Normativa Referente a la Educación

The development o f a fire within a compartment represents an event o f both theoretical

and practical interest, displaying a variety o f nonlinear behaviour. O f particular

importance is the flashover transition, whereby a small fire undergoes a rapid

intensification to engulf the whole compartment in flames.

In an effort to understand and predict such phenomena, mathematical models o f compartment fire growth have been constructed. As with models o f any physical system

they contain a number o f control parameters, which combine to determine the form o f

behaviour exhibited by the model. In this thesis we have sought to map out and explore the influence o f the controlling parameters on the qualitative nonlinear behaviour o f

three different simplified compartment fire models by applying some o f the techniques o f nonlinear dynamics. By doing so, we have obtained an overall picture o f the types o f behaviour to be expected, and an idea o f what the result o f varying a given

parameter on the development o f a compartment fire will be.

Chapter 2 surveyed basic fire dynamics and combustion phenomena before examining

elements o f compartment fire modelling. Attention was focused on the flashover transition, one mechanism for which was associated with the criticality occurring at a

fold bifurcation point. Two key nonlinear features o f compartment fire models were

identified as the fuel to ventilation control transition and radiation feedback from hot

gasses and compartment surfaces to the fire.

In chapter 3, bifurcation theory was introduced as the prim ary tool for exploring the

behaviour exhibited by the theoretical compartment fire models. The variation in response o f a nonlinear model, indicated by some representative quantity (e.g.

temperature) as a particular (distinguished) param eter varies, can be traced out to

produce a bifurcation diagram. The effect o f the remaining (auxiliary) control

parameters on the form o f this bifurcation diagram can then be mapped out by locating the boundaries or varieties which separate the different qualitative type o f bifurcation

diagram, exhibited by the model (i.e the hysteresis and isola transcritical varieties

described in section 3.2.1). Such varieties are defined in terms o f singularities o f the

function(s) defining the model and its derivatives.

In section 3.2.2 we have shown that these definitions can be extended to include the transition points between two different functions, which are valid in different regimes,

to obtain the analogous transition hysteresis and transition isola/transcritical varieties.

Such definitions can be applied to the fuel to ventilation control transition exhibited by

compartment fire models. Path following techniques were also presented as a numerical means o f obtaining bifurcation diagrams. When used in conjunction with the Jepson and

Spence approach (suitably modified to allow for the application to two functions and the

possibility o f transition points) to climb the hierarchy o f singular points, these path following techniques can be employed to locate varieties numerically.

In chapter 4 we applied the methods o f chapter 3, to construct param eter maps and classify the possible bifurcation diagram types for three different simplified compartment fire models. In the first o f these, the simplified radiation feedback model,

three basic parameters controlling, fuel area, ventilation and radiation feedback were considered. Treating the fuel area parameter as distinguished produced a transition hysteresis variety which divided the feedback/ventilation parameter space into two qualitative types of bifurcation diagram. The first o f these, located at low feedback and

ventilation levels displayed a monotonie response, with ju st a fuel to ventilation control

transition. The second type found at higher ventilation and feedback levels displayed

a characteristic ‘S ’ type response exhibiting flashover at a lower fold bifurcation point. Analytical expressions for the fold bifurcation point (on the fuel controlled branch)

demonstrated the effect o f the auxiliary parameters on both the critical area and

temperature required to trigger flashover.

Treating the ventilation parameter as distinguished produced both transition hysteresis and isola/transcritical varieties. It was shown that for higher ventilation and feedback

form behaviour, with the peak response located at moderate ventilation levels between

the extremes o f very high and very low ventilation. The effect o f physical boundaries on the bifurcation diagram, produced by the maximum compartment floor area and introduction o f a critical temperature flashover threshold (as an additional criteria for

flashover), were also examined. Two flashover types were identified and located: (a)

direct flashover where the fuel controlled solution branch crosses the critical

temperature threshold directly, and (b) jum p flashover where the fuel controlled solution

branch exhibits a fold bifurcation.

It was also shown that the transient behaviour produced by considering flame spread

over a fuel, confirmed the qualitative predictions o f the steady state analysis. In addition

the introduction o f fuel consumption produced a sharp temperature peak on the occurrence o f flashover.

F o r the Thomas vitiation model we examined the influence on compartment fire behaviour o f a vitiated compartment atmosphere. With ventilation param eter distinguished and fuel area and radiation feedback param eter auxiliary, analytical solutions for the bifurcation varieties (isola/transcritical, transition hysteresis and

isola/transcritical) were derived. Construction o f the param eter map and classification o f behaviour types, indicated that vitiation introduced a new bifurcation diagram exhibiting back-to-back folds in a ‘mushroom’ form. This form implied the possibility o f two types o f criticality being exhibited; firstly a jum p at a fold bifurcation as

ventilation was reduced, as for breaking wave form and an additional fold bifurcation

leading to a jum p as ventilation was increased. This new type o f criticality was identified with the backdraft phenomenon, where a sudden in-rush o f oxygen, due to

a window or door being opened, triggers a rapid intensification o f the compartment fire.

For the Takeda model we examined the influence o f combustion efficiency as a function of the characteristic residence and mixing times o f the gases in the compartment.

Varieties dividing both the feedback/ventilation and feedback/fuel area param eter planes

locating hierarchical singularities. Classification o f the bifurcation diagram types revealed the possibility o f upper fuel controlled solution branches and restriction o f

hysteresis to moderate ventilation levels and fuel areas, due to reduced combustion efficiency.

In section 4.2 .3 it was shown that incorporating the effect o f fuel consumption into the

Takeda model resulted in a sharp secondary temperature peak ju st prior to complete fuel exhaustion, followed by abrupt extinction jum p. This feature was explained by a rise in combustion efficiency, in the ventilation controlled regime, due to an increase in

residence time o f the gases in the compartment produced as a result o f the fuel mass

loss rate being reduced by fuel consumption.

Based upon the analysis o f the simplified radiation feedback model and the concept o f the minimum fuel area necessary to trigger flashover, in chapter 5 a flashover index

was derived, to be used as a means to evaluate the flashover potential o f a given compartment. The predictions o f flashover for an example compartment were shown to be comparable with that o f the existing method, whilst providing more information

on the influence o f different fuel types, and the influence o f radiation feedback, at the expense o f requiring a larger amount o f input data and greater complexity o f

calculation.

Finally, in chapter 6, a comparison o f experimental data obtained for two thermoplastic fuels was made with both the qualitative and quantitative predictions o f an extended

version o f the simplified radiation feedback model. The experimental data and observations (illustrated by the graphic photograph sequences) confirmed the critical

nature o f flashover and correlated well with the predicted theoretical nonlinear

behaviour, such as lower fuel controlled and upper ventilation controlled solution

branches, ‘S ’ type responses, flashover criticalities for varying fuel areas and ‘breaking w ave’ responses for varying ventilation. The transient features we observed in chapter

4 (i.e primary and secondary temperature peaks with flashover and extinction jum ps)

The present analysis has been mainly restricted to exploring the quasi-steady state and multiplicity behaviour o f some simplified compartment fire models, with only a brief

examination o f transient behaviour. Future research directions could include the extension o f the analysis to provide a more comprehensive examinination o f dynamic

phenomena and transient responses, incorporating fuel consumption and flame spread behaviour. It should also be possible to examine the flashover phenomena in a spatial

context considering the transition from localised to non-localised burning, and the

ignition o f the second and subsequent combustible items during the course o f a compartment fire.