Thermal diffusivity

Thermal diffusivity describes how rapidly a temperature change spreads through a material by conduction. It is the thermal conductivity divided by the product of density and specific heat capacity. The quantity combines the material's ability to conduct heat with the energy required to change the temperature of a given volume.

Thermal diffusivity is expressed in square metres per second. It appears in calculations of changing, or transient, temperature distributions inside solids. A larger value means that a temperature disturbance spreads through a given distance more rapidly under otherwise comparable conditions. It does not, by itself, specify the surface temperature or the amount of heat supplied by a fire.

In fire engineering, the quantity helps estimate heat penetration and interpret whether a component behaves as thermally thick during a stated exposure. Such estimates depend on duration, dimensions and the assumptions made about material properties. Temperature-dependent properties and material changes can limit calculations based on one fixed diffusivity.

Conducting heat and storing energy

Thermal conductivity measures the connection between a temperature gradient and the heat conducted through a material. Conductivity alone does not describe the resulting rate of temperature change. The material also stores energy as its temperature rises. The density and specific heat capacity provide the corresponding storage terms in the heat-conduction equation.

Specific heat capacity expresses the energy needed to change the temperature of unit mass. Multiplying it by density gives heat capacity per unit volume. Thermal diffusivity compares conductivity with this volumetric heat capacity. A material can therefore conduct heat more readily while also requiring more energy to raise the temperature of the material being heated.

The defining relationship divides thermal conductivity by volumetric heat capacity. The introductory book uses the symbol λ for conductivity, whereas the SFPE chapter uses k. Both notations represent the same property in this relationship. The expression below uses k consistently and writes c for the specific heat capacity used in the solid calculation.

Thermal diffusivity combines heat conduction and energy storage per unit volume. Based on Rockett and Milke (2002), p.1-29.

α = k / (ρ c)

Symbol	Description	Unit
α	Thermal diffusivity	m²/s
k	Thermal conductivity	W/(m K)
ρ	Density	kg/m³
c	Specific heat capacity	J/(kg K)
ρ c	Volumetric heat capacity	J/(m³ K)

The units distinguish diffusivity from a speed. Square metres per second is a length-squared divided by time, rather than metres per second. A penetration distance therefore does not follow by multiplying diffusivity directly by time. The characteristic distance instead involves the square root of the diffusivity-time product in the simplified transient estimate.

The definition also shows how changing one property affects the result. At fixed density and heat capacity, increasing conductivity increases diffusivity. At fixed conductivity, increasing volumetric heat capacity decreases diffusivity. These are relationships within the stated property model, rather than a claim that actual materials can change one property without changing their other properties.

Its role in the transient conduction equation

Conduction through a solid involves a balance between heat entering, heat leaving and energy stored locally. Rockett and Milke derive the temperature equation by applying that balance to a small material volume. Their general expression can include direction-dependent conductivity and internal heat addition. A simpler form follows when the relevant properties are constant and there is no internal heat source.

For heat flow in one direction through a constant-property solid, the temperature's rate of change equals diffusivity multiplied by its second spatial derivative. The second derivative describes how the temperature gradient changes with position. The expression therefore connects local energy accumulation with the difference between conductive heat flows entering and leaving neighbouring regions.

∂T / ∂t = α ∂²T / ∂x²

Symbol	Description	Unit
T	Material temperature	K
t	Time	s
x	Distance in the direction of heat conduction	m

The coefficient does not provide a complete temperature history on its own. The initial temperature throughout the solid must also be known. Conditions at the boundaries must describe the imposed temperature or heat transfer over time. Rockett and Milke emphasise that these initial and boundary conditions distinguish one physical heating problem from another.

Different boundary descriptions can therefore give different temperatures for the same diffusivity. Holding a surface at a prescribed temperature is not the same condition as supplying a prescribed heat flux. The SFPE formulation specifies one of those conditions on a given boundary, rather than independently imposing both. The material coefficient remains the same while the defined heating problem changes.

Steady-state conduction is a separate limiting case. Once the temperature distribution no longer changes with time, the time derivative vanishes. Conductivity still determines heat flow through the established temperature gradients, but transient energy storage no longer changes the temperature field. This is why diffusivity cannot replace conductivity in the thermal-resistance expression for a layer.

Characteristic penetration distance and heating time

Van Rossum introduces a characteristic penetration distance after a sudden surface-temperature increase. The estimate takes the square root of thermal diffusivity multiplied by elapsed heating time. It indicates the depth over which the temperature disturbance becomes appreciable. The expression is a practical scale estimate, rather than the position of an abrupt boundary between hot and cold material.

d ≈ √(α t)

Here, d is a characteristic penetration distance in metres and t is heating duration in seconds. The temperature decreases gradually with depth in the described heating situation. A point deeper than d is therefore not automatically unchanged, and a point shallower than d is not necessarily at the surface temperature. The estimate compresses a continuous temperature distribution into a useful length scale.

The square-root dependence gives a simple comparison between durations. With diffusivity held fixed, four times the exposure duration gives twice the characteristic penetration distance. This follows from taking the square root of four in the same expression. It describes the simplified thermal scale and does not establish a linear advance of a burning or charring front.

The corresponding characteristic time for conduction over a selected distance is proportional to that distance squared divided by diffusivity. Rockett and Milke derive this time scale by expressing the conduction equation in dimensionless form. Their selected length represents a dimension of the object in the principal heat-flow direction. Here that length is written as L.

t₀ = L² / α

This characteristic time is not automatically the time at which a component reaches a specified temperature. The actual heating history still depends on the boundary conditions and the selected temperature criterion. It is a scale for comparing conduction through different lengths or materials. In the constant-property expression, doubling the selected length multiplies that time scale by four.

Worked example: wooden and concrete walls

Van Rossum compares thick wooden and concrete walls exposed to the same fire for one hour. Both walls initially have a temperature of 20 °C. The example uses rounded thermal properties to explain the difference between surface heating and penetration. Those properties define the calculation and are not presented as universal values for every wood or concrete product.

Material in the example	k, W/(m K)	ρ, kg/m³	c, J/(kg K)
Wood	0.13	500	1,600
Concrete	1.7	2,300	900

For the wooden wall, the product of density and heat capacity is 800,000 J/(m³ K). Dividing the conductivity of 0.13 W/(m K) by that product gives 1.625 × 10⁻⁷ m²/s. For the concrete, the corresponding division gives about 8.21 × 10⁻⁷ m²/s. The values are calculated from the same property set throughout the example.

α_wood = 0.13 / (500 × 1,600) ≈ 1.63 × 10⁻⁷ m²/s

α_concrete = 1.7 / (2,300 × 900) ≈ 8.21 × 10⁻⁷ m²/s

One hour corresponds to 3,600 s. Substituting that duration into the penetration estimate gives about 0.024 m for the wood and 0.054 m for the concrete. These distances are approximately 24 mm and 54 mm. The calculation retains the unrounded diffusivities internally and rounds the final distances, reproducing the scale of the book's results.

Calculated penetration scales using the fixed properties of the wooden and concrete wall example. Based on Van Rossum (2027), §5.2.

d_wood = √[(0.13 / (500 × 1,600)) × 3,600] ≈ 0.024 m

d_concrete = √[(1.7 / (2,300 × 900)) × 3,600] ≈ 0.054 m

The concrete diffusivity in this example is about five times the wooden-wall value. Its characteristic penetration distance is about 2.25 times greater, because distance depends on the square root of diffusivity. The result does not mean that the concrete surface must be hotter. Van Rossum uses this comparison specifically to distinguish the deeper spread of temperature change from the speed of surface heating.

The calculated distance is also not a charring depth or a fire-resistance rating. The example concerns a thermal penetration scale using fixed properties. Van Rossum separately explains that wood can char and that high-temperature material changes alter the heating process. Interpreting the estimated 24 mm as a directly calculated char layer would therefore assign a meaning the equation does not provide.

Diffusivity, thermal inertia and component thickness

Thermal inertia describes another aspect of material heating. The SFPE conduction chapter uses the product kρc for this parameter, while Van Rossum presents its square root for the surface-response comparison. Diffusivity instead uses the ratio k/(ρc). These combinations contain the same underlying properties but describe different features of the transient thermal response.

In the worked example, concrete has a larger square-root inertia parameter as well as a larger diffusivity. Its surface can consequently heat more slowly while the temperature disturbance penetrates further. There is no contradiction between those observations. One concerns the temperature developed at the heated face, while the other concerns how a temperature change spreads inside the material.

The distinction between thermally thin and thick materials adds the component dimensions and the exposure duration. A physically thick component can initially retain a relatively cool interior while its exposed surface heats. The same component may show a different thermal response over a longer exposure. The classification therefore does not belong to a material name independently of its geometry and heating history.

Van Rossum gives an approximate screening thickness equal to twice the characteristic penetration distance. A component much thicker than this scale behaves thermally thick for the stated period in that simplified assessment. The book explicitly treats the boundary as approximate because the temperature disturbance has no sharp front. The comparison supplements the diffusivity rather than supplying a universal material classification.

The book illustrates that screening calculation with a concrete wall 200 mm thick during the first 15 minutes of heating. This separate example uses a diffusivity of 9.88 × 10⁻⁷ m²/s and a duration of 900 s. Twice the square root of their product is approximately 0.060 m, or 60 mm. The wall thickness is substantially greater than this indicative scale, supporting a thermally thick description for that period.

A200 mm concrete wall exceeds the approximate 60 mm screening scale for the stated 15-minute exposure. Based on Van Rossum (2027), §5.2.

The result describes the early thermal response of the stated wall. It does not assign the same classification for every later duration, because the penetration scale increases with time. Van Rossum contrasts this early response with a steady-state calculation that assumes a temperature distribution has already developed through the entire thickness. Using that established distribution too early can overestimate the heat flow towards the unexposed side. The duration attached to the diffusivity calculation is therefore part of its interpretation.

Surface heat transfer adds a further condition when a uniform-temperature approximation is used. Rockett and Milke compare the internal conduction time with the time associated with heat transfer to a plate's surface. Rapid conduction relative to surface heating supports approximately uniform temperature through the plate. Penetration distance alone is therefore not a substitute for checking the boundary assumptions of a lumped-temperature model.

Selecting properties and recognising limits

The introductory book's worked comparison and its general material-property table use different rounded concrete inputs. The comparison uses conductivity 1.7 W/(m K) and specific heat capacity 900 J/(kg K). The separate table uses 2.0 W/(m K) and 880 J/(kg K), with the same listed density. These produce different diffusivities and belong to distinct stated datasets.

A calculation remains traceable when the conductivity, density and heat capacity come from the identified property set. Combining the diffusivity from one example with the component properties from another can conceal that distinction. The values in the wall comparison are illustrative inputs, while the table supplies another set of reference values. Neither should silently replace the other within a reproduced example.

High temperatures introduce changes beyond the constant-property expression. Van Rossum describes temperature-dependent conductivity and heat capacity, together with the energy consumed by moisture evaporation. Melting, charring, material loss and intumescence can also change the material or its geometry. A fixed diffusivity does not independently account for those evolving processes.

The SFPE heat equation makes room for local energy effects that the simplified equation omits. Its discussion includes evaporation in one region and condensation in another, with corresponding local heat removal and addition. Such processes cannot always be represented by one unchanged temperature-diffusion coefficient. The relevant energy terms and property descriptions belong in the more detailed thermal model.

The assumed direction of heat flow also limits a simplified estimate. Rockett and Milke begin with conductivity that may differ between directions, before adopting simpler material assumptions. A one-dimensional expression represents temperature variation along one selected direction. More complicated geometry or directional behaviour can require a temperature field with additional spatial dimensions and corresponding material information.

References

• Rockett, J. A., & Milke, J. A. (2002). Conduction of heat in solids. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., chapter 1-2, pp. 1-27–1-43). National Fire Protection Association.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.