On mathematical modelling of the self-heating of cellulosic materials : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, New Zealand

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ON MATHEMATICAL MODELLING OF THE SELF-HEATING OF CELLULOSIC MATERIALS

A thesis presented in p arti al fulfilment of the requirements for the degree of Doctor of Philosophy in M athem atics

Robert Antony Sisson June 1991

at M assey Unive rsity, New Ze al and

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A B STR A CT

This thesis consi ders mathemati cal mo delling of self-heating of cellulosi c materials, and in parti cular the effe cts of moisture on the heating chara cteristi cs. Following an introdu ctory chapter containing a literature review, Chapter 2 presents some

preliminary results an d an in dustrial case study. The case study, whi ch dis cusses a ' dry' body self-heating on a hot su rfa ce, investigates the following questions : (i) how hot can the surfa ce get before ignition is likely? (ii) how well does the (slab like) bo dy approximate to an infinite slab? an d (iii) how vali d is the Frank

Kamenetskii approximation for the sour ce term? It is shown that the minimal stea dy state temperatu re profile is stable when the temperature of the hot surfa ce is below a

certain criti cal value, an d boun ds for the higher stea dy state profil e are deriv ed.

Chapter ³presents the thermo dynami c derivation of a rea ction- dif fusion mo del for the self-heating of a moist cellulosi c bo dy, in clu ding the effe cts of dire ct chemi cal oxi dation as well as those of a further exothermi c hy drolysis rea ction an d the

evaporation an d con densation of water. The mo del contains three main variables: the temperature of the bo dy, the liqui d water con centration in the bo dy, an d the water vapour con centration in the bo dy. Chapter 4 investigates the limiting case of the mo del eq uations as the thermal con du ctivity an d diffusivity of the bo dy be come l arge. I n parti cular it is shown that, in this limiting case, the mo del can have at least twenty-five distin ct bifur cation diagrams, compare d with only two for the well known mo del without the effe cts of moisture content. In Chapter 5 the maximum p rin ciple an d the methods of upper an d lower solutions are use d to derive existen ce, uniqueness an d multipli city results for the stea dy sta te solutions of the spa tially

distribute d mo del. Finally, in Chapter 6, existen ce an d uniqueness results for the time depen dent spatial ly distribute d mo del are derive d.

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I would like to express my gratitude to my supervisors Professor Graeme Wake and Mr Adrian Swift for their unfailin g help and enthusiasm throughout my work. Thanks also to Or Alex McNabb and Mr Aroon Parshotam for many helpful comments, Mr Richard Rayner for his help in producing the graphics, Miss Fiona Davies for her typing of this thesis, Joanne for her constant support, and Patricia for her friendship.

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CHAPTER 1 1 . 1 1 .2

1^.3

CHAPTER 2 2. 1 2.2

CHAPTER 3

3^.1 3.2 3.3 3.4 3.5

C ONTENTS

INTRODUCTION Physical background

Fommlation of the model for self-heating by a single exothe nnic reaction

I nterpretation

PRELIMINARY RESULTS AND A CASE STUDY Preliminary results

Industrial case study

T HERMODYNAMIC DER IVAT ION O F A MODEL FOR T HE SELF- HEATING OF DAMP CELLULOSIC MA TERIALS

Heat producing reactions Assumptions

Derivation of the equations

Dimensionless formulation of the equations

The steady state equations for the spatially distributed model

3 6

24 24 42

85 85 86 87 98

1 03

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CH APTER 4 4. 1 4.2

4.3 4.4 4.5

4.6 4. 7

CHAPTER 5

5. 1 5.2 5.3

CHAPTER 6

T HE SPATIALLY UN IFORM MODEL Introduction

Questions of existence, uniqueness and multiplicity of solutions

The nature and stability of steady state solutions Hopf bifurcations and periodic solutions

Plotting degeneracy and bifurcation curves : the pseudo- arclength method

The degeneracy curves in AI, L space The distinct bifurcation diagrams

EXISTENCE, UNIQUENESS AND MULTIPLICITY RES ULTS FOR T HE SPATIALLY DISTRIBUTED STEADY STATE MODEL

Existence results Uniqueness results

Results on the multiplicity of solutions

EXISTENCE AND UNIQUENESS RESULTS FOR THE TIME DEPE NDENT PROBLEM

CONCL UDING COMMENTS REFERENCES

1 06 1 06

1 0 7 1 1 7 123

1 33 135 143

153 153 1 70 1 84

1 96

204 205