

should have the form of smooth, roughly paroblic, curve with values of

zero at 429 and 943 K and a maximum of 1 or 2 K midway between 429 and

943 K. The qualifications on this result are that the In and K

2 CrO^

melting points are assumed to be known on the IPTS scale to 0.1 K and

the instrument is adjusted so that is less than 0.1 K, at these

two melting points. In the calculations, the resistance-temperature

characteristics of the platinum resistance thermometers were assumed to

be those for strain-free platinum and the measured programmed variation

in the supply voltage of our DSC was used. The results of experiments

8b (Pb) and 50 (Zn) of table C—6 were not included in Fig. C-6 because

C.4-12
0)u3u

CO

0)o.s

(U3

COeoo

0433

•HEozo

‘Hoc3AouE3C

•HE3o

>. u

•HO

COO.3y3

<U

00I O3

}-i 3

00

•H

St * I o

CO

<t

CN

CN• o

II rH* I O. s U 00 <f iH

iH 3U

O.

O

X 00

Cd COI •>

aE-i

srI O o E X

00 CO

r—I • 3 r—I U0 <r

00 CN CO CN I • oo H

II St a. 1 CJ o

rH •> r—«r X

rH+ MH*

CU in OS CN ^ 0- +

O. E U 00 O CO rH CJ

HI

O

Or CO X CO

Cd I

C.4-13
of the abrupt shift that they would be caused in the plot of

versus T (AT„_._ for the tin melting point is reliable). This decision

p STAT

was also supported by indications of premelting due to impurities in the

Pb and Zn endotherms and the agreement of Si02» expt. 29,

with measurements of others to better than 1 K.

The energy measurement equations of the DSC instrument, entries A

and 5 of table C-5, were solved to obtain the approximate differential

equations satisfied by S and (T + T )/2. The aim was two fold: (1) to

S IT provide apriori support for and some idea of the limits of the eq. C-17a

fdr the ordinate transfer function and eq. C-18 for the lag of T^

behind T^ in a scanning run, and (2) to determine the conditions for the

assumption that DT^ is the same for a measurement and blank run which

reduces to DT = w. This equality for DT is used to derive all the

s s equations in table C-4 and appendices C.3.4. and C.3.5.

The assumptions used to simplify entries 4 and 5 of table C-5

for the initial part of the analysis are given in entry 1 of table C-8a.

The lid and cup of each holder are assumed to be uniform, entry la, and

the total heat capacity of each holder plus its contents is assumed to

• I • I be the same. Following O'Neill et al [176], Q and Q were approximated

S IT ^ by entry Ic. (0) , j = s,r, is the rate of absorption of either holder

I i at a constant initial temperature T.(0), j = s,r. Experimental measurements

• f i • show the error in assuming Q varies linearly with temperature is about

2% if 9 ^ 25 K. The resistance of the electrical heaters of the sample

and reference holders are assumed to be equal, and the manual temperature

calibration procedure discussed previously is assumed to, in effect,

make f = f of entry 1 of table C-5.

a p

'’d = d/dt
Table C-8a. Approximate Equations for DSC, Formulation of Equations

1. Assvunptions

a)

b)

c)

T» = T , T » T . Jl s* m r

= E.C. =C: C indep. of temp.

j j 2 2 s,il,l,2 r,m,3,4

• f =

Qj (0) - 5 j “ s,r; indep. of temp.

9, » T. - T. (0)

2 2 2

d) electrical heater resistance of each holder is

•Sie) temperature calibration adjusts f^ = f^, = 0

2. Energy Balance Equations

V (0)2

2

a) t - 0: 0 - Q (0) + + 1/2

I AV(0)

o

= -1, j = s; = +1, j = r

b) t > 0: CD0. = -h 0. + k (Y-0 -0 ) + N.k.^(9 -0 ) 2 0 2 a ^ 3 r' j jd^ s v'

j = s,r

r) k = —^ G f • k = — ^ a 2Rg a a* jd 2

d) P(t) = k’(Y-9^-9J^ +

^d^®s“®r^^

AV(0) G.f.N,; : 4R^ d d j

s r

“I

“

‘‘d =

£) Y = 2[Tp - T (0)]

g) W_ = P, - C, (DT.-DT ) - C.DT. - cioT ; W = - E.C

s 2 11s 2 2 2 s r i ] 3,4

; j = s,r

+ p(t) + ;

= s,r

(DT.-DT ) 1 r

C.4-15
Table C-8b. Approximate Equations for DSC, Solution of Entry 2, Table C-8a

1. Equation for S-S(O) = AS; AS= ” ^r^^AV^^c

a) (t’d + 1)(t'd + 1)KAS = (1 + t'd)(W -W ) ^ Q ^ S IT

b) =

C ^Jo

2k (1+r )’ a k

a a a

d) K =

2k, (1+r,)’ ^d k

’ ^d

“ ^^sd ^rd^'^^

d d a k ,+k , I

sd rd . , o ,, , , f,,,G,f, f.

AV d d AV

2. Equation for (0 + 0 ) when r « 1

S 3T ^

a) (t'd + 1)(t’d + 1)(0^ + 0 J = t'(1 + t'd)(W^ + W )/C +

a cl sraf dsrt k k

(1 + T^D)^ + a + T^D)

a ff-63-ep^ + ^a

b) Rea.rrangement of 2a when r^ « 1:

I f {tadT,D + [Tad+T,(1 + „V (0)

AV AV

)]D + [1 + 1/2 "

* ’ 1

’ VA^

+ T (1 + T D)(DY- -^) - ^

a + T D) {KAS [1 + -

. ^ Vo

T, D KAS};

a

] +

\

- ^A^0> = Va (^-®s-®r)/2

C.4-16
The static energy balance for the sample and reference holders, at

time t = 0, entry 2a, were subtracted from entries 4 and 5 of table C-5

to give a dynamic energy balance equation for each holder at time t > 0,

entry 2b. In entry 2a, V (0) and AV(0) are the static output voltages

of the average and difference amplifiers. The parameters R^, I , G , hi o a

G,,dadf and f , are defined in table C-5. The second through fourth terms

on the right side of entry 2b are the dynamic terms contributed by P^. +

P„ + 1/2 P^ of entries 4 and 5 of table C-5. -W and -W are the rates

B — D s r of increase of the enthalpy of sample and reference containers and their

contents minus the corresponding rates of increase assuming their temperatures equal T and the heat capacity of the sample does not change,

s (The value of 6 , is assumed to be small enough that C , Cn , C, , C , C , j ® s X, 1’ r m

C^, and C^ are constant.)

Using operator algebra, entry 2b of table C-8a can be rearranged to

give entries la and 2a of table C-8b. S is the ordinate output of the

instrument if the longest time constant of the difference amplifier,

ft T^-, is zero. S(0) is the value of S at t = 0. The relation between S

and 9 -8 follows from entries 2 and 6 of table C-5 and entry 2c of

table C-8a. Entry la of table C-8b corresponds to the DSC energy

measurement equation given in entry 6 of table C-5. Entry la indicates

' ^ the ordinate transfer function is given by

(T^D + 1) (T^D + 1)0 'V [1 +(T^ + T^)D]0

Entry 2a is the expression for the average temperature of the

sample and reference holders assuming 1* Using the definitions of

I t ^a* ^a* ^d ^d table C-8a, entry 2a of table C-8b can be rearranged
for r, « 1 to give entry 2b. The nonlinear entries on the left side

d

can be dropped since experimental measurements show AV /V (0) is of the — A A

order of .02 for 6 25 K. The nonlinear term in AS can be dropped if

KAS/8 is much less than the static power supplied by the constant current

generator of fig. C-3, which is usvially the case (e.g. at 355 K, 1/2

2

Rjj is 80 mW so KAS must be much less than 0.64 W) . Dropping the

nonlinear terms from entry 2b, one has the result that T leads (T +• ' p

• s

f T )/2 by T w + AT-_._. After starting transients have decayed out and

1C 3l ^ XaX no thermal event is occuring in a scanning run (i.e. AS and are zero).

In the DSC-2 an additional increment of constant power that is proportional

to w is supplied to each holder to null out the T w temperature log.

Si Also one has the result that the absolute fractional error, |dT -wI/W

s is less than [dkAS | /4kaW"*’. The error is small because k is large as

O'Neill [176] has pointed out.

If h^, C, and of the sample calorimeter exceed the corresponding

quantities of the reference calorimeter by the fractional amounts of 5^,

6 , and 5/2, the following terms must be added to the right side of

C R. entry lb of table C-8b if rad= r , = 0

-[t’cSo + + k'(Y-e_-e_)^

a' R s r

K is multiplied by 1-5/2 and both T and T, are multiplied by 1 + (5 +

R a Q c 5_)/2 if 5 and 5_ are small. The first line of the added terms contributes

R c R

"**The equivalent condition for the QDTA instrument is lDKS|/h ,w.

C.4-18
J,

.1

a constant term of 5 Cw due to the mismatch in heat capacities and a

I c term proportional to wt due to the mismatch in the dynamic rate of heat

I

loss and heater resistances of both calorimeters. These terms support

the idea that the linear extrapolation of base lines as shown in fig. C-4

correctly accounts for heat loss mismatch. The 6^Cw term is what one

i should expect and explains why the time constant for <5^Cw to appear in

the ordinate signal is nearer to than T^. Using an argument similar

I to that used to show that the nonlinear terms are small in entry 2b of

table C-8b, one can show that nonlinear terms in the second line of

added terms are smaller than those of entry 2b, table C-8b, by a factor

of 6^ + 6^ and hence can ordinarily be dropped.

It should be pointed out that if the temperature of the holder cups

:| is, as we showed, essentially independent of the effect of the lid

can be accounted for by adding the analogous terms into W . The assumps tion that the sample holder lid dT/dt = w, which is basic to the measureI ment equations, can be solved and compared with suitable experiments.

This has not been done as of yet.

j The major drawback to the model is that the time constants of the

I electrical heaters and resistance thermometers of the holders have not

I been explicitly taken into account. Physically, one can see that they

are in fact the main components of and T^. Introduction of the

' appropriate equations for these time constants produces a 6 x 6 matrix

that has not been reduced to a form whose meaning is simple to analyze

in physical terms. After this is done the limitations of the preceding

or zero th order model of the DSC will then be known in a experimentally

measurable form.

C.4-19
The form of W^(t) for ^ melting endotherm is of interest for calibret tion of the temperature scale, determination of h^^, and evaluation of

time constants. _If the restriction that the temperatures of the sample

and sample container be equal is removed it can be shovm that of

entry 2a and 2b of table C-8b is

W^(t) = wH (t-t^) - wA(l - exp[-(t-t^)/T^]) ; ^ C-24a

where

H

**12**ls

^izKs

C-24b

A = [Cj^r^ + C2<r* - rSa + r^]/t(l + r^)^l

I I - h. . h, Cf ^ Is ^ 1 ^

Is ^ _ 1 r“f;r=‘i;Xi , ^

*>^2 »ls

+ "12

C-24c

C-24d

The superscript i and f refer to before and after melting, respectively.

t is the time of melting (i.e. wt apart from a constant is the program

m m

temperature at the melting point) . The remaining symbols are as defined

I in Appendix C.3. One can see from eq. C-24b that is equal to h^^

only if r^ of eq. C-24d is negligible. From the analysis of samplelimited melting given by O’Neill [176], one concludes on physical grounds

that for pure metals the melting point endotherm,. should become

progressively more concave down as melting proceeds. wH^ is the initial

slope as t-t^ becomes small but still is greater than of eq. C-24d (if

A is not negligible). For a pure substance there should be a. relatively

sharp break in the output signal upon melting. (W =0 before melting and is

s given by eq. C-24a afterwards). Since is small any deviation from

this behavior should be due to impurities as Flynn has implied [190].
This was verified by observation of a sharp break (i.e. curvature of

break is consistent with T, < 0.5 sec) in the output endotherm for very

a pure tin in experiment 28 of table 11.

The values of T (calc.) and T (meas) of columns 7 and 8 of table C-6,

s s while agreeing within experimental uncertainty, should obey eq. C-25

if the melted sample and the sample container have different but uniform

temperatures. One can show that T (meas) and X (calc) "box” the value

s s

f

» of X , (0, + C_)/h. , that prevails if the temperature of sample and

s 1 2 Is

sample container are uniform.

C, + d (C + C2)(l + r^)

X (meas) < ; < j| x (calc)

s « « s c-25

Is Is

f

» If X (meas) = X (cald) , then r = 0, H = h^ , and the temperature of the

sample after it starts melting and the temperature of the sample container

are equal. It is evident from table C-6 that, experimentally, x^(cald)

is nearly equal but slightly greater than x (meas). One possible explanas tion is that the effect of x^ is influencing the decay curve (i.e., the

assumption that it does not may be in error). In any event, it seems

f

» reasonable that r is evidently negligible and H = h^^ as was mentioned

earlier

.

C.4-21
M

iXo'f’i- Ic' 8 'fi^ t afia»to'':.M^

«li' «ilfc'“ ’I

. \ '.*lg«' &HP

'

' ''*’’ "''® w. i"' '*

it,

'

bakMi' siS^e 4o ' Id

. .. _®. '.

.«^''V.'r, ‘'f.l eti.'»

)Tril s&aifi3xio9< »ii

T-’-'-fet,^^^-; » (»»»f);T ,;;^alig!

®^;# '

, ^'" '

' «!.. .A ' ,.

-. 1, , ii

' t. -»>*’ • -i

' ’ * i « ' .

••* '•

.. - - -\«£i

300'U'A1IO:J dliltidft Ii|i

f'

t'::^

*i'ii<fe ,...ir»i»5 arfj'&wjM

»"-.s,?ii'

. ,

.' . « A: .i*(|i.' .£5^,4?-..#^^^

.^bdwblanaa mo/w Mb U b

:bqod*f

.rji

i m.f-^

•fm

4'^

' -<®

' V- i?

- * ,;4fevi'^ m':

,ii"

' .”', _ ,3^'

. *4

It^’'

m
C. 5. Error Analysis for Four Assumptions

In this appendix we consider four assumptions of the ASTM method:

(1) that the value of g(h^) at the temperature, T^, of maximum rate is

known and does not vary with the nominal scanning rate, w, (2) that

deviations of the rate of change of temperature of the sample from w does

not cause g(h^) to vary with w or the temperature of the sample at

maximum rate, (3) that maximum in the observed ordinate signal corresponds

to the maximum in the rate of decomposition of sample, and (4) the rate of

change of the sample temperature at equals w. The third and fourth

assumptions are assumptions 1 and 2 of table 11 of the text and are discussed in the final section of this Appendix.

C . 5 . 1 . The Value of g(hj) at the Maximum dr|/dt

The basic kinetic equation is

dn/dt =» f(n)k(T) = f(n) A exp(-E/RT) (11)

The derivative with respect to time is set equal to zero at the maximum:

d^n/dt^ = 0 - (df/dn)^(dn/dt)^k(T^) + f (dk/dT)^(dT/dt)^

Substitution of w for dT/dt, and the expressions for k(T) and dh/dt yields

g(n^) = -(df(n)/dn)^ = (wE/RA)T^”^ exp(E/RT^) ^ C-26

The method assumes w, E and A are constant and g(n^) = 1 when A is calculated

using equation C-26.

For the general rate law for simple reactions,

f(n) = (1-n)^ (8)

= 1 for first order reactions, by definition. For other values of n,

g(rij) is nearly unity and almost independent of T,.d

Later the question of other rate laws is considered.

This is shown below.
To show the approximation equation 11 is integrated to obtain G(ri) a

readily calculable function and then the product g(h^)G(ri^) is reduced to

an approximate numerical form.

G(T)) 5

0

T

dn/f(n) = /

T

o

k^dT

dT/dt

where G can be written as

T

(A/w) f exp(-E/RT)dT

T

o

C-27a

G(n) = (AE/wR)[P(X)- P(X^)]

with

^ -2

P(X) 1 -/„X ^exp(-X)dX

-and

X = E/RT; X = E/RT

o o

An appropriate assymptotic expression for P(X) is [161]

P(X) = X”^ exp(-X)[l - 2/X + 6/X^ ]

C-27b

C-27c

C-27d

C-28

Combining C-26 and C-27b, with chosen to make P(X^) negligible gives

g(n^)G(n^) = X^exp(X^)PCX^) = l - 2/X^ + 6/X^ C-29

The product gG is evaluated using equation 8 to yield

g(rid)G(Tld) = (g(Tld) - n)/(l - n); n 7^ 1 C-30a

-^n(l-n,); n = 1 C-30b

u

Thus, from equations C-29 and C-30

g(n^); n = 1 ^ C-30b

where

0 = 2Cl-n)(l/X. - 3/Xj + ...); n 7

^ 1 C-31b

d d

0 = 0; n = 1 C-31c

C.5-2
Since X, is large (30 or more) g(rij) is near one, and the effect on A is

d n small, g(rij) is nearly independent of temperature over the limited range

d

encountered in the scanning experiments (that is, does not vary noticeably

with w) but does change with reaction order. For example, with X = 35,

g(ri^) = 0.95, 1, 1.05 and 1.10 for n = 0, 1, 2 and 3.

Using the approximation given above, it can be shown that D in equation

13

becomes

D

D = 1 + 2RT^/E

1 + 2/Xj + 2(l-n)(X"^ - 6Xj"^...)

a da C-32

again introducing only a small effect on the value for E derived using the

ASTM method.

Each functional form for f(ri) must be investigated separately in

order to determine whether or not the assumptions about g(ri) hold. Consider

f(r|) for an autocatalytic but "nearly" simple reaction:

f(n) = (n + n^Xi-n) c-33a

where is small and one can neglect its temperature variation over the

temperature range of the scanning runs. In place of eqs. C-30a and C-30b

one then has

(1 - y)n.

g(n^)G(n^) = -2yiln[—

C>33b

where

g(ri^)

5^ ' TST" ’ - 1 - c-33c

C.5-3
When f(ri) has its maximum value, 2ri^ = 1 + Then g(n^) is not one

(e.g. if Hq * 0.05 and 0 = 0, then g(O^) 0.16) and thus A derived from

eq. 15 would be a factor of five too small. However, for purposes of calculation of the explosion temperature, what is required is Af(ri^) where

f(n^) is approximately 1/4. Hence, the net error in assuming that f(O^)

1 and Ag(ri^) A would approximately cancel in calculating a critical

explosion temperature, T^. This is true only if T^ lies in the range of

the scanning runs and, if not, to the extent that the temperature dependence of can be neglected. Intuitively, it seems clear that if the

temperature dependence of cannot be neglected, one should expect to

2

detect a temperature dependence in a plot of £n(w/T, ) versus 1/T,

d d

The preceding analysis is consistent with all the assumptions of the

test method (e.g. uniform sample temperature, simple reaction that is not

autocatalytic if one wishes to determine A) and identifies the fact that

at least a necessary condition on an experiment is that error in E caused

by deviation of dT/dt from a constant value of w be less than the imprecision in E.

The magnitude of the change in E and A produced when n 1 so 0 # 0

in eq. C-31a is evaluated for our experimental data on m-azidobenzoic acid

in footnote 6 of table 9. The least squares program (LSTSQR) of table 9

fits eq. C-26 with g(in^) given by eq. C-31 directly to the experimental

values of pairs of w and T^. The results in the main part of table 9 were

calculated for n = 1 (see tables 6, 7, and 8) so that 0=0 for these

results. The form of the equation used, which is cited at the bottom of

table 6, can be derived as follows. Taking natural logarithms of both

sides of eq. C-26 and rearranging with g(n^) as given by eq. C-31 yields
i = |[-Jln (|) + Jin A] - |[£n w + 2 Jln(l/T) - U(l-0)]

To eliminate round-off errors in the least squares procedure the above

equation was modified to

1/T - 1/T- = a - b[Jln(w/w ) + 2 Jln(T^/T) - iln(l-0)]

o o o

a = -1/T + b[Jln b + ilnA-Znw - 2 iln(l/T )] o o o

b = R/E

1/T and Jin w can be selected arbitrarily; we let 1/T be the average of

o o o values 1/T, and Jin w be the average of Jin w of the input experimental

d o data. The nonlinear least squares problem was linearized using the formulation of Deming [201] using 1/T-l/T^ and Jln(w/w^) as the variables and the

initial values of a and b computed assuming 0=0. The random error in

w is assumed to be zero. The variance of A was computed from the variances

of a and b and their covariance and the differential of the relation for

the parameter a given above.

C . 5 . 2 . Assumption of Constant Scanning Rate in Evaluating g(0j )

The assumption of a constant scanning rate over the entire exotherm

is used in equation C-27a and underlies the treatment defining the position

of the maximum. This assumption cannot hold exactly because self-heating

increases the rate of temperature rise and the amount of reaction that

\ / occurs before the peak. Only the temperature regime near T, is important.

d ^ An error bound is developed below.

Equation 11 in the form

Zn(w/T^^) = Jln(g(n^)) - E/RT^ - Jln(E/RA)

C.5-5
is differentiated with respect to 1/T^

dJln(w/T^^)/d(l/T^) = -E/R + (l/g(n^)) (dg(n^)/d(l/T^)) C-34

The second term on the right hand side is evaluated using a procedure

similar to that used earlier in this appendix for equations C-27 thru C-29

except that G(ri^) in eq. C-27b is redefined to allow for self-heating

G(ti) = (AE/wR)[P(X) - I - P(X^)] C-35a

where the new term, I, is the self-heating correction

I = (R/E)/^ exp(-E/RT)dV C-35b

o and V is defined in eq. C-22a.

T - Tp + -(T^ + T^)w + V C-22a

V is the correction to T^, the nominal programmed temperature, due to self

heating. Integrating eq. C-35b by parts yields

R ^d.

I = |[V exp(-X^) - exp(-X)(-dX)]

o since V at X^ is zero in our experiments (because the heat capacity of

both reactants and products are small in comparison to the heat capacity

of the sample container)

Using eq. C-27, ignoring P(X^), the analog to eq. C-29 becomes

X

g(Tld)G(T1d) = 1 - e ‘^Xd^I - . . .

' C-35

d X^

Assuming the general rate law for simple reaction applies, G(f|^) in eq. C35 can be eliminated for n 1 using eq. C-30a to obtain:

g(rid) = 1 - 0 - C-36a

where 0 is defined by eqs. C-31b and C-31c and ^ is

X

ijj = (l-n)e n 1 C-36b

d ^ ^ 0 n =» 1 C-36c
Thus, again, there is no correction to g(n^) = 1 for a first order reaction.

Using eqs. C-36a, b, c, the second term on the right side of eq. C-34

becomes

j (i/g(n^Hdg(n^))da/T^) = [-d(i>/d(i/T_j) - d*/da/T^)i/a-<(i-i(/)

where

-d4>/d(l/T^) = |(l-n) 5^

= 2/X^ - 12/X^

-diJ;/d(l/T^) = - |(l-n)5i|;

C-37a

C-37b

C-37t

C-37d

s

- <i>d C-37e

Y » exp(-X + X^) (-dX)

o d“^

The quantity V is the correction for self heating.

Inserting eqs. 37a, b, and d into eq. C-34 yields

C-37f

dJln(w/T^^)/d(l/T^) = (n-l)(6^ - a^)C-38

Thus, the effect of self heating on g(ri,) for simple order reactions is to

a decrease E for n > 1, etc. An upper bound to 6^ can be obtained from eq.

C-37d. Since (dV/dT)^ is positive and Y is positive (because both V and

' j are positive) , one obtains '

V , X, (0-0^)

C-39

(0-0^)^ is 0-0^ in watts at T^. For the experimental results given in

table 9 of the text one finds the maximum value of X,/T, is at most 0.09

d a

so that if n < 2, one can be assured that the error in E will be less than

C.5-7
2% if V- is always kept less than 0.2 K for all scanning rates (i.e. up to

a-1 ’ -1

20 Kmin ). For our sample containers, is 20 mWK so if ordinate

signal is kept less than 4 mW or 1 mCal/sec. the error in will certainly

be less than 2%. Many of our experiments violate this condition but the

condition of eq."C-39 is clearly conservative.

C.5.3. Corrections to Assumptions 1,2 of Table 11

Assumption 1 is that the maximum in the observed signal corresponds

to the maximum in the rate of decomposition. To determine a correction we

first consider the condition that must be obeyed by the ordinate signal

difference 0-0 at the maximum value of dri/dt. Then we expand 0-0 in a

power series in the program temperature difference, where

is T when 0-0 is a maximum, (0-0 ) . Inserting this relation into the

p o o m

condition on 0-0^ when dri/dt is a maximum determines the nominal program

temperature (T ) , at which this dri/dt occurs. The correction that must be

p d

added to (0-0 ) to evaluate the self heating correction at (T ) , is then

o m p d

evaluated. The latter plus (T ) ,-(T ) is the correction needed for pa pm

assumption 1. The correction for assumption 2 is generated as a part of

the analysis for assumption 1.

Define T by

s T

S

where is defined by entry 2c of table C-4

C

2 = C2 + (C^-C^)^); C2

, C

2 independent of temperature

One can then express the rate of heat evolved by the sample, P

2

, from

entry 2c of table C-4 in terms of the obseirved ordinate signals 0, 0^, 0^

corresponding to S, S^, using the ordinate transfer function, eq. C-17a

+ i

We assume DO DO^ = 0 over the limited temperature range in questionn;

D = d/dt.

C.5-8
C-40a X ^x. = K[1 + (T^ + Tg)D + TgT^D ](0-0^). - K(0 -0^) (1-n)

From entry 1 of table C-4, is related to Dr) by

?2 - [-«3(V - (C^ - C2)(T2-Tj^)]Dn C-40b

The symbols of eqs. C-40a, b have been defined in Appendix C.3.2. It

should be noted that is the heat capacity of all the products of the

reaction and is greater than the heat capacity of the products left in

the partially open sample containers in our experiments.

2

At the maximum of Dri, D r| = 0 so that DP

2

at the maximum of Df),

(Dri)^, is given by (DP2^d from eq. C-40b

(DP2)d = “(C^-C2)(Dn)^[w +

^

(D +

h-iIs

2 Differentiating eq. C-40a, using again (D n)^ = 0, and assuming and

f h^g are independent of temperature gives

(DP2)^ = K[D + (T^ + Xg)D^ + + w(C2-C2)(Dti)^ +

KCDCj/h’^) (D + t/) (0-0^)^

Equating eqs. C-41a and C-41b, one has for the condition on 0-0

o

» f [(1 + -V-)D + [T + T (1 + -V-)]D^ + T T D^]K(0-0^) , = Zw

, SQ- sa od

hi h.

Is Is

C-41a

C-41b

C-42a

where

z = (cP-C2)(Dn)^; z' C-42b = -z + zee^-c^) (Dn)^

Equation C—42a indicates that occurs at a program temperature

* 2 - before Tppm. = (T ) . (Both Z and Z are positive. Since D (0-0 ) , is

' o d

negative for a fairly large range of T below and above (T ) , D(0-0 ) , p pm* o'^d

must be positive rather than negative for eq. C-42a to be satisfied.) Eq.

t C-42a can be simplified since Z /h^^ is small in comparison to one. One

» » If

can show Z /h^^ « 1 by writing Z /h^^ in the form

C.5-9
writing Z the form

cP-c^ +—c^-c^

—u

1

Is

where f can be approximated by

K(0-0 ) wC1 ^ _ f o in ^ f X ^ p Xs

- n . Is Is

-1

For nitrocellulose, the measured value of Q(T ) is about 2.2 kJ g K

Using the numerical values for nitrocellulose in table 4, values have

i f been calculated for f in Table C-9; f is less than 0.02. Since C

2

, C^,

_ f I C

2

« in our experiments Z 1* The term involving Z cannot be

neglected at this stage.

Suppose now that is expressed as a power series in about its

maximum (0-0 ) , for T < (T ) then

o m p — pm

K(0-0 ) = K(0-0 ) + a.X - a„X^ + a_X^...; X = T -(T ) <0 C-43

o oml2 J ppm

Since D(0-0 ) = 0, a^ = 0, and since D (0-0 ) < 0, a_ > 0. o m 1 o m 2

Assuming that the cubic term can be dropped in eq. C-43, one obtains

f upon substituting eq. C-43 into eq. C-42a with Z =0:

Z

(T ) ,-(T ) = X,

p d pm d

„ - w(t +T,) Za^ s d

C-44

Numerical values of w(T +T.) and a_ are listed in table C-9. The values

s d 2 ,

. of a

2

were determined from the intercept of a plot of [K(O-O^)^ - K(0-

2 0^)]/(-X) = a^ + a

2

(-X)... versus (-X) . To determined the condition for

dropping Z, one notes that

cP-c^

z ^

^

) fh

Is

2a,

C.5-10
Table

C-9.

Correction

for

Aeaumptlona

1»2

of

Table

11

o

d

00o

0^0I-I

• • • o o o

C.5-11
From the numerical values of fh^^/Za^ in table C-9, one concludes that

^for the term involving Z to be less than 0.05 K, must be

+

less than 0.2 . Hence, the term involving Z can certainly be dropped.

Approximate values of listed in table C-9, show that dropping

the cubic term in eq, C-A3 when determining is warranted.

The correction to the self heating term is given by

K(0-0 ) o m 2 2

(W(T +T,))^

s d

C-45

Is Is Is

>. Since this leads to a maximum correction of 0.01 K, it can be dropped

in our experiments. Thus the net correction for assumption 1' is given

by eq. C-43. If we add this correction to eq. C-22c, we have the

result that numerical values of T, determined by the ASTM method are

d

too large by From column 8 in table C-9 AT = 0.4 K for w = 20 K min

Assumption 2 is that the rate of change of temperature of the sample (at

the maximum is equal to w. The corrected value of (dT/dt) , is obtained

d

. -1

by differentiating eq. C-22a and C-22b:

(DT) . = w +

DK(O-O^)^ + T^D K(O-O^)^

Is

=« w +

Is

-A. 2a wC

= w(l + -yi)

h, h.

Is Is

C-47

C-48

Numerical values of (DT„-w)/w = Aw and DT, are listed in table C-9. It

2 d

should be noted that the value of DT, (DT)^, at K(00^)^ using eq. C-47,

with (0-0 ) , in place of (0-0 ) and DK(0-0 ) = 0 is

o d o m o m

(DT)

m

w +

Td2a2W

Is

The curvature of the ordinate signal is to first approximation the same

at its maximum value and at the value for the maximum in dri/dt.

For scanning rates above 1.25 K/min, this is the condition that the term

involving Z be less than 0.1 w(tg+T(j) which is .06 K at 20 K/min. Corrections less than 0.05 K are certainly not warranted in our experiments'.

C.5-12
C. 6. Design of a Closed Reusable Capsule

The closed reusable capsule consists of a flat cylindrical cup and

a screw cap lid made from 750-X Inconel which are sealed by means of a

gold disc. The cup and lid are made from 750-X Inconel. Their dimensions

are given in the cross section sketches in fig. C-9.

The gold disc is a flat plate of 0.25 mm thickness and 6.71 mm diameter.

The disc is located in the top of the screw cap as indicated in fig. C-9,

(A) . To place the lid in this position and also clear the internal

screw threads, the disc is punched from an annealed gold sheet in the

form of a shallow dish of 6.55 mm diameter. After annealing, the dish

is placed in the inside top of the screw cap and flattened into a plate.

The depth of the original gold dish is adjusted so the final diameter

of the final plate just matches that required to fit the screw cap lid.

The lid is screwed on to the cup hand tight using jigs that grip

the base and cup. The assembled height of the capsule is 3.49 mm.

Since the inner height of the DSC sample holder is 3.28 mm, new sample and

reference holder covers were fabricated from 0.15 mm thick platinum

sheet. . Dimensions are given in the cross section sketch in fig. C-10.

Further development work on the capsule is required to achieve the

following three aims: (1) the capsule should have a blowout disc to prevent rupture of the cup or lid (i.e. due to overloading the cell with ;a

material which produces gaseous decomposition products) . (2^ The capsule

temperature shold be uniform. (3) The sample should be placed in the

capsule in such a way that the temperature of the sample is kept as

unifoinn and as close to the capsule temperature as possible.

The first of these aims can probably be accomplished by drilling

a hole in the lid. The hole diameter and disc thickness would set the

blowout pressure.

C.6-1
The necessity of the second aim can be established as follows.

Suppose that to a good approximation the temperature of the screw cap

and cup are uniform but different and that the sample temperature is

uniform and equal to the cup temperature. Also, assume that the transfer

of heat between the capsule cup and the sample holder lid is negligible.

If and are the heat capacity and temperature, respectively, of

the capsule lid and C^, are the corresponding quantities for the

capsule cup, one can show the equations for column #2 of table C-4 are

valid provided the following changes are made:

a) is replaced by

b) h* is defined as

Is

h. + hg e c Is Z5 s5

c) T' is defined as

s T'

s + ‘'£5

(l+r)4^

(Tj,-T^) + (C5DT5 + C2DT2-P2)r

h^g and h^^ are the heat transfer coefficients between the cup and sample

holder and the cup and capsule lid, respectively, h^^ and h^^ are the

heat transfer coefficients between the capsule lid and sample holder and

capsule lid and sample holder lid, respectively. is the temperature

of the sample holder lid. In order for the DSC temperature scale calibration to apply to a decomposition r\in, r must be as small as possible and,

thus, h^^ » h^^ + This is equivalent to making T^ and T^ equal.

It should be noted that one cannot prove that r equals zero by

showing that a plot of the onset temperature of a transition versus the

scanning rate is linear. We have not found a sufficient test.

Method and tests for achieving the third aim given above have been

discussed in section 5.6.

C.6-2
CB)

Figure C-9, Sketch of a Diametral Cross Section of Closed

Reusable Capsule.

(A) Capsule screw cap (B) Capsule cup (C), Position of gold

sealing disc. All dimensions in ram (in.) "^tolerance of -.00,

+ .03 mm, ^"^tolerance of +.00, -.03 ram.
:{:

^ J,y-^ ^ ^

/

/

2.06>(.o8l)

i_ _BIr

T

,IS(-006)

8.55 (.328) >1

Figure C-10 Sketch of Diametral Cross Section of

Modified DSC Holder Lids.

All dimensions in mm (in.).

C.6-4
D. Table of Symbols

Latin alphabet symbols are listed first (including symbols preceded

by A), the Greek symbols, and finally any special symbols. Symbols

without a listed dimension are dimensionless.

English Symbols -

A Preexponential factor, time ^

B

Bi

c

Dimensionless adiabatic temperature rise: dimensionless

temperature rise of material above its constant enviornment

temperature, T , if heat lost by the sample is zero and initial

the sample temperature equals T^. See eq. 6a.

B when the constant enviornment temperature equals the critical

value, T^. See eq. 6b.

Biot number of material: ratio of heat transfer per unit area

of the sample surface to its immediate environment divided by

a mean heat transfer coefficient per unit area for conductive

heat transfer within the sample. See eq. 5. The immediate

environment is assumed to be either at a constant temperature

in the isothermal model or to vary at a constant rate in

programmed heating model (Appendix B.8); dimensionless.

-1 -3

Heat capacity per unit volume of material, J*K m

C

C

o

C.

J

c?

Total heat capacity of material. ^Heat capacity of sample container in sections 5.4, 5.5, J*K

Heat capacity of sample container in Appendix C.l, J«K

Total heat capacity; j = 1, sample container; j = 2, sample; j = 3, reference container; j = 4, reference material; j = s, sample holder (QDTA) or sample holder cup (DSC) ; j = r, reference holder (QDTA) or reference holder cup; j = £, sample

holder lid (DSC); j = m, reference holder lid (DSC);

Total heat capacity of sample just before the decomposition

reaction or change in state occurs (e.g. fusion). Assumed to

be independent of temperature. J*K“^.

Total heat capacity of sample just after the decomposition or change in state (e.g. fusion) is complete. Assumed to be independent of temperature in table C-4. J*K~^.

Total heat capacity of the products of a reaction. All of

products may or may not be retained in the sample container.

d Ordinary differential. Used as a subscript in section 5.2 and

Appendix C.5. to denote temperature at which the rate of decomposition of the sample is a maximum.

D-1
D

E

f(n)

f(n^)

f(V

f .

f

p

f

1

g

1

g(n^)'

Ordinary differential operator with respect to time, t, D = d/dt. Factor in proposed ASTM test method, see eq. 13.

(Meaning clear from context.)

Arrenhius activation energy, J mol

Dependence of rate of decomposition on fraction, r), of a sample decomposed when sample temperature is uniform.

f Functional dependence of 5^ or on reactant consumption.

For simple reactions, see eq. 9, or see the dependence of

(6 e)” of table B~3 on the order, n, of the reaction,

c Value of f(n) at the maximum value of the rate of decomposition of the sample in a programmed heating experiment.

Maximum value of f(ri).

Proportionality facotr of the DSC average holder temperature

detector (Appendix C.3.3.), V*K~^

Proportionality factor of the DSC differential holder temperature detector (Appendix C.3.3.). Product of the thermoelectric

coefficient of the QDTA temperature difference couple divided

by the gain of the difference signal amplifier (Appendix

C.3.I.),

Adjustable proportionality factor of the DSC programmer temperature (Appendix C.3.1.) V*K"^

i = 1,2,3; Proportionality factors of QDTA energy measurement

equation (see entry 7 of table C-3)

Ratio of DSC instantaneous ordinate signal divided by output

voltage supplied to sample and reference holder heaters by the

differential power circuit amplifier (see entry 6 of table C5).

The ratio of the gradient of the temperature of the material

at and normal to the surface of the material divided by (TT )/r. T and T are the average and the surface temperatures

of the material^ respectively, and s is a characteristic

dimension of the material. Heat transfer within the material

is assumed to be by conduction only. (See discussion of eq.

B-31a).

Volume of the material divided by the product of its surface

area and a characteristic dimension. (See eq. B-27).

The negative of the derivative of f(r|) with respect to ri at

the maximum rate of decomposition of the sample: -[df (r))/dr|]^.
The integral; /^dn/f (H)

The value of G(ri) at the maximum rate of decomposition of the

sample in programmed heating.

Heat transfer coefficient between the material and its container

or environment if the temperature of the outer surface of the

material and container or environment are equal. Used only in

Appendix B.3. W-K"^

Overall heat transfer coefficient between the material and its

environment in the isothermal model. Appendix B.3., in the

programmed heating model (Appendix B.8.)). Heat transfer

coefficient between sample and its container in adiabatic

calorimetry (Appendix C.I.). W*K"1

Heat transfer coefficient between the sample container and its

environment in adiabatic calorimetry (Appendix C.I.).

Heat transfer coefficient between the surface of the material

and its environment in the isothermal model. Appendix B.3.)

and programmed heating model (Appendix B.8.), W«K”^

Heat transfer coefficient between i and j, i i , h. , is

ii assumed to be equal to h.^. The subscript i (or j) nave the

following meaning. i = i, sample container; i = 2, sample; i = 3, reference container; i = 4, reference material; i = s,

sample holder (QDTA) or reference holder cup (DSC); i = r,

reference holder (QDTA) or reference holder cup (DSC); i = b,

block (QDTA), i = 5,, sample holder cup (DSC); i = m, reference

holder cup (DSC); W*K“^

Total heat transfer coefficient between the sample container

and its environment (QDTA and DSC), W«K“^

Difference heat transfer coefficient of QDTA energy measurement

equation (see entry 8 of table C-3)

,

Total heat transfer coefficient of QDTA energy measurement

equation (see entry 7 of table C-3), W*K"^

Enthalpy of sample at uniform sample temperature T^, J

Constant current supplied to each DSC holder by constant

current generator, A.

Specific rate constant of the decomposition reaction at a uniform sample temperature T, time"!

Specific rate constant when uniform sample temperature equals

to uniform constant environment temperature, time”'^.
K Calibration constant of QDTA (Appendix C.3.2) or DSC (Appendix

C.3.3.); X is temperature dependent for QDTA. Multiplication

of the observed ordinate signal in volts by K yields the rate

of increase of energy of the reference holder plus contents

^ minus the sample holder plus contents in units of watts; W*V

Also the unit of temperature.

a Order of simple reaction, see eq. 8.

0 Observed ordinate signal in a QDTA or DSC in a measurement

(i.e. fusion, transition, decomposition experiment: difference

between actual signal and baseline signal constructed as shown

in Fig. 11, V.

0^ k = i,f. 0 just before (i) or after (j) thermal event (i.e.

fusion, transition, decomposition) takes place. In Appendix

C.3.4, 0^ is extrapolated, V, linearly as shown in Fig. 11.

0.

2 j = o,e Observed ordinate signal in a blank QDTA or DSC run

with products of decomposition present in sample container (0) or with an empty sample container (e) , V.

0° Observed ordinate signal in a blank QDTA or DSC run with empty

sample holder, V.

0^

2

j = o,e; k = i,f. 0. at temperatures just before (k = i) or just after (k = f) tAe thermal event of the corresponding

measurement experiment, V.

The maximum 0“0^ in a decomposition run, V.

The rate of heat generation by the sample, W.

P.

2

j = A,B,D. Components of electrical power supplied to DSC

holders, see entries 1 and 2 of table C-5, W.

P(X) See Appendix C.5.1, eq. C-27c, X = E/RT.

Total rate of heat generation by sample in isothermal model.

Appendix B.3., W. >'

'^2 Rate of heat loss by sample in isothermal. Appendix B.3., W.

Q

_3 Heat of decomposition per unit volume of sample, J*m

t Q Total heat of decomposition of sample, J.

\

Total heat of decomposition of sample referred to reference

temperature T , J.

K

• f Rate of heat absorbed by DSC sample holder, j = s, or reference

holder, j = r, from block, W.

D-4
Rate of heat absorbed by DSC sample holder cup, j = s, sample

holder lid, j = ^, reference holder cup, j = r, reference

holder lid, j = m; W.

Gas constant, J*K ^mol

Resistance of electrical heater of each DSC holder,

2

Surface area of sample in Appendix B, m ; instantaneous

ordinate signal of DSC or QDTA (section 5, Appendices C.3 to

C.5); the observed ordinate signal lags S and is related to S

by eq. C-17a to a good first approximation; V.

k = i,f. S corresponding to 0 , k = i,f, V.

j = o,e; S corresponding to 0^ , V.

o

S corresponding to 0 , V.

j=*o,e;k=i,f. S corresponding to 0^, V.

time,s.

Total time to explosion, isothermal model. Appendix B.4., s.

Time for temperature of material to rise from the environment

temperature, T , to the explosion temperature, T , of the

material see Appendix B.4., s.

Time for temperature of the material to rise from its initial

value, T. , tot he environment temperature, T , see Appendix

B.4., s,^°

Temeprature of the material, K.

Temperature of the block (QDTA) , K.

Temperature of the material in isothermal model when the

environment temperature, T , has its critical value, T^. See

eq. B-6, K.

°

Temperature of the material at its maximum rate of ’decomposition,

K.

Explosion temperature of the material. Appendix B.4. . Temperature

of outer environment of adiabatic shield. Appendix C.I., K.

Initial uniform temperature of material at zero time of step

jump case (i.e. isothermal model). Appendices B.1-7; initial

uniform temperature of the material and environment, programmed

heating. Appendix B.8, K.

Temperature of sample holder lid (DSC) , K.
T

m

Maximum value of the temperature of the material, isothermal

model, Appendices B.1-B.7; temperature of the reference holder

lid (DSC), Appendix C.3., K.

T

o Uniform temperature of sample container. Appendix C.l. only.

Uniform- temperature of the environment, K.

on

Observed onset temperature (QDTA or DSC) of transition or fusion at scanning rate.w, K.

T

P

Instrument programmer temperature (DSC) , temperature of the

environment of the material in programmed heating. Appendix

B.8., K.

T

r Temperature of reference holder (QDTA) , reference holder cup

(DSC), K.

TR

Reference temperature of the heat of decomposition, K.

T

s Temperature of sample holder (QDTA) or sample holder cup

(DSC), K.

t T

s Apparent environment temperature of sample container (QDTA,

DSC), K.

^"^STAT

The static correction to be added to the abcissa or program

temperature, K.

i = 1,2, 3, 4. Temperature of the sample container, sample, reference container, reference material for QDTA or DSC, K.

T Critical explosion temperature: critical value of environment

temperature, isothermal model; calculated from eqs. B-7, 8, 9

for no reactant consximption and eqs. B-7, 47, 48 for reactant

consumption (f(n ) would be given by eq. B-54b, for example).

For the unidimensional heat conduction model. Appendix B.6.,

T^ is calculated from eq. B-7, eq. B-30 divided by f (q ), and

eq. B-31 with f(ri ) deleted for the case of reactant consvimption

m

In the programmed heating model, T^ is a parameter having the

same formula as T^ for the isothermal model with reactant consumption taken into account, K. .

V Correction to the nominal programmer temperature to take into

account the effect of self-heating (or absorption of heat

apart from heat capacity effects). See eq. C-22a, Appendix

C.3.4., K.

w Rate of increase of environment temperature with respect to

time: a) of sample environment in programmed heating. Appendix

B.8. and b) of sample and reference holders in the QDTA and

DSC instruments, see section 5 of text. Appendices C.3 to C.5,

K*s“^.

D-6
Critical value of w below which thermal explosion occurs in

the programmed heating model, Ks

Greek Symbols

a

01e

a

o

0

3*

Y

y*

f 6

I 5 c

f

5

a c

A

n

Heat transfer coefficient h divided by surface area of the

sample, W*K“^m“^.

Heat transfer coefficient h divided by surface area of sample.

Heat transfer coefficient h divided by surface area of sample,

RT /E. o RT^/E.

1/B.

Critical parameter for isothermal model, sections B.1-B.7.

See eq. B-9 for definition for sections B.2 through B.4, eq.

B-47 for definition for section B.5.

t Value of 6 when the environment temperature is at its critical

value

.

Dimensionless parameter in programmed heating model having the

same formula as 6 . c Critical parameter for unidimensional heat conduction model

for thermal explosions; constant uniform environment temperature.

See eq. B-31, for no reactant consumption.

Value of 6 when the environment temperature is at its critical

value

.

Dimensionless parameter in programmed heating model having the

same formula as 6 . c e-1/5 . -

Fraction of initial material decomposed.

Value of n when f(ri) is a maximum.

D-7
n

o

o

n

0

9e

0 .xn

9 . m

9

P

K

X

X

a

T

o

X

.q

X

r

X

s

Parameter in f(n) for simple autocatalysis, see eq. B-22, p.

B-14, temperature dependence is ordinarily ignored. Actual

dependence on temperature is given in footnote.

Fraction of material decomposed during time t°. Appendix B.4

only . ^

Dimensionless temperature of material when its temperature is

T; isothermal model: E(T-T )/RT^, programmed heating model:

E(T-T^)/Rt2. ° o

0 when T equals T , the explosion temperature, used in Appendix

B.3 only.

^

9 when T equals T. . xn Maximum value of 9 in isothermal model, used in Appendix B.3.

Diemsnionless temperature of environment in programmed heating

model. Appendix B.8: E(T, + wt - T^)/RT2.

2 Thermal diffusivity, m s.

Thermal conductivity, WK ^m”^.

9/(1 + 36).

Time constant of the temperature lag, x w, of the sample

holder temperature behind the instrument programmer temperature

(DSC) at a scanning rate of W*Ks”l; s.

Longest time constant of the differential power circuit plus

empty sample and reference holders (DSC) . The product of T

,

and the rate of change of the observed ordinate signal equals

the lag of the observed ordinate single behind the instaneous

ordinate signal, see eq. 17a, s.

Slope of a plot of the onset temperature of fusion or transition,

T^^(w), versus the scanning rate, w; the sum of T^, x^, ' see eq. C-21c, Appendix C.3.3, s.

I Thermal relaxation time of fhe material C/h in notation of

Appendices B.l to B.8 (^2/^23 terms of the notation of

Appendix C.3 to C.5., s.

Reaction time when the sample temperature equals T ; 1/k

where k is the specific rate constant, s. °

^o

Time constant of the temperature lag, T w, of the sample

container behind the sample holder temperature at a scanning

rate of w Ks~^ if P were zero, but everything else is the same

(e.g. (C^ + C^ + (c| - Ci)n)/h^g), s.

D-8
k = i,f£ Tg when the heat capacity of the sample is (k =

i) or C^Ck = f ) , respectively, s.

Adiabatic induction time, see eq. B-37.

Dimensionless scanning rate, programmed heating model. Appendix

B.8, see eq. B-64.

Critical value of
t. ,, .. •

mi

,T»Td,.«.AJl'.Ak^

t<r

:f!i

.4--:'' ‘Sr4

u,ie.<i-.i)k

y

'Wi.-*,

,_ V. ',«. >^«*>-*. . in-' •,;;^,mJ

rm

^ 2' 'll?-- 21-

' •'i

-3 .V-iJf2'-;fj

'i>- riv* <*>»#*?'* -*iyi% ,-, U4

j> » iuMMlXA

1JISC«-

mmf

l,1£U-- -X-- m

h

'ft* ^

”

' r>

' ;-«•- '&^'-

'., *'.»' k'3 ^ -' Y"'. '^'* • i

fcisif

» . -WSesi 'i*--^ -&l» -iTM ,.»> «',

^'S;i f

>y'Aj)«il!lllWftJi jt. ^' 44

TKj

fe-,: Vjkf

IB

*|r,

.a>

Fsd

,v,« .'•,',iiS

.-»..j£Ci'<l
E. References

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.

E-1
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,

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E-3
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E-4
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E-5
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,

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E-16
NBS-H4A (Ri£v. a-7e>

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 80-2018 (DOT)

«to. 1. No.

4. TITLE AND SUBTITLE A Third Appraisal of Methods for Estimating Self-Reaction

Hazards

5. Publication Date August 1980

' -'C.

.

7. AUTHOR(S)

Kenneth L. Churney and David Garvin

8. Performing Organ. Report No. NBSIR 80-2018 (DOT)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON. DC 20234

11. Contract/Grant No.

DOT-AS-40028

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fsrr«e(, city, state, ZIP) U.S. Department of Transportation

Research and Special Programs Administration

Materials Transportation Bureau

Unfihington, D.C. 20'i9.0

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15. SUPPLEMENTARY NOTES

DOT Report No. DOT/RSPA/MTB-7816

I ~~1 Documerrt describes a computer program; SF-185, FIPS Software Summary, is attached.

18. ABSTRACT (A 20(hword or loom feetuml aummory of moot oignificant information. If document includes a significant bib!ioi)iaphy or

literature suroey, mention it here,) A study has been made of thermal explosion theory and thermal hazard tests. From

this definitions have been developed, suitable for use In a transportation context, of

thermally unstable materlals-package configurations and thermal instability hazards.

A thermally unstable materials package configuration is one: (1) composed of substances that can decompose or react with the evolution of heat, (2) for which a thermal

mechanism of Initiation of a temperature rise Is present, and (3) in which a resulting

rapid, large increase in the temperature of the material can occur. A thermal instability hazard exists if decomposition of the material-package configuration results in

destruction of the package or neighboring objects or poses a safety threat. Typical

destructions are explosion, rupture of package, and fire.

The properties of the material and package needed to predict the hazard are identified and methods for their measurement, using heat conduction and adiabatic calorimetry

are analyzed. It Is concluded that, for least ambiguous scale-up of results to bulk

shipping conditions, measurements should be made on large samples of the material under

conditions such that heat transfer within the sample is large in comparison to heat

transfer between the sample and its environment. Tests, such as the explosion temperature test, should be carried out to check the scaled-up hazard predictions.

An experimental study on three thermally unstable materials, nitrocellulose (I),

m-azidobenzoic acid (II), and azldotrlphenylmethane (III), was carried out using (con'ti

17. KEY WORDS (eu to twelve entries; alphabetical order; capitalixe only the first letter of the first key word unless a proper name;

eepareied by eemicoione) calorimotry , azldotrlpheuyImothane , decomposition reactions, differential scanning calorimetry, heat conduction calorimetry, kinetics of

heat evolution, m-azidobenzoic acid, nitrocellulose, quantitative differential thermal

analysis, thermal explosion theory, thermal hazards, thermal instability.

18. AVAILABILITY pyl Unlimited

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20402, SD Stock No. SN003-003-

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NBS-114A

16. ABSTRACT (continued)

a differential scanning calorimeter (DSC) . The procedure used is that of a proposed ASTM method of test for determining Arrhenius kinetic constants

from the variation of the temperature of the maximum rate of heat evolution

pre-exponential factors. A; are: AE = (I) 166.9 + 6.1 (II) 139.9 + 5.9, and

(III) 145.3 + 4.4 kJ«mol“-^; log^Q A, min"^) = (I) 18.0 + 0.7, (II) 15,1 + 0.7,

and (III) 15.0+0.6. A systematic error analysis clarifies and tends to

support the test procedure. The main drawbacks to the test are its nonapplicability to complex systems and the small temperature range of the

data, which makes the extrapolation of results to lower temperatures

uncertain.
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