In order to implement a model describing the ignition and subsequent combustion of a polymer a statement has to be made about when the flame appears. A complete description of the mechanism leading to the establishment of a flame over a burning surface requires consideration of mass and heat transport in the gas-phase (Blasi 1993). Instead, ignition and extinction are interpreted in terms of a critical mass flux of volatiles sufficient to support a nascent flame; a flame appearing (disappearing) when the flux of volatiles into a boundary layer above the surface of the polymer increases (decreases) though a critical value. The advantage of this approach is that no model is required for transport processes in the gas-phase, so that simple predictions of flammability emerge.
\dot{m}''cr \approx 2.5e-3 kg m-2s-1
was deduced.
Although the concept of a critical mass flux is widely accepted, there have been few experimental investigations into its validity (Koohyar et al 1968; Melinek 1969; Tewarson 1982; Deepak and Drysdale 1983; Rasbash et al 1986; Thomson and Drysdale 1989). The critical values reported in these investigations are summarised in table 1. The paper by Rasbash et al (1986) is noteworthy as the first systematic investigation into the validity of the critical mass flux concept. The lower values measured by Thomson and Drysdale (1989) were attributed to the sensitivity of the mass flux to the convective heat transfer coefficient at the polymer surface. Hence experimentally determined values for criticality may be context-specific.
| Author | Critical Mass Flux (kg m-2s-1) | Comment |
|---|---|---|
| Bamford et al (1946) | \dot{m}''cr \approx 2.5e-3 | Pilot ignition of wood. |
| Koohyar et al 1968 | Scattered over an order of magnitude | Vertical slabs of different woods. |
| Melinek 1969 | \dot{m}''cr \approx 5.1e-3 | Analysis of the data of Koohyar et al 1968. |
| Tewarson 1982 | 1.9e-3 <= \dot{m}''cr <= 3.9e-3 | Thermoplastics - natural convection. |
| 2.5e-3 <= \dot{m}''cr <= 4.5e-3 | Thermoplastics - forced convection. | |
| Deepak and Drysdale 1983 | \dot{m}''cr \approx 4-5e-3 | PMMA. |
| Rasbash et al 1986 | \dot{m}''cr \approx 3-6e-3 | Thermoplastics - pilot ignition. |
| Thomson and Drysdale 1989 | 0.8e-3 <= \dot{m}''cr <= 2.0e-3 | Thermoplastics. |
Atreya and Wichman (1989) have presented an experimental and theoretical investigation into the heat and mass transfer processes that occur during piloted ignition of thermally thick cellulosic materials. By combining the concept of a nearly constant limit diffusion flame temperature at extinction with that of a nearly constant heat of combustion of air for most hydrocarbons they derive simultaneous equations for the mass flux and surface temperature at ignition. (The energy released upon complete combustion of a unit mass of fuel is called the heat of combustion of the given fuel. This energy when calculated per unit mass of the stoichiometric amount of air required for combustion is termed the heat of combustion of air). They deduced a critical value
\dot{m}''cr \approx 1.8e-3 (kg m-2s-1).
The existence of a minimum fuel flow rate as a criterion for
piloted ignition and extinction of a steady diffusion flame has been
investigated by
Tzeng et al (1990) using
a one-dimensional analysis of a thin gaseous slab that is
periodically raised to the adiabatic flame temperature of the
stoichiometric mixture. This work substantiates the
hypothesis that conditions at extinction of a steady diffusion flame
are very close to those at piloted ignition, which is one of the
modelling assumptions used by
Atreya and Wichman (1989).
The existence of critical mass pyrolysis rate for flame extinction has been investigated by Delichatsios and Delichatsios (1997). Extinction conditions were derived by separating the dynamics of gaseous reactions from the energy balance in the solid using a simple physical interpretation. The conditions were validated by comparison with experimental data.
The simplifying hypothesis that the ignition and extinction of a flame can be modelled by a critical mass flux assumption is supported by experimental and theoretical studies. This assumption has therefore been used as a modelling strategy in investigations into the flammability of thermally thin polymers (Ohlemiller and Shields 1993; Nelson et al 1995, 1996a, 1996b, 1996c 1997). Ohlemiller and Shields (1993) is a comparison of the one- and two-sided burning of materials. Nelson et al have used the critical mass flux approach to investigate the flammability of thermoplastics (1995, 1996a, 1996b) and to model the effectiveness of solid-phase active fire-retardants (inert and heat-sink) (1995, 1996c 1997) by assuming that the incorporation of the additive leaves the criticality condition unchanged. In these papers the dynamics of the flame are not modelled, the critical mass flux hypothesis is used as a `switch' to turn a flame on and off.
Staggs and Nelson (2001) model of thermally non-thin thermoplastics. Results are presented which quantify the effect that the thickness of the test sample has on the mass-loss rate, or equivalently heat-release rate, curve. From these the authors conclude that thermally thick samples are characterized by a region of steady burning which is independentend of the initial sample thickness.
The critical mass flux hypothesis has also been used to investigate the burning behaviour of thermally thick charring materials (Babrauskas and Parker 1987; Ritchie et al 1997). The latter paper does not model the flame as a `switch', instead the flame temperature is calculated using an enthalpy balance about a control volume encompassing the flame.
References