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THEORY OF THERMAL EXPLOSIONS
A thesis presented in partial fulfilment of the requirements for the degree of
Doctor of Philosophy in Mathematics at Massey University
Michael Rowlinson Carter 1976
ii
ABSTRACT
When a heat-producing chemical reaction takes place within a confined region, then under certain circumstances a thermal explosion will occur. In investigating from a theoretical viewpoint the conditions under which this happens, it is necessary to study the behaviour of the solution of a certain non-linear parabolic initial-boundary value
problem.
A frequently used approach is to study the problem indirectly, by investigating whether positive steady-state solutions exist ; the
underlying assumption is that positive steady-state solutions exist if and only if a thermal explosion does not occur . The main theme of this t hesis is the development and application of an alternative direct approach to the problem, involving the construction of upper and lower solutions for the parabolic problem and the application of appropriate comparison theorems . The assumption here is that a thermal explosion will not occur if and only if the solution of the parabolic problem remains bounded for all positive time .
Following three chapters of introductory material, Chapter 4
contains a survey of some of the important known results concerning the e xistence of positive steady-state solutions, especially those dealing with the effect on the theory of different assumptions as to the rate at which heat is produced in the reaction.
The comparison theorems that are used in the alternative approach, which are modified versions of known results , are proved in Chapter 5 .
In Chapter 6, the equivalence of the two criteria mentioned above for t he occurrence or non-occurrence of a thermal explosion is
established under fairly general conditions . Also in this chapter , a critical value A* is defined for a parameter A appearing in the problem , s uch that a thermal explosion will not occur if the value of A is
smaller than A*, but will occur if the value of A is greater than A*·
In Chapter 7, upper and lower solutions are constructed for the t ime-dependent problem under a variety of assumptions as to the rate at which heat is produced in the reaction, and these are used to obtain a number of theorems concerning the behaviour of the solution of the problem, especially as the time variable tends to infinity . The
information obtained from these theorems is related to and compared with t hat known from investigations of the existence of positive steady-st ate solutions . In conclusion , a theorem is proved concerning the effect of reactant consumption on the theory. This is examined in the light of
some recent research, and an apparent defect which is thereby revealed in t he usual criteria for the occurrence'of a thermal explosion is discussed.
The theorems of Chapter 7 are employed in Chapter 8 to obtain rigorously derived bounds for the critical parameter A*, for a number of different shapes of the region in which the reaction takes place;
these bounds are compared with known estimates for A* obtained using an empirically derived formula .
The thesis concludes, in Chapter 9 , by using the methods of Chapters 7 and 8 to obtain some results concerning the case where the boundary condition is non-linear .
ACKNOWLEm EM ENT
I would like to thank Dr . Graeme C. Wake for his ready advice and unfailing encourageme nt and enthusiasm during the preparation of this thesis .
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INDEX TO THEOREMS 1. INTRODUCTION 2. AN EXAMPLE
3. DEFINITIONS AND NOTATION
4. THE STEADY-STATE PROBLEM - A SURVEY
5. COMPARISON THEOREMS FOR THE TIME-DEPENDENT PROBLEM 6. RELATIONS BETWEEN SOLUTIONS OF THE TIME-DEPENDENT AND
STEADY-STATE PROBLEMS Monotone Iteration
Asymptotic Behaviour of Solut ions
7 . CONSTRUCTION OF UPPER AND LOWER SOLUTI ONS FOR THE
TI ME-DEPENDENT PROBLEM Notat ion
Construction of Lower Solutions on V m Construction of Upper Solutions on V m Extension to Other Domains
Discussion
The Effect of Reactant Consumpt ion 8. BOUNDS FOR THE CRITICAL PARAMETER
Preliminaries
Bounds for the Critical Value X*
Comparison with Known Values 9. NON-LINEAR BOUNDARY CONDITIONS
APPENDIX
LIST OF REFERENCES
vi 1 6 8 14 21
28 29 41
52 52 5 3 6 3 72 77 80 86 86 89 91 102 116 127
INDEX TO THEOREMS
Theorem Number Page
1 22
2 24
3 35
4 41
5 41
6 43
7 46
B 47
9 48
10 50
11 53
12 54
12A 74
12B 104
13 55
14 57
15 58
16 60
16 (Corollary) 62
17 63
17A 75
17B 103
18 66
1BB 103
19 68
20 70
20B 104
21 80
