T.GRAHAM, G.MAKHVILADZE and J.ROBERTS Department of Built Environment,
University of Central Lancashire, UK Abstract
This paper examines the nature, role, and critical relationship between non dimensional parameters for flashover. A two-zone model for the development of a fire in a ventilated enclosure is proposed and used. The treatment of the model performed with the aid of non-dimensional variables results in the formulation of critical conditions for flashover;
these critical conditions are obtained for small and high thermal inertia of the walls.
Keywords: Compartment fire, enclosure, zone model, critical condition, flashover
1 Fire development and flashover in a compartment
Typical fire development in an enclosure is characterised by a temperature history such as that sketched in Fig. 1.
Fig. 1. Stages in a compartment fire temperature history.
Fire Engineering and Emergency Planning. Edited by R.Barham.
Published in 1996 by E & FN Spon. ISBN 0 419 20180 7.
Following ignition there is a slow growth period, which is limited by the pyrolysis rate of the fuel supply (fuel controlled combustion). Early in the fire development the combustion products are usually segregated in a well-stirred ceiling layer whose properties are roughly homogeneous. The hot smoke and warming room boundaries radiate heat back to the fire. Radiative feedback can enhance the reaction rate so that the fire accelerates towards the fully developed stage.
The transition (which is usually very rapid) to a fully developed fire is called
‘flashover’. Flashover fires are disastrous [1], and an increasing problem now [2] because of the use of modern materials in buildings. It is a major objective of research to understand and to predict this process so that precautions can be made during building design. Apart from flashover, the regime of quasi-steady low-intensity fire is possible (lower curve) with small heating of the upper layer.
Thomas [3] and Thomas et al [4] presented a theoretical treatment of the flashover phenomena using the fundamentals of classical thermal explosion theory. Further development of this approach was made by in [5,6] with the application of modern nonlinear dynamics. In this treatment the flashover point is represented by a fold catastrophe.
In this paper we suggest the two-zone model for the compartment fire and, in line with the classical explosion theory approach we deduce the characteristics for the flashover phenomenon, namely the analytical expression for the critical temperature and critical conditions in terms of determining parameters.
2 The zone model
Zone models are known to assume that the compartment is divisible into two homogeneous regions: a hot/smoke zone and a cool/lower zone. Each zone is represented by average thermodynamic/gas properties. In this approach we lose local detail in return for simplicity, but gain the ability to interpret complicated bulk phenomenon with physical clarity. Another advantage is that one could specify a small number of the most important (determining) parameters and obtain quite simple relationships, describing the main features of the system.
Main assumptions:
• the compartment can be divided into two zones which may be represented by average temperatures,
• the fluid motion is very slow in comparison to the velocity of sound,
• flashover takes place during the early development of the fire, within the fuel controlled combustion regime; during early fire development the density of the lower zone may be assumed to be its initial value (ρL≈ρ0),
• the wall surfaces surrounding the zones can be described by two temperatures, the lower zone and wall surfaces below the thermal discontinuity are at the initial temperature.
The equation of energy conservation takes the form of a heat balance:
(1) Fire engineering and emergency planning 20
The left hand side being the change in internal energy of the hot layer, t is the time, T is the smoke/hot zone temperature, m is the total mass in the hot layer, cp is the specific heat capacity (at constant pressure). On the right hand side G is the net heat gains and L is the net heat losses from the hot zone. The initial condition is T(t=0)=To, where To is the reference (ambient room) temperature.
We treat the case of a developing fire and reasonably assume that flashover occurs during the early development of the fire, and do not account here for factors such as fuel/air exhaustion. As a consequence of the assumption ρL≈ρ0, at the beginning of flashover the neutral plane and the thermal discontinuity plane coincide.
The heat gains for the smoke layer are given by:
(2) where χ is the efficiency of the combustion process (the fraction of the theoretical heat that would reach the smoke layer), ∆hc is an effective heat of combustion, ∆hvap is the effective heat of vaporisation of the solid fuel, is the mass burning rate of the fuel, Af
is the pyrolysing area (surface area of fire), is the incident heat flux to the fuel surface from the fire, αU(T) is the radiation feedback coefficient from the hot layer at temperature T, σ is the Stefan-Boltzman constant. The right hand side of the equation contains two terms; the first being the heat gain to the smoke layer from a free-burning fire, and the second term describes the radiation feedback to the fire from the upper zone [3] and compartment walls.
The heat losses from the smoke layer are given by:
(3)
Here is the total mass flow of smoke/gas out of all vents and doorways, Av is the total area of the vent, Au is the surface area of wall surrounding the smoke layer (including any windows), D is the fractional height of the thermal discontinuity plane (D=ZD/Hv, ZD is the height of the thermal discontinuity plane above the bottom of the vent, Hv is the height of the vent), hc is the convective heat transfer coefficient for the hot wall surfaces, Tw is the surface temperature of the walls surrounding the hot zone, hv is a convective heat transfer constant for the vent, αg is the emissivity of the gas layer, σ is the Stefan-Boltzman constant, AL is the surface area of walls surrounding the cool/lower zone (including referred parts of any vents), Tf is the surface temperature of the fuel bed.
The right hand side of (3) contains six terms. The first is an enthalpy flow out of the vent. The second and third terms are convective heat losses to the walls surrounding the hot layer, and out of the vent respectively. The last three terms are the radiative heat transfer from the hot zone to the hot wall surfaces, the cool zone and the vent, and the fuel bed areas respectively.
The parameter D varies from 0 to 1 and can in principle be found from the mass balance equation (in line with the assumption ρL=ρ0 we ignore the mass influx
Critical conditions for flashover in enclosed ventilated fires 21
into the room through the lower part of the vent under the neutral plane). However, this makes further calculation algebraically difficult, and is not necessary at this stage in the study. We shall assume that D is a constant (does not depend on T) and we will give results for different values of D.
Henceforth we shall use dimensionless time introduced by the following formula , where t*=mcpTo/Es is the characteristic time of heating of the upper layer by heat from a freely burning fire, the value is the characteristic heat flux (per unit time) to the smoke layer for a freely burning fire (that is to a smoke layer at the ambient temperature), and dimensionless temperature, θ=T/T0. With these dimensionless variables equation (1) with the initial condition takes the form:
(4)
where
Here εout is a dimensionless scale for enthalpy flow out of the vents of the compartment, εR,j(j=W, L, f) is the dimensionless scale for radiative heat transfer from the hot zone to the hot walls, to the lower zone and to the fire bed, εC,k(k=H, L) is a dimensionless scale for heat convected from the hot layer to the wall surfaces and the vent surfaces. Here on we shall ignore the term εout(θ−1) in comparison with the heat gains due to heat flux from the fire, because , but ∆hc>>cp(T−T0). Assuming also that
and determining the wall temperature Tw as in [5]
(θw=1+β(θ−1), 0≤β≤1) we have:
(5) where
(6)
The simplest model is described by the four determining parameters a1, a2, a3, and a4. Solutions to (5) in the non-dimensional variables, also the initial condition and critical condition are given by some function of these parameters and time:
Fire engineering and emergency planning 22
(7)
3 Critical conditions
An intersection of the curves of gain function G and loss function L represents quasisteady behaviour. Generally speaking, there are at most three solutions to the balance condition (see Fig. 2, which is called Semenov’s diagram in classical thermal explosion theory) but the intermediate solution is unstable and not observed in practice.
This is because any small perturbations will result in a large change in temperature. The number of intersections may change.
During fire development the losses and gain function curves move relative to each other. With the approximations used here the gain curve does not actually move and Fig.
2 shows just a variation in losses.
Consider the behaviour of heat losses as the walls are heated. The heat conducted away is proportional to the temperature difference; so losses begin to fall. A quasisteady state in the fuel controlled regime (point P) moves towards higher temperatures (Fig. 2).
In the case of flashover this movement of the point continues until a tangency of the curves occurs at the point P* followed by a rapid increase in temperature.
Fig. 2. Semenov’s diagram, critical points in development.
Small perturbations beyond the critical point lead to a catastrophic jump to the only remaining quasi-steady state. This jump or bifurcation to a higher temperature represents flashover. Extinction is represented by a jump to a lower temperature (for example when fuel is depleted and heat gains decrease there would be such an extinction). The phenomenon is well known in thermal explosion theory [7].
Differentiating the heat conservation equations gives the tangent to the curves. The critical points are found where the tangent is zero. If the tangent is positive then gains
Critical conditions for flashover in enclosed ventilated fires 23
exceed losses and the fire develops, conversely a negative tangent represents a decaying fire. Our aim is to study the lower critical point.
3.1 Walls of large thermal inertia
Consider different cases of β (which describes the thermal response of the walls). In the case of a large thermal inertia there is a delay for the wall surface temperature to rise before flashover and β=0.
The coefficients become:
a0=1, a1=εK−εR,L−εR,f−εR,W, a2=εC,H+εC,L, a3=a4=0
(8) and we have only two determining parameters. Equation (5) takes the form:
(9) which gives the critical conditions:
(10)
Hence the critical temperature is
(11)
where is the total
inner surface area of the compartment. Clearly if θ>θ* there is no stationary solution, gains exceed losses and flashover takes place. Equation (11) is a simple expression in terms of the ratio of convection heat losses from the smoke layer, in comparison to the net radiation heat gains.
Fire engineering and emergency planning 24
Fig. 3. Critical temperature as a function of dimensionless fire area.
The dependence of the critical temperature θ* as a function of the non-dimensional fire area is presented in Fig. 3 for the value b2=34.25 (from the data in [5]).
The region of flashover is above these curves. The greater the fire area, the easier for the fire to flashover. The greater the value of b1 (the more intensive the convective heat losses, or the greater the vent area), or the greater the value of b2 (the less intensive the heat release of the fire), the greater the critical temperature and the greater the difficulty for the fire to achieve flashover. All these considerations are physically reasonable.
Noting that usually Av<<Au, hv≈hc, we have from (11)
(12)
In this limit, the critical temperature θ* does not depend upon the vent area.
It is physically clear that θ*≥1(a2≥4a1), i.e.
(13)
This shows how the non-dimensional fire area is limited by the ratio of convective heat losses to radiative heat gains for a quasi-stationary regime of fire to exist. This is a necessary but insufficient condition for the existence of the quasi-steady state.
Substituting θ* from the second equation in (10) back into the first, gives the critical relationship between the non-dimensional parameters a1 and a2 which can be presented in the parametric form:
Critical conditions for flashover in enclosed ventilated fires 25