# Combustion of Polymers

## The critical mass flux concept

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.

### Experimental evidence

The concept that a critical mass flux can be used as an ignition condition was introduced by Bamford et al (1946). Based upon a combined experimental and theoretical investigation into the pilot ignition of wood a critical value
\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.

Table 1. Experimental values for the critical mass flux.
AuthorCritical 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.

### Theoretical evidence

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.

### Applications of the critical mass flux hypothesis to modelling polymer combustion

#### Thermally thin samples

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.

#### Thermally non-thin samples

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

1. A. Atreya and I.S. Wichman. Heat and mass transfer during piloted ignition of cullolosic solids. ASME Journal of Heat Transfer, 111(3):719--725, 1989.
2. V. Babrauskas and W.J. Parker. Ignitability measurements with the cone calorimeter. Fire and Materials, 11:31--43, 1987.
3. C.H. Bamford, J. Crank, and D.H. Malan. On the combustion of wood. Part I. Proceedings of the Cambridge Phil. Soc., 42:166--182, 1946.
4. C.Di Blasi. Modelling and simulation of combustion processes of charring and non-charring solid fuels. Progress in Energy and Combustion Science, 19(1):71--104, 1993.
5. D. Deepak and D.D. Drysdale. Flammability of solids: An apparatus to measure the critical mass flux at the firepoint. Fire Safety Journal, 5:167--169, 1983.
6. M.A. Delichatsios and M.M. Delichatsios. Critical mass pyrolysis rates for extinction of fires over solid materials. In Fire Safety Science: Proceedings of the Fifth International Symposium, pages 153--164. International Association for Fire Safety Science, 1997.
7. A.N. Koohyar, J.R. Welker, and C.M. Sliepcevich. The irradiance and ignition of wood by flame. Fire Technology, 4:284--91, 1968.
8. S. Melinek. Fire Research Note 755, 1969.
9. M.I. Nelson, J. Brindley, and A.C. McIntosh. The dependence of critical heat flux on fuel and additive properties: A critical mass flux model. Fire Safety Journal}, 24(2):107--130, 1995.
10. M.I. Nelson, J. Brindley, and A.C. McIntosh. Polymer ignition. Mathematical and Computer Modelling, 24(8):39--46, October 1996a.
11. M.I. Nelson, J. Brindley, and A.C. McIntosh. Ignition properties of thermally thin materials in the cone calorimeter: A critical mass flux model. Combustion Science and Technology, 113-114:221--241, 1996b.
12. M.I. Nelson, J. Brindley, and A.C. McIntosh. Ignition properties of thermally thin thermoplastics - The effectiveness of inert additives in reducing flammability. Polymer Degradation and Stability, 54(2-3):255--267, 1996c.
13. M.I. Nelson, J. Brindley, and A.C. McIntosh. The effect of heat sink additives on the ignition and heat release properties of thermally thin thermoplastics. Fire Safety Journal, 28(1):67--94, February 1997.
14. T. Ohlemiller and T. Shields. One- and two-sided burning of thermally thin materials. Fire and Materials, 17:103--110, 1993.
15. D.J. Rasbash, D.D. Drysdale, and D. Deepak. Critical heat and mass transfer at pilot ignition and extinction of a material. Fire Safety Journal, 10:1--10, 1986.
16. S.J. Ritchie, K.D. Steckler, A. Hamins, T.G. Cleary, J.C. Yang, and T. Kashiwagi. The effect of sample size on the heat release rate of charring materials. In Fire Safety Science: Proceedings of the Fifth International Symposium, pages 177--188. International Association for Fire Safety Science, 1997.
17. J.E.J. Staggs and M.I. Nelson. A critical maxx flux model for the flammability of thermoplastics. Combustion Theory and Modelling, 5:399-427, 2001.
18. A. Tewarson. Experimental evaluation of flammability parameters of polymeric materials. In S.M. Atlas and E.M. Pearce, editors, Flame Retardant Polymeric Materials, volume 3, pages 97--153. Plenum Press, New York and London, 1982.
19. H.E. Thomson and D.D. Drysdale. Flammability of plastis II. Critical mass flux at the firepoint. Fire Safety Journal, 14(3):179--188, 1989.
20. L.S. Tzeng, A. Atreya, and I.S. Wichman. A one-dimensional model of piloted ignition. Combustion and Flame, 80:94--107, 1990.

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Page Created: 29th January 2002.
Last Updated: 21st August 2002.