# Conduction

**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.

*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

## Steady-State Conduction

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.

*L*and thermal conductivity

*k*. This wall is exposed to two different boundary temperatures on each side of the wall,

*T*

_{2}and

*T*

_{1}. Assuming a one-dimensional heat transfer and constant thermal conductivity, then the heat equation reduces to

*h*is the heat transfer coefficient,

*T*

_{S}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

## Boundary Conditions

*x*=

*L*, the boundary conditions are one of the three types:

- 1.
Dirichlet boundary condition, where the boundary has a prescribed temperature and is expressed as

*T*|_{x = L}=*T*_{surface} - 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.
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

*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

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.

*h*is the convection heat transfer coefficient (W/m

^{2}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.

*L*is length of the medium being considered (m), t is time (s) and

*α*(m

^{2}/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

## The Infinite 1-D Solid Slab Analytical Solution

*T*(

*x*, 0) =

*T*

_{i}.

*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.

## 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).

## Cross-References

## References

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