## Research Repository

# Stochastic Modelling and Optimal Control of Compartment Fires

Tat, Mehmet Ali
(2002)
*Stochastic Modelling and Optimal Control of Compartment Fires.*
PhD thesis, Victoria University.

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## Abstract

Compartment fires are defined as fires in enclosed spaces. They are labeled as oxygen driven fires and are non-stationary growth phenomenon. A gap exists in the knowledge of deterministic fire growth models and stochastic fire growth models. In this thesis we develop non-stationary stochastic models in an endeavor to bridge the gap. The class of Epidemic models for infectious diseases are non-stationary growth models. In the first part of the thesis the Deterministic Simple Epidemic, Deterministic General Epidemic and the Stochastic General Epidemic models are investigated to develop equations for the growth of compartment fires by drawing analogies between the epidemic variables and the compartment fire variables. The Percolation and Contact processes are investigated for the spread of compartment fires. A mechanism for converting deterministic differential equations to stochastic differential equations based on the theory of Martingales is presented. In part two of the thesis, two deterministic models based on the risk assessment model of the National Research Council Canada (NRCC) are developed and calibrated. One of the deterministic models is a fuel driven model and the other is an oxygen driven model. The oxygen driven deterministic model is converted to a stochastic model based on the theory of Martingales, and used as an input to calculate a fire severity measure called Heat Load. Statistical tests are applied to the Heat Load data set to determine its distribution. A non-parametric statistical test, W Test, is used to calculate the upper quartiles of the heat load. A third model based on the NRCC model is built. This model is closer to the Epidemic models and its parameters do not require tedious optimisation algorithms to calculate. They are evaluated from the initial conditions of the physical process. In this model we make the assumption that the gas temperature inside the compartment is a function of the burning rate and develop a two variable model based on the burning rate and oxygen fraction. A change of variable is applied to simplify the differential equations, the equations are solved implicitly and their parameters evaluated using the initial conditions. The temperature equation is modelled using a first order differential equation with the burning rate and is solved separately. Finally part three of this thesis investigates automatic sprinkler systems and the mathematical theory of optimal control. Optimal control theory is applied to automatic sprinkler systems to model sprinklered compartment fires. To reduce water damage inside a compartment due to sprinkler activation from small fires, we model the water spray rate. Two cases are considered, the first when the water damage is proportional to the total amount of water and the second when the water damage is proportional to the integral of the square of the water flow rate. Pontryagin's principle is used to solve the integrals and obtain the water spray rate equations.

Item Type: | Thesis (PhD thesis) |
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Uncontrolled Keywords: | stochastic modelling; optimal control; fires; automatic sprinkler; compartment fires; oxygen driven fires |

Subjects: | RFCD Classification > 290000 Engineering and Technology Faculty/School/Research Centre/Department > School of Engineering and Science |

Depositing User: | Mr Angeera Sidaya |

Date Deposited: | 30 Nov 2005 |

Last Modified: | 23 May 2013 16:37 |

URI: | http://vuir.vu.edu.au/id/eprint/227 |

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