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Open kitchen designs are found in small units in tall residential buildings of Asian-Oceania regions for better space utilization. As many combustibles are stored in small residential units, fire originated in the open kitchen can grow and spread fast. Consequently, flashover can occur to give a big fire and result in severe casualties and property damage. Nonlinear dynamics can be applied to predict critical heat release rate to flashover in the unit with an open kitchen and will be illustrated in this paper. Based on a two-zone model, temperature of the hot smoke layer was taken as the system state variable. An evolution equation was developed with selective control parameters. Onsetting of flashover using a nonlinear dynamical system was demonstrated in the example residential units. Effects of the floor dimensions, the radiation feedback coefficient and thermal properties of wall material on the onset of flashover were then examined and analyzed. The developed nonlinear dynamical model for studying the onset of flashover gives a better understanding of the various control parameters.

The fire safety of tall buildings associated with smoke movement and control, big fire due to wind action, firefighting accessibility and long evacuation time should be watched in densely populated areas in the Asia- Oceania regions [

In Hong Kong, there are residential units in tall buildings with a floor area of less than 30 m^{2}. Open kitchen design was employed to make this tiny unit more spacious. However, these open kitchen designs [^{−2}. Kitchen is a hazardous area with naked flames cooking and usually enclosed with fire resisting construction. Cooking fire was the leading cause [

A fire originating from the open kitchen spread beyond the cooking area can easily ignite combustibles stored in other parts of the unit. The fire can grow quickly to have flashover. The onset of flashover is usually considered as an indication of untenable condition, playing an extremely important role in disastrous fires [

Efforts have been made to understand and predict flashover [^{2}, or flames come out of the openings.

Flashover is characterized by a sharp increase in burning rate and gas temperature. Thermal instability is considered to be one of its mechanisms [

The process of a fire in a small residential unit was considered as a dynamical system. Based on a two-layer zone model, the evolution equation was developed for the upper hot smoke layer. The upper smoke layer temperature was chosen as the state variable because it is important for predicting hazardous conditions. Change of the system state with time is of interest. The system behavior is dominated by control parameters, such as heat release rate, room configuration, opening geometry, and others. When one or more control parameters change, the system state responds accordingly. Normally, a small perturbation only causes a relatively slight variation in the system state. However, the system can experience violent change or bifurcations [

An example residential unit adopted [_{d} and height H_{d} is located at the center of one wall. A fire source at the open kitchen is assigned at the room centered at the floor level. There will be no fire resisting wall enclosing

the kitchen in that residential unit. Main assumptions made in the fire model are listed below:

・ Density of the smoke layer is kept constant at ambient density r_{0}.

・ The temperature of the lower air layer and its bounding surfaces are kept at the initial temperature T_{0}.

・ Surface temperature of the fire source is assumed to be the ambient value T_{0} and its emissivity is taken to be 1.

・ Before flashover, the fire is assumed to be quasi-steady and the height of the smoke layer interface is constant and kept at 0.5 H [

・ The height of the neutral plane coincides with the height of the smoke layer interface.

・ In ventilation-controlled stage, the air entering into the residential unit is assumed to be completely consumed.

・ The wall surface is assumed to be black body and the emissivity of the smoke layer is assumed to be 1.

The evolution equation was developed based on the energy conservation for the upper hot smoke layer. It takes a similar form as described [_{p}, the average temperature of the hot smoke layer T at time t, net heat gain rate G_{E} and net loss rate L_{E} of the hot smoke layer.

The terms G_{E} and L_{E} are functions of smoke layer temperature. G_{E} is determined by the fraction of the heat release rate of the fire _{E} can be written as:

For fire plumes resulted from common fuels, the radiant part

The heat release rate of a room fire can be calculated for fuel-controlled fire and ventilation-controlled fire. For a fuel-controlled fire, there is enough air for combustion and the heat release rate _{com}.

For a ventilation-controlled fire, excess fuel is released and the heat release rate is dominated by the mass flow rate of air

Further, _{vap}.

In a compartment fire, the hot smoke layer and heated boundary surfaces radiate heat back to the fuel surfaces, which accelerate the gasification rate of the fuel. This radiant feedback has been recognized as playing an important role in the onset of flashover [_{0}.

In a ventilation-controlled fire, the gas temperature is most often very high and the smoke gas isroughly mixed evenly. The inflow of air through openings can be obtained [_{d} and height H_{d} of the opening:

The energy lost from the hot smoke layer L_{E} is composed of mass flow through the opening, convection and radiation heat loss to the solid boundary and radiation loss to the opening.

where Z is the height of the smoke layer interface from the floor level; and h_{t} is the convective heat transfer coefficient._{w} are the surface temperature and surface area of the upper parts of the solid boundaries enclosing the hot smoke gas respectively. A_{w} is given by:

There are four items on the right hand side of Equation (8). The first item is the radiative heat loss from the smoke layer to the lower part of the compartment and vent. The second item and the third item are the radiative and convective heat loss to ceilings and the upper part of the walls, respectively. The forth item is the enthalpy flowing out through the vent.

For simplicity, surface temperature of the heated walls

Note that

As demonstrated in _{d}; the height of neutral plane from floor Z_{N} where the pressure deference across the opening is zero; and the acceleration due to gravity g.

For simplicity, Z_{N} was assumed to be coincided with Z, equation (11) can be rewritten as:

According to the dynamical theory, equilibrium points of the dynamical system meet:

where T_{equ} is the system state value at equilibrium points, its corresponding eigenvalues l can be obtained by:

If the value of l is equal to zero at certain equilibrium point, then flashover is considered to happen here.

The control parameters and constants are set values as listed in

Onsetting of flashover was further demonstrated [_{E} and energy loss rate L_{E} were plotted against the upper smoke layer temperature respectively as shown in

For the case with floor geometry of 6 m by 5 m, as shown in

Parameters | Values | Parameters | Values |
---|---|---|---|

s | 5.67 × 10^{−}^{8} W∙m^{−}^{2}∙K^{−}^{4} | μ | 0.15 |

c_{p} | 1003.2 J/kg∙K | r | 30 |

C_{d} | 0.7 | T_{0} | 300 K |

g | 9.81 m∙s^{−2} | U_{c} | 0.7 |

h_{c} | 7 W∙m^{−2}∙K^{−1}∙W/m^{2}∙K | W_{d} | 1 m |

H | 3 m | Z | 1.5 m |

H_{com} | 4.2 × 10^{7} J/kg | 1 | |

H_{d} | 3 m | 1/3 | |

H_{vap} | 1.008 × 10^{6} J/kg | 1.18 kg∙m^{−3} |

there is a small increase in temperature at point D, the fire will jump rapidly from an equilibrium state to a new stable state E, i.e. flashover occurs. When

Eigenvalues in

To consider the effect of unit geometry on the onsetting of flashover, critical conditions for five units with different floor dimensions but same room ceiling height (3 m) were evaluated. The five cases were labeled as C1 (6 m by 3.5 m), C2 (7 m by 3 m), C3 (5 m by 5 m), C4 (6 m by 5 m) and C5 (7.5 m by 4 m) respectively.

In these cases, the radiation feedback coefficient is kept at a constant value of 0.15, though they vary with the compartment geometry and the fire process. The values for critical temperature, critical heat release rate and equilibrium temperature after flashover discussed before [

Three floor area values 21 m^{2}, 25 m^{2} and 30 m^{2} were investigated. For units with a larger floor area, the critical heat release rate for flashover is lower. In the fire model developed, the thermal radiation feedback is closely related with the smoke layer interface area, which is equal to the floor area. If the other conditions are the same, as the floor area increases, the fire base receives more energy feedback from the upper part of the compartment. But for a large compartment, it may take a longer time to reach a higher temperature.

C1 and C2, C4 and C5 are of the same floor area but different aspect ratio respectively, giving different internal surface areas. For residential units with the same floor area, when the internal surface area is small, the values of critical temperature and critical heat release rate are lower than those with a larger internal surface area. When the internal surface area of an apartment is small, less heat is conducted away from the enclosure boundary, allowing more energy to be stored in the unit. Therefore, a smaller heat release rate would onset flashover to give a higher compartment temperature.

Radiation feedback from the smoke layer to the fire source is assumed to be the driving force to the onset of flashover in this study based on thermal instability. The heat feedback process is complicated and affected by the geometry of the enclosure, the concentration of the participating media such as carbon monoxide and soot, the thickness and temperature of the smoke layer and other factors. Heat radiated to the fuel is simplified in the

Cases | Floor dimensions (mm) | Solid surface area enclosing hot smoke ^{2}) | Critical temperature (K) | Free burning heat release rate | Critical heat release rate | New stable Temperature (K) |
---|---|---|---|---|---|---|

C1 | 6 ´ 3.5 | 48 | 647 | 656 | 1898 | 818 |

C2 | 7 ´ 3 | 49.5 | 652 | 669 | 1953 | 814 |

C3 | 5 ´ 5 | 53.5 | 604 | 542 | 1694 | 772 |

C4 | 6 ´ 5 | 61.5 | 566 | 439 | 1444 | 774 |

C5 | 7.5 ´ 4 | 63 | 569 | 447 | 1475 | 772 |

present study using the radiation feedback coefficient m in equation (6), which implicitly incorporates the effect from emissivity and view factor. Only radiation from the smoke layer is considered and radiation from hot surfaces is neglected.

Taking the residential unit in C4 as an example, effect of parameter m on the onset of flashover was evaluated. The other parameters are kept the same as values in

As demonstrated in

Heat lost in a room fire through the wall depends on the material properties of the boundaries. Effect of wall properties on onsetting flashover was studied [

If the thermal inertia of the wall is large, the wall temperature parameter U_{c} in Equation (10) is small and more heat is lost through the walls due to a larger temperature difference between the hot smoke and the solid boundary. If the thermal inertia is small enough, U_{c} is nearly 1 and the wall has a good insulation performance.

Heat lost through the wall Q_{L} can be roughly estimated [

where h_{c} is an effective heat transfer coefficient.

For simplification, the heat transfer through the solid boundary is presumed to reach a steady state, and then the value of h_{c} can be determined by:

where value of C_{1} is 0.4 for compartment fires [_{w} is the thermal conductivity and d_{w} is thickness of the wall.

Actually, walls of buildings are made of layered material, it was assumed in studying flashover [

Typical values for thermal properties and the steady-state value of h_{c} for timber, gypsum plaster and concrete used before [

Fire hazards associated with open kitchen in small residential units of tall buildings storing large amount of combustibles should be watched. A nonlinear dynamic model [

It is observed that the solid boundary surface area or the floor aspect ratio will change the critical heat release rate and temperature for onsetting flashover, even for the same unit floor area. Consequently, architects can change the geometry of the residential unit with an open kitchen to give favorable conditions to mitigate the occurrence of flashover. The thermal properties of wall material also have effect on the critical conditions for flashover. Fire hazards of units with an open kitchen located in buildings enclosed with material of small thermal inertia should be given more caution.

Flashover in an open kitchen fire can be demonstrated by the model developed [

Materials | Timber | Gypsum plaster | Concrete | |
---|---|---|---|---|

Thermal conductivity k_{w}/W∙m^{−}^{1}・K^{−}^{1} | 0.12 | 0.8 | 1.6 | |

Density r_{w}/kg・m^{−3} | 540 | 1700 | 2400 | |

Specific heat capacity c_{w}/J∙kg^{−}^{1}・K^{−}^{1} | 2500 | 840 | 750 | |

Steady state effective heat transfer coefficient h_{c}/W∙m^{−}^{2}・K^{−}^{1} | 0.48 | 3.2 | 6.4 | |

Critical heat release rate Q/kW | C1 | 978 | 1070 | 1177 |

C2 | 978 | 1073 | 1186 | |

C3 | 868 | 964 | 1077 | |

C4 | 755 | 855 | 977 | |

C5 | 756 | 855 | 983 | |

Critical temperature T/K | C1 | 544 | 552 | 562 |

C2 | 544 | 553 | 563 | |

C3 | 516 | 525 | 535 | |

C4 | 488 | 498 | 509 | |

C5 | 488 | 498 | 510 |

assumptions made on the complicated heat transfer process can be taken as a starting point for future development.

The work described in this paper is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China for the project “Aspects of Open Kitchen Fires in Tall Building and Protection Alternatives” with account number B-Q27R.

Wan Ki Chow,Jing Liu, (2016) Application of Nonlinear Dynamics in Studying Flashover Fire in a Small Open Kitchen. Journal of Applied Mathematics and Physics,04,914-924. doi: 10.4236/jamp.2016.45100