Fire Safety Science

G. Cox, B. Langford, G. Cox, B. Langford

• Publisher: Routledge
• Year: 1991
• ISBN:
  9781135378509
• Edition: 1
• Topic: Architecture
• Subtopic: Architecture General

This book provides an essential reference on the current state of the art in this field covering topics as diverse as physics, chemistry, toxicology and human behaviour. It contains nearly one hundred scientific papers on all aspects of the subject. Many papers are included which illustrate the current state of development in the mathematical modelling of fire phenomena using computing.

Fire Physics

A Thermal Model for Piloted Ignition of Wood Including Variable Thermophysical Properties

Marc Janssens

National Forest Products Association

1250 Connecticut Avenue NW, Suite 200

Washington, DC 20036, USA

Abstract

A simplified thermal model of piloted ignition is formulated. The model equations are then solved numerically for the thermally thick case using a finite difference technique. A systematic analysis of some solutions leads to a functional relationship between ignition time t_ig and irradiance

suitable for correlation of piloted ignition data. This suggests plotting ignition data in a graph of (t_ig)⁻⁰⁵⁴⁷ versus

. The critical irradiance

is then found as the intercept with the abscissa of a straight-line fit through the data. An apparent kρc can be obtained from the slope of the regression line. Theoretical calculations show that this apparent kρc for wood products is evaluated at a temperature approximately halfway between T_∞ and T_ig. The suggested correlating procedure is applied to measurements for six oven dry wood species obtained in the Cone Calorimeter.

KEYWORDS: wood, piloted ignition, mathematical model, thermal inertia, Cone Calorimeter

Nomenclature

c	: Heat capacity (J∙kg^−1∙K^−1)
C	: Constant
C′	: Constant
F	: Characteristic function of time in equation (12)
h	: Convection coefficient (W∙m^−2∙K^−1)
k	: Thermal conductivity (W∙m^−1K−^1)
n	: Exponent
	: Heat flux (kW)
r	: Reflectivity
t	: Time (s)
T	: Temperature (K)
x	: Space coordinate (m)

Greek

α	: Absorptivity
γ	: Non-dimensional convection coefficient
ε	: Emissivity
η	: Non-dimensional space coordinate
θ	: Non-dimensional temperature
ρ	: Density (kg*m’^3)
τ	: Non-dimensional time
ψ	: Non-dimensional radiation coefficient
φ	: Non-dimensional irradiance

Subscripts

a	: Average
c	: Convective
e	: External
dry	: Oven dry
ig	: At ignition
r	: Reference
s	: Surface
∞	: Ambient

Superscripts

“	: Per unit area

Introduction

Piloted ignition of wood has been studied extensively over the past 40 to 50 years. The time to piloted ignition, t_ig, of a certain material is primarily a function of the incident heat flux. Ignitability at a given heat flux level depends on the thermal properties of the material, in particular the thermal inertia kρc. In previous work, small samples of wood were usually exposed to the radiant heat flux produced by a gas panel or an electric heater, and t_ig was measured as a function of the irradiance,

. Many investigators correlated such data using a power law of the following form

where C is constant for a given material and

is the critical irradiance below which piloted ignition under practical conditions no longer occurs. Lawson and Simms suggested n=2/3 and correlated C with kρc values obtained from the literature [1]. Buschman correlated n,

, and C with literature values for kρc [2]. Magnusson and Sundström suggested an inverse correlating procedure, i.e., a technique to derive an apparent kρc from the correlation of piloted ignition data [3]. However, this proposal was not very practical because it required measurement of surface temperature. Quintiere and Harkleroad developed another procedure to obtain an apparent kρc, without the need for such tedious temperature measurements [4].

The thermal properties used by Simms and Buschman were evaluated at ambient temperature. Both k and c of wood products, however, are strongly dependent on temperature. Intuitively one expects an apparent kρc to correspond to a temperature somewhere between T„ and the surface temperature at ignition, T_ig, perhaps closer to the latter. The results presented below confirm that this is indeed the case.

Mathematical Model of Piloted Ignition

A large number of mathematical models of the piloted ignition problem have been developed with varying degrees of complexity. Some models include gas phase diffusion and mixing [5]. Others consider only the solid phase, but include pyrolysis and other chemical reactions [6]. The model considered in this article is less sophisticated, but still includes many features not addressed by other thermal models. It is based on the following assumptions

• Heat flow in the solid is one-dimensional, i.e., perpendicular to the exposed surface.
• Chemical effects prior to ignition are negligible, i.e., no pyrolysis.
• Convective heat transfer between fuel vapors and the porous solid is negligible.
• Ignition occurs when the surface reaches a given material-dependent temperature. This criterion is acceptable for engineering purposes, as shown experimentally in [7].
• The material is opaque. Although wood is in fact not completely opaque, especially at small wavelengths, it is much less transparent than many other materials.
• Kirchoffs law is valid for the total α, ε and r, i.e., α = ε = l−r.
• The values for α, ε and r are constant between the start of exposure and ignition.
• The heat losses from the surface are partly radiative and partly convective with a constant convection coefficient.
• The specimens behave as a semi-infinite solid. All wood samples tested have a thickness of more than 16 mm and may be considered thermally thick [8].

Under these assumptions, the piloted ignition problem becomes a purely thermal problem. It has the following mathematical form

Energy Conservation

Boundary Condition (x=0)

Boundary Condition (x→∞)

Initial Condition (t=0, x ≥0)

The absorbed part of the critical irradiance,

is equal to the heat losses from the surface at ignition because below this irradiance level the surface temperature can never reach T_ig, even for t→∞. Consequently, a total heat transfer coefficient from the surface at ignition h_ig can be defined as

The objective is to find a solution of (2)–(5) in a form suitable for the correlation of experimental data and to infer an apparent value for kρc. The latter is a constant representative average over the temperature range between T_∞ and T_ig. In order to obtain such a solution, k and c are assumed constant. The following non-dimensional variables can then be defined

The resulting non-dimensional model equations are

Energy Conservation

Boundary Condition (η = 0)

Boundary Condition (η →∞)

Initial Condition (τ = 0, η ≥ 0)

Solution with Linearized Heat Losses

The solution of (8)–(11) with linearized surface heat losses (γ=1 and ψ=0) can be found in standard textbooks on heat conduction e.g., [9], p.276. At the exposed surface η=0, this yields a fairly simple expression

Equation (12) shows that θ_s<φ is independent of irradiance. The complimentary error fun...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• International Association for Fire Safety Science
• In Memoriam
• Symposium Committees
• Session Chairs
• Contents
• Invited Lectures
• Fire Physics
• Statistics and Risk
• Fire Chemistry
• Translation of Research into Practice
• Structures
• People and Fire
• Special Fire Problems
• Smoke Movement
• Detection
• Suppression
• Posters
• Author Index
• Subject Index
• Cumulative Author Index of Papers Presented at the First, Second and Third Symposia