J.E.J. Staggs

Modelling thermal degradation of polymers using single-step first-order kinetics

A theoretical framework for characterising single-step Arrhenius degradation kinetics in terms of a characteristic temperature and temperature range is developed. It is demonstrated that for the purposes of practical calculation, the reaction order may be assumed to be unity and also that a first-order approximation to an nth-order TG curve remains a good approximation over an order-of-magnitude variation in heating rate. This fact implies that when the pyrolysis of much larger samples of material is modelled, the error involved in using first-order kinetics is small. The equivalent first-order approximation is then applied to a global in-depth model of polymer degradation in order to predict mass loss rates in bench-scale experiments such as the cone calorimeter test. The mass loss rate curves obtained from the equivalent first-order approximation are found to compare well with the full nth-order model. Finally, an estimate of the average or steady mass loss rate is developed which fully accounts for the interaction between the degradation kinetics, the external heat flux, the heat losses and the latent heat of vaporisation.

A simple model of polymer pyrolysis including transport of volatiles

Mathematical models and physical understanding of polymer pyrolysis are key tools in predicting the fire safety performance of polymeric materials. Without a suitably realistic view of the processes occurring in the solid phase during gasification, flame spread or ignition phenomena cannot be accurately or reliably predicted. Most models of polymer pyrolysis neglect to include two key factors: change of volume during pyrolysis and transport of volatile gases. In this paper, a simple model of volatile transport is incorporated into a model of pyrolysis, which includes change of volume. The qualitative effect of volatile transport is discussed and is found to be particularly important at low heat fluxes, moderate solid thermal conductivity and moderate decomposition temperature range. The shortcomings of the simple model are also discussed and suggestions for further work are included.

Heat and mass transport in developing chars

The physics of heat and mass transport in chars is explored in the context of a developing char layer. The most important features of the problem are discussed and a section of the relevant literature is briefly reviewed. A mathematical model is then developed and analysed in order to illustrate some of the important phenomena. The model relates to the simplified case of a polymer which degrades to produce char and volatile gases in a single step at a critical temperature. Transport of gas through the char is modelled by Darcys law and for simplicity it is assumed that there is no overall volume change as char is formed. Conservation of mass, Darcys law and the perfect gas law are combined to give a non-linear diffusion equation describing the evolution of pressure in the developing char. This equation, together with equations expressing conservation of energy in the char and the virgin polymer and appropriate boundary/initial conditions complete the description. Exact solutions are derived for isothermal conditions in the char and these are used to frame the numerical results for the full model. A key parameter in the model is a dimensionless number λ akin to the Lewis number, namely the ratio of pressure diffusivity to thermal diffusivity. Generally, for realistic parameter values, λ is very large and the results show that under such conditions the developing char offers no resistance to the flow of gases. In other words, one may effectively assume that the mass flux of volatile gases through the char is uniform. Numerical solutions of the full model are presented and discussed and the question of integrity of the char is briefly explored.

Modelling random scission of linear polymers

A mathematical model for the random scission of linear polymers is presented. The model takes the form of a set of ordinary differential equations which describe the evolution of the MW distribution as a function of the fraction of bonds broken. An exact solution, valid for large initial MW, is derived from the equations and compared with results from a Monte-Carlo type simulation. Isothermal thermogravimetric experiments using polyethylene are used to suggest a relationship between the rate of bond breaking and temperature and this is then used to compare model predictions for the rate of degradation with constant heating rate thermogravimetric experiments. Excellent agreement is found between theoretical predictions and experimental results for the case of a standard sample of polyethylene with number-average MW 2015.

Modelling the combustion of solid‐phase fuels in cone calorimeter experiments

The process of ablation often forms a key part in many mathematical models describing the combustion of a solid-phase fuel. The phrase was first used to describe the thermal erosion of glaciers over 140 years ago. In recent times it has been applied to the thermal degradation of a solid when exposed to a large flux of heat. Two fundamental assumptions in the treatment are that mass is lost only from surface regions and that the temperature of the surface remains constant throughout the period of mass loss. The two critical parameters in this model are the critical temperature Tp at which mass loss occurs and the heat required to convert matter from solid to gas at temperature Tp. In this paper we report mathematical models designed to simulate the loss of mass of a solid fuel (such as polyethylene) in cone calorimeter experiments. The models do not compromise on the thermal properties of the solid fuel or the heat loss mechanisms and consequently they require a numerical method of solution. Initially we explore a relatively simple ablation-based model. Although conceptually simple in approach, the model reproduces many qualitative features observed in experiments. We go on to consider a different approach, where the thermal degradation of the solid is governed by a set of kinetic rate laws. We show that this approach removes many of the unrealistic features and assumptions of ablation-based models. Furthermore, we demonstrate that the agreement between theory and experiment is improved by this approach. We also show how the mathematical models may be used as aids in the interpretation of cone calorimeter data. For example, we show that a steady state measurement of mass loss rate is a more reliable indicator of a material response than a peak measurement. In fact we show how peak measurements are strongly affected by the thermal properties of the sample holder, but steady state measurements are insensitive to the particular choice of sample holder. Copyright

Estimating the thermal conductivity of chars and porous residues using thermal resistor networks

A conceptually simple approach is used to estimate the thermal conductivity of a composite solid. The approach uses a three-dimensional (3D) network of thermal resistors to model a unit cube of material. Fixed temperature boundary conditions are applied on two opposing faces of the cube and the other four faces are well insulated. The net heat flux through one of the fixed temperature faces is then used to compute the effective thermal conductivity. The results are compared with analytical estimates and experimental results for Al2O3 with spherical inclusions and the differences between 2D and 3D results are investigated. Correlations are also given for the numerical results. The cube itself may be composed of any number of different phases (provided of course that this number is much less than the total number of thermal resistors in the network) but the discussion is restricted to two phase systems with some reference to a three phase system.

A simplified mathematical model for the pyrolysis of polymers with inert additives

A mathematical model for the pyrolysis of a solid consisting of a homogeneous mixture of a polymer and an inert, porous filler is discussed. Change of volume of the solid during pyrolysis is modelled and it is assumed that volatile species, formed from the thermal degradation of the polymer, escape as soon as they are formed. The filler and polymer may have different thermal properties. This simplified model represents an upper bound for the early mass loss rate behaviour for the case where the escaping gases diffuse through the filler residue. Although results are presented for the entire mass loss period, for the case of heavily filled materials we expect them only to be valid for the early stages, when the physical effects of the filler residue are negligible. For lightly filled materials, the model should give better results for most of the mass loss period. Change of volume during pyrolysis is shown to have little effect on the initial mass loss rate for moderate external heat fluxes, but has a considerable effect at later times. Furthermore, the model predicts that for increased ignition resistance, optimum filler properties depend on initial filler loading. Low-to-moderate filler loadings require a filler with a similar density to the host polymer, but with higher thermal conductivity and specific heat capacity; highly loaded systems require a filler with high density as well as high thermal conductivity and specific heat capacity.

A discussion of modelling idealised ablative materials with particular reference to fire testing

Abstract An investigation is made of a mathematical model of an idealised ablative material of finite thickness, which is mounted horizontally and exposed to a constant, uniform heat flux. The effect of a substrate or sample mounting is also investigated. Numerical and approximate solutions are developed for a perfectly conducting and a perfectly insulating substrate. These two extreme cases represent idealised upper and lower bounds for the mass loss rate of an ablative material mounted on a real substrate. Three ablation regimes (thermally thick, thermally non-thin and thermally thin) are identified and quantified in terms of the dimensionless parameter l l c , where l is the initial thickness of the sample and lc is a critical thickness defined in terms of the difference in ablation temperature from ambient temperature, incident heat flux and thermal conductivity. The approximate solutions are valid for the thermally thin and non-thin regimes. It is observed in the numerical solutions that the substrate or sample holder has negligible effect on the mass loss rate in the thermally thick regime. However, the sample holder may have a large effect on the mass loss rate for the thermally thin and non-thin regimes.

A Theoretical Investigation into Modelling Thermal Degradation of Solids Incorporating Finite-Rate Kinetics

We consider a mathematical model of thermal degradation of thick solids which is formulated in terms of a general kinetic rate law. The model is applicable to many fire-test situations, notably the standard cone calorimeter test where a horizontal sample is exposed to a uniform heatllux. The model incorporates a mechanism for the transfer of mass through the material as vaporisation occurs - a feature neglected in many mathematical models published to date. Attention is focused on a single-step nth. order Arrhenius reaction for the thermal degradation. Results are presented which quantify the effects of incident heat flux, heat losses and kinetic mechanism on the temperature distribution in the solid and the mass loss rate. In particular we demonstrate that thermal history strongly affects mass loss rate for materials that degrade according to reactions of non-unitary order. The consequences of this and other effects on the ignition characteristics of solids are briefly discussed.

A theory for quasi‐steady single‐step thermal degradation of polymers

A theory for approximately steady thermal degradation of solids is developed from a superset of nonlinear integral-differential equations. The theory extends previous work, using a degradation model that is more consistent than previously published models and fully accounts for surface radiation losses. The thermal decomposition of the solid is assumed to follow a single-step first-order Arrhenius reaction. A quasi-steady regime is identified and approximate solutions are compared with experimental results for PMMA and numerical results obtained by integrating the full model. The numerical solutions are found to compare well with experimental results and the approximate solutions compare well with the numerics. Furthermore, it is found that the quasi-steady mass loss rate gives a good estimate of the average mass loss rate even during thermally thin degradation. To simplify interpretation and to aid the analysis, the degradation kinetics are re-cast in terms of a critical temperature and a critical temperature range. Application of the theory to practical situations and other modelling approaches is also discussed.