A simplified analytical model for radiation dominated ignition of solid fuels exposed to multiple non-steady heat fluxes

Abstract

Heat fluxes from fires are strongly time-dependent. Historically, the thermal ignition theory in its classical form has neglected this time dependency until recent years, where theories have been developed to include time-varying incident heat fluxes. This article proposes a simplified general model formulation for the heating of solid fuels exposed to four different heat flux behaviors, considering the penetration of radiation into the medium. The incident heat flux cases developed where: Constant, Linear, Exponential and Polynomial, which represent different situations related to structural and wildland fires. The analytical models consider a spatially averaged medium temperature and exact and approximate solutions are presented, based on the critical ignition temperature criterion, which are valid for solids of any optical thickness. The results were validated by comparison with various models presented in the literature, where the model granted in this work was capable to adjust to all of them, especially when high heat fluxes are involved. Therefore, the proposed model acquires a significant engineering utility since it provides a single model to be used as a general and versatile tool to predict the ignition delay time in a manageable way for solid fuels exposed to different fire conditions.

Introduction

The problem of the ignition of solid fuels subjected to radiant heat fluxes has received the attention of the fire safety science community for at least 75 years because of its role in several fire initiation and fire growth processes. The seminal work by Bamford et al. [1] opened the way for the development of theories which decoupled solid- and gas-phase processes, thereby promoting the development of analytical solutions which can be treated to yield simple expressions for pencil and paper calculations amenable for the practitioners. The overall problem of increasing the fire safety in the built environment, and the threat of nuclear warfare prompted further work into this topic by different researchers in Britain and America [2], [3], [4], [5], [6], [7]. Lawson and Simms [2] were the first to theoretically consider the conductive heating of an inert solid exposed to radiant heat fluxes through the heat diffusion equation. Their work assumed the following: (i) infinite Damkhler number, which allows to treat the gas phase as an extinction problem and not an ignition problem, making a pilot necessary; (ii) ignition occurs when the lean flammability limit is attained at the pilot location; (iii) the entire solid is inert during the heating stage. These assumptions can be merged into the general assumption used in all the works which follow this approach that flaming ignition is attained when the solid reaches a critical or ignition temperature, $T_{ig}$. The solution to this classical heat conduction problem [8] yielded good agreement with experimental results (particularly for intense incident heat fluxes ${\dot{q}}_{inc}^{″}$ such that the ignition delay time, $t_{ig}\sim 0$), and paved the way for a physical interpretation of experimental data. This approach corresponds to the thermal ignition theory based on the radiant heating of an inert material.

From the 1950s through the 1970s, two distinct experimental approaches are recognized in terms of the thermal ignition problem. The first considers the use of radiant panels or flames to produce a continuous heat insult over the surface of the sample and is applicable to the fire problem because the objective is not to develop a scenario approach that resembles a fire but to develop a canonical experiment that allows to explain the ignition problem and extract effective thermal properties [1], [2]. The second approach subjected samples to pulses of radiation of much larger magnitude than in the former case, characteristic of atomic blasts [3], [6].

The historical development of fire science prompted members of the combustion community to focus on the fire problem, aided by the development of new bench-scale experimental apparatuses in the late 1970s whose operating principles and experimental goals were guided by the theoretical development of both the ignition and flame spread problems [9]. Based on the thermal ignition theory, the suitable correlation of test results from the new apparatuses permitted the analyst to obtain apparent material properties applicable to several fire safety problems, like ignition temperatures, heat release rates, and thermophysical properties. A key feature of the “modern” interpretation of the experimental data is the recognition of the linear relationship between $t_{ig}^{-1/n}$ vs. ${\dot{q}}_{inc}^{″}$ (for $t_{ig}\to 0$), where $n$ depends on the thermal thickness of the sample (given by the Fourier number of the solid fuel) and it takes the value of 1 for thermally thin samples and 2 for thermally thick fuels. This correlation is readily obtainable from the solution of the heat transfer problems for the inert materials, and while the Lawson and Simms developed the correlation (cf. Eq. (6) in [5]), its use to obtain apparent properties was pioneered by the University of Oklahoma group in the early 1970s [10, Eq. IV-6]. Note that many researchers cite the work of Quintiere [9], [11] as being the first to establish the correlation of ignition data in modern form, but to the best of the authors’ knowledge, the earliest work to present these correlations was that by Mikkola and Wichman [12], [13]. The following decades were marked by the widespread use of the thermal ignition theory and correlations of $t_{ig}^{-1/n}$ vs. ${\dot{q}}_{inc}^{″}$, which have now become the standard form of analyzing ignition data by the fire community [14], [15], [16], [17], [18], [19], [20].

A common feature in all of the early works was the assumption that the radiation was treated only as a boundary condition, thereby considering the solid fuel as optically opaque. While this assumption is adequate for materials like wood, it may cease to be appropriate when dealing with PMMA and other materials which have in-depth absorption of radiation [21], [22]. Simms recognized this issue early on [23], including the effect of diathermancy or the divergence of the radiative heat flux term ($\nabla {\dot{q}}_r^{″}$) in the energy equation, but later neglecting it to obtain an analytical solution. This problem only began to be addressed in the 2010s [19], [24], although the first analytical treatment of the radiative transfer equation (RTE) to obtain $\nabla {\dot{q}}_r^{″}$ in the context of the thermal ignition theory was recently developed by our group [20], [25]. The authors used the novel treatments developed by combustion researchers working in participating media (with applications in soot, spectroscopy, and laser diagnostics). It is significant to note that the approximations of the RTE were developed more than 100 years ago by the astrophysical and atmospheric science communities [26], [27], [28], [29].

Heat fluxes from fires are strongly time-dependent. However, the thermal ignition theory in its classical form neglects this time dependency. Recently, the theory has been expanded by different groups to include the temporal evolution of the incident heat flux, with different methodologies being proposed to treat this problem by considering different expressions for the time-dependent flux, which include increasing, decreasing, periodic or cyclical functions of time [18], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. What has not been done by the community so far is to create a generalized closed-form analytical solution, formally treating the RTE with radiation penetration into the solid and including the time dependence in the incident radiative flux. The purpose of this study is to expand the previous work carried out by the authors on ignition theory and propose a general analytical formulation for the ignition delay times of solids fuels exposed to different heat flux behaviors, from the constant case to more complex scenarios like exponential and polynomial functions of time, considering the divergence of the thermal radiation, which represent different situations related to structural and wildland fires. Additionally, a series of functional relationships of $t_{ig}^{-1/n}$ vs. ${\dot{q}}_{inc}^{″}\left(t\right)$ are presented in order to systematically correlate experimental data, which can be easily applied to previous works [18], [34], [41].