# Encyclopedia of Wildfires and Wildland-Urban Interface (WUI) Fires

Living Edition
| Editors: Samuel L. Manzello

# Conduction

Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-51727-8_224-1

## Definition

Conduction is the form of heat transfer that occurs within a stationary medium due to the presence of a temperature gradient.

## Basic Concepts

Fire science requires a fundamental understanding of heat transfer. There are three basic mechanisms of heat transfer: conduction, convection, and radiation. Within the field of fire science, usually all three mechanisms are important, but for the different stages of fires or types of fires, one of the three mechanisms generally predominates over the other. Conduction is the form of heat transfer that occurs within a stationary medium due to the presence of a temperature gradient. Within fire science, this is often the predominant form of heat transfer in ignition and fire spread for solid combustible elements, as well as for the study of insulation materials.

In conduction, the heat flux is proportional to the spatial derivative of temperature and is governed by Fourier’s law of conduction. For a one-dimensional homogeneous and isotropic object, this law reduces to
$${\dot{q}}_x^{\prime \prime }=-k\ \frac{dT}{dx}$$
where $${\dot{q}}_x^{\prime \prime }$$ is the heat transfer per unit area, $$\frac{dT}{dx}$$is the spatial derivative of temperature in the one-dimensional x, and k is the proportionality constant between the two and is known as the thermal conductivity of the solid with units of $$\frac{W}{m\cdot K}$$.

## Thermal Conductivity

Thermal conductivity can be dependent both on local temperature and orientation of the solid. For materials like water, air, and most metals, the thermal conductivity is independent of orientation (isotropic), but for other materials like biomasses, it is also dependent on orientation. The most common material in fire applications where thermal conductivity is dependent on orientation is wood; as if the heat is perpendicular to the structural grains of the wood or parallel to them, it will change the spatial heat transfer rate for a given heat flux and therefore the effective thermal conductivity. Values of thermal conductivities have a very large range, depending on the material. A summary for material categories is given in Fig. 1. Fig. 1Values of thermal conductivities depending on the material type. Values are extracted from (Bergman et al. 2011)

Although most heat conduction problems related to fire are transient in nature, most of the systems involved will eventually reach some equilibrium in the time before either the material involved begins to decay or the heating source dissipates. This equilibrium condition is known as the steady-state heat conduction.

To analyze the steady-state conduction condition, we normally consider an infinite plane wall of thickness L and thermal conductivity k. This wall is exposed to two different boundary temperatures on each side of the wall, T2 and T1. Assuming a one-dimensional heat transfer and constant thermal conductivity, then the heat equation reduces to
$${\dot{q}}_x^{\prime \prime }=\frac{k}{L}\left({T}_1-{T}_2\right)$$
However, most materials of interest for fire science applications are more complex than this. Usually we encounter a composite of materials, say, for example, a wall covered with a layer of wood such as in Fig. 2 or more generally an infinite slab made up of various layers exposed to different boundary conditions. If we again assume we are looking for the equilibrium condition, we can then calculate the net heat flux through the wall for a system in equilibrium by equating the steady-state heat fluxes across each layer:
$${\dot{q}}_x^{\prime \prime }=\frac{k_1}{L_1}\left({T}_0-{T}_1\right)=\frac{k_2}{L_2}\left({T}_1-{T}_2\right)$$ Fig. 2The composite wall infinite slab. Each layer has a different thermal conductivity, and the two walls can be exposed to different temperature boundary conditions (Thot and Tcold in this example)
Normally, for temperatures we usually only have information on the surrounding environmental temperatures, and not the internal temperatures as they are harder to measure. Therefore we need to account for the convective boundary conditions happening at the surfaces of the solid exposed to the fluid surrounding it. To do this, we use the empirical expression known as Newton’s law of cooling, namely
$${\dot{q}}_x^{\prime \prime }=h\left({T}_s-{T}_{\infty}\right)$$
where h is the heat transfer coefficient, TS is the solid surface temperature, and T the temperature of the surrounding fluid. Convection is discussed in much more detail in its own entry in this encyclopedia. Therefore, accounting for convection, our heat flux balance reduces to
\begin{aligned} {\dot{q}}_x^{\prime \prime }&={h}_{\mathrm{cold}}\ \left({T}_{\mathrm{cold}}-{T}_0\right)=\frac{k_1}{L_1}\left({T}_0-{T}_1\right)\\&=\frac{k_2}{L_2}\left({T}_1-{T}_2\right)={h}_{\mathrm{hot}}\ \left({T}_2-{T}_{\mathrm{hot}}\right) \end{aligned}
which can be written as
$${\dot{q}}_x^{\prime \prime }=\frac{T_{\mathrm{cold}}-{T}_{\mathrm{hot}}}{\frac{1}{h_{\mathrm{cold}}}+\frac{L_1}{k_1}+\frac{L_2}{k_2}+\frac{1}{h_{\mathrm{hot}}}}$$

## Boundary Conditions

In the examples above, to resolve the conduction heat transfer problems deriving from the original heat partial differential equation, boundary and initial conditions were used. The initial conditions for most problems are the initial temperature of the system when a heat source is applied, so in the case of fire, it might be when any external heat flux is imposed. For a given boundary, say x=L, the boundary conditions are one of the three types:
1. 1.

Dirichlet boundary condition, where the boundary has a prescribed temperature and is expressed as T|x = L = Tsurface

2. 2.

Neumann boundary condition, where the boundary has a prescribed heat flux, expressed as $$-k\ {\left.\frac{dT}{dx}\right|}_{x=L}={\dot{q}}_x^{\prime \prime }$$, with the special condition when $${\dot{q}}_x^{\prime \prime }=0$$ known as the adiabatic condition

3. 3.

The mixed boundary condition, which equates convective and conduction heat transfer at a surface and is expressed as $$-k\ {\left.\frac{dT}{dx}\right|}_{x=L}={h}_{\mathrm{fluid}}\left({\left.T\right|}_{x=L}-{T}_{\mathrm{fluid}}\right)$$

Since the heat equation is a partial differential equation that is second order in space and first order in time, for a one-dimensional boundary value problem like the examples in the previous section “Basic Concepts,” boundary conditions and one initial condition are needed to have a unique solution to the problem.

## Transient Conduction

Having covered the simplified case of the limit conditions when the system approaches equilibrium, the steady state, let’s now focus on the more general form of the heat equation. Most fire-related conduction problems are transient, so they require a solution to the partial differential equation known as the heat equation, namely
$$\frac{\partial^2T}{\partial {x}^2}+\frac{\partial^2T}{\partial {y}^2}+\frac{\partial^2T}{\partial {z}^2}+\frac{{\dot{q}}^{\prime \prime \prime }}{k}=\frac{\rho c}{k}\frac{\partial T}{\partial t}$$
where $${\dot{q}}^{\prime \prime \prime }$$ is the volumetric heat release rate, ρ is the density of the material, and c is the heat capacity of the material. $$\frac{k}{\rho c}$$ is the thermal diffusivity of the material, α, so we can rewrite the equation as
$${\nabla}^2T+\frac{{\dot{q}}^{\prime \prime \prime }}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

The volumetric heat release rate $${\dot{q}}^{\prime \prime \prime }$$in fire science is often found in problems of spontaneous ignition of a solid fuel, where a material generates its own heat due to a high environmental surrounding temperature, as well as in problems where solid fuels are exposed to a very hot environmental temperature causing phase changes (Bowes 1984). In general, the volumetric heat release is important for any problem that involves exothermic or endothermic changes, such as pyrolysis, phase change, or other chemical decompositions.

To resolve the transient conduction equation for practical problems, simplifications are often necessary. In its full three-dimensional form, for a complex geometry, it usually requires numerical means, generally by use of finite-difference schemes. There are however some useful analytical solutions for simpler geometries and other ways to simplify multidimensional problems to a single dimension, therefore being able to consider the problem via geometries such as plane-infinite slabs. Non-dimensional quantities are very useful when trying to solve heat conduction problems analytically and can be used to find solutions to specific conditions: The main non-dimensional numbers used are the Biot number (Bi) and Fourier number (Fo). The Biot number represents the ratio of internal conductive resistance to external convective resistance, therefore providing a measure of the temperature difference inside the solid relative to the temperature difference between the surface and the surrounding fluid. It is expressed as
$$Bi=\frac{hL}{k}$$
where h is the convection heat transfer coefficient (W/m2 K), k is the thermal conductivity of the solid (W/m K), and L is the characteristic length~(m). For cases when Bi<0.1, the resistance to conduction within a solid is much less than the resistance to convection across the fluid boundary layer, and it allows us to use the lumped thermal approximation to model the overall transient thermal response of a solid. The lumped thermal approximation, also known as lumped capacitance approximation, allows us to treat a solid as thermally thin, therefore ignoring any temperature gradients within a solid. A simple problem where such an approximation is useful in fire is when a solid experiences a sudden change in its environmental conditions, say, for example, a solid being engulfed in a fire, as most of the heat transfer occurs at the surface of the solid.
The Fourier number is the ratio of the rate of heat conduction to the rate of thermal energy storage in a solid. It identifies the degree of penetration of heating or cooling effect into a solid and represents the non-dimensional time in heat transfer problems. It is expressed as
$$Fo=\frac{\alpha t}{L^2}={t}^{\ast }$$
where L is length of the medium being considered (m), t is time (s) and α (m2/s) is the thermal diffusivity as defined earlier, and t* is non-dimensional time. By defining a non-dimensional temperature $$\frac{\theta }{\theta_i}=\frac{T-{T}_{\infty }}{T_i-{T}_{\infty }}$$ and performing an energy balance at the surface of a solid, we obtain the relation between temperature Biot and Fourier number given by
$$\frac{\theta }{\theta_i}=\exp \left(-\mathrm{Bi}\cdot \mathrm{Fo}\right)$$

## The Infinite 1-D Solid Slab Analytical Solution

A common solution to transient heat conduction problems used in fire science, for solids exposed to a hot environment, is the solution to a solid slab exposed to a hot environment such as the one presented in Fig. 3. Fig. 3Infinite slab exposed to an environmental temperature of T∞
This problem is symmetrical and can be expressed by the heat equation in one dimension
\begin{aligned} \frac{\partial^2T}{\partial {x}^2}=\frac{1}{\alpha}\frac{\partial T}{\partial t} \end{aligned}
with boundary conditions, $$\frac{\partial T}{\partial x}\Big|{}_{x=0}=0\therefore$$$$T(0)={T}_i,\frac{\partial T}{\partial x}\Big|_{x=L}=-\frac{h}{k}\left(T\left(L,t\right)-{T}_{\infty}\right),$$ and initial condition T(x, 0) = Ti.
Using the earlier-defined non-dimensional numbers and defining $${\theta}^{\ast }=\frac{\theta }{\theta_i}$$ and $${x}^{\ast }=\frac{x}{L}$$, the governing equation and boundary conditions can be expressed as
$$\frac{\partial^2{\theta}^{\ast }}{\partial {x}^{\ast 2}}=\frac{\partial {\theta}^{\ast }}{\mathrm{\partial} {Fo}}$$
with boundary conditions $${\left.\frac{\partial {\theta}^{\ast }}{\partial {x}^{\ast }}\right|}_{x^{\ast }=1}=-Bi\ {\theta}^{\ast}\left(1,{Fo}\right)$$ and $${\left.\frac{\partial {\theta}^{\ast }}{\partial {x}^{\ast }}\right|}_{x^{\ast }=0}=0$$ and initial condition θ(x, 0) = 1.

Since θ* is a function of only x*, Bi, and Fo, this means that the transient temperature distribution has a prescribed form that is independent of the environmental temperature, initial temperature, and thermal properties. That means that, by calculating a Biot number and Fourier number for the prescribed problem, one can then look up the temperature distribution for a given location within the solid with respect to time. There are tables of such values in most fundamental heat transfer books, including but not limited to Bergman et al. (2011). Analytical solutions that use the same non-dimensional approach as the infinite slab problem exist for other simple geometries, such as radial and cylindrical geometries, although they require a few more steps in deriving than the one shown here. They can be found in most fundamental heat transfer textbooks (Bergman et al. 2011).

## Importance of Conduction to Wildland and WUI Fires

Firebrands are often transported by wind/flow and land in areas further away from the main fire. Conduction is very important for its contribution to heating and ignition both in wildland and wildland-urban-interface (WUI) fires, mainly through firebrands. Firebrands can cause fires to spread, as well as ignition of common building materials, vegetation, or target fuels (Manzello et al. 2009). Work by Manzello et al. studied ignition of mulch beds via firebrands, showing that contact between the firebrand and mulch bed was critical for ignition of the mulch bed, via conductive heat transfer (Manzello et al. 2006). Furthermore, firebrands can have quite high surface temperatures causing a significant transport of heat from the firebrand to the surface of the target fuel.

Firebrands are often smoldering upon landing, so they can conductively transfer heat not only from their elevated temperature and heat capacity but also by the heat generated by the exothermic reactions occurring during the smoldering, which can then ignite the target fuels (Caton et al. 2017). An example of such a case can be seen in Fig. 4. Fig. 4Fuel bed heat transfer via conduction from glowing firebrands. This can cause a significant transport of heat due to the firebrand high surface temperatures

## Summary

Conduction is the form of heat transfer that occurs within a material because of a temperature gradient across the material. In fire science applications, most conduction problems are of a transient thermal nature and often require the resolution of the full heat equation in its partial differential form, but steady-state solutions are often very helpful when trying to find the limit conditions, especially preignition. Non-dimensional numbers such as the Fourier and Biot number are the primary means of assessing if the heat transfer mode of a material is primarily a conductive or convective one and if a problem for a solid material can be simplified. This is key when trying to understand insulation properties of materials and is important in wildland and WUI fires especially as a primary mode of ignition, especially for solid fuel beds in contact with firebrands. This is a brief and incomplete introduction to conductive heat transfer; some much more exhaustive sources on the topic include fundamental textbooks (Bergman et al. 2011; Drysdale 2011; Bejan and Kraus 2003).

## References

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6. Manzello SL, Cleary TG, Shields JR, Yang JC (2006) Ignition of mulch and grasses by firebrands in wildland – urban interface fires. Int J Wildland Fire 15(3):427–431
7. Manzello SL, Park SH, Cleary TG (2009) Investigation on the ability of glowing firebrands deposited within crevices to ignite common building materials. Fire Saf J 44(6):894–900