Thermally thin and thick materials

Thermally thin and thick materials are solids whose heating can be approximated by, respectively, a nearly uniform temperature or a temperature that varies through the depth. These descriptions depend on the component, its material properties and the heating conditions. They are not permanent classifications assigned solely from a material's name or its physical thickness.

A thermally thick component can retain a relatively cool interior while its exposed surface heats. A thermally thin approximation treats internal heat conduction as sufficiently rapid for an average temperature to represent the material. The appropriate description therefore affects the equations used to calculate temperature and interpret ignition or heat transfer through a construction element.

Two comparisons are useful but answer different questions. Penetration depth indicates whether heating has appreciably reached a distant boundary during the stated period. The Biot-number comparison examines whether internal conduction is rapid relative to heat transfer at the surface. Reaching the back of a component does not automatically establish a uniform-temperature model.

Physical thickness and thermal response

Physical thickness is a geometrical measurement. Thermal thickness describes the temperature response under an identified exposure. A material with relatively slow internal heat transfer can develop a substantial difference between its surface and interior temperatures. A highly conducting thin component can redistribute heat quickly enough for the temperature variation through its thickness to be small.

Thermal thickness describes the temperature distribution under stated heating conditions. Based on Rockett and Milke (2002), pp.1-33–1-37.

Conduction transfers energy through the solid as its temperature changes. The rate of that redistribution depends on conductivity together with density and specific heat capacity. Surface heating also depends on how energy is supplied by the surrounding gases or radiation. The thermal classification consequently combines material behaviour with the boundary conditions of the heating problem.

Duration changes the comparison. Van Rossum explains that a construction element can behave thermally thick early in an exposure but show increasing influence from its full thickness later. Heat penetrates progressively into the component. A description attached to the first part of a fire therefore does not automatically remain applicable throughout the whole event.

The thermally thick idealisation can be represented by a semi-infinite solid during the period when the distant boundary has little influence. Semi-infinite does not mean that an actual wall has infinite dimensions. It is a mathematical approximation in which the heated region is small compared with the relevant available thickness. The SFPE conduction treatment uses this idealisation for depth-dependent transient heating.

Comparing component thickness with penetration depth

Thermal diffusivity combines conductivity and volumetric heat capacity in the ratio α = k/(ρc). Van Rossum uses it to estimate a characteristic thermal penetration distance after a sudden change in surface temperature. The estimate grows with the square root of exposure duration. It therefore supplies a length scale for comparison with the component's actual thickness.

d ≈ √(α t)

The introductory book proposes approximately twice this characteristic distance as an indicative boundary thickness. A component substantially thicker than that scale behaves thermally thick for the stated period in the book's screening approach. The estimate is approximate because the temperature profile does not contain a sharp front separating heated and unheated material.

L_limit ≈ 2 √(α t)

Symbol	Description	Unit
d	Characteristic penetration distance	m
α	Thermal diffusivity	m²/s
t	Heating duration	s
Llimit	Indicative screening thickness	m

The comparison always includes time. With fixed diffusivity, increasing the duration by a factor of four doubles both the characteristic distance and the screening thickness. An element that was well beyond the early screening scale can therefore become less remote from the heated region later. This follows from the square-root relation, without assigning a sharp physical transition.

A penetration estimate does not independently specify the temperature at every depth. The boundary temperature or heat flux and the initial temperature remain necessary for that calculation. It also does not demonstrate that all points within the penetration scale have the same temperature. The SFPE heat equation requires a complete heating problem to establish the actual distribution.

The distinction matters when using the term thermally thin. Van Rossum's screening rule identifies when heating involves much of a component's thickness. The uniform-temperature approximation discussed by Rockett and Milke additionally compares internal conduction with surface heat transfer. These related ideas are therefore most useful when their respective assumptions are stated.

The Biot-number comparison

The Biot number compares the internal conduction scale with the surface heat-transfer scale. Rockett and Milke derive it for a plate heated by convection on one face with the opposite face insulated. In that arrangement, the relevant length is the plate thickness. A small number supports a nearly uniform temperature because conduction within the solid is rapid relative to heating at its surface.

Bi = h L / k

Symbol	Description	Unit
Bi	Biot number for the stated plate model	dimensionless
h	Convective heat-transfer coefficient	W/(m² K)
L	Plate thickness for this one-face heating model	m
k	Thermal conductivity of the plate	W/(m K)

The material and exposure both affect this ratio. Increasing conductivity reduces the number, making internal temperature redistribution more rapid relative to the surface process. Increasing thickness or the surface heat-transfer coefficient increases the number. The SFPE discussion therefore associates thermally thin behaviour with sufficiently small thickness or sufficiently high conductivity under the specified surface conditions.

Rockett and Milke express the same comparison using two time scales. Internal conduction has a characteristic time proportional to density, heat capacity and thickness squared, divided by conductivity. Surface heating of the uniform plate has a characteristic time proportional to density, heat capacity and thickness, divided by the convection coefficient. Their ratio is the Biot number.

t₀ = ρ c L² / k

t_c = ρ c L / h

t₀ / t_c = Bi

Here, ρ is density in kg/m³, c is specific heat capacity in J/(kg K), and both time scales are in seconds. The comparison describes how quickly the two processes act. It does not replace the initial temperature, gas-temperature history or rear-boundary condition. Those details still determine the temperature predicted by the model.

A uniform-temperature plate model

For the stated plate with an insulated rear face, convection supplies energy through the exposed face. The lumped model stores that energy through a change in the plate's average temperature. The approximation treats that average as representative throughout the thickness. It is appropriate only when internal temperature differences remain small enough for the intended calculation.

For a constant surrounding gas temperature, the SFPE solution approaches that temperature exponentially. The plate starts from its specified initial temperature and progressively reduces its temperature difference from the gas. The characteristic heating time is t_c from the previous section. The relationship below is an algebraic rearrangement of the source's constant-temperature solution.

T(t) = T_g + (T_i − T_g) exp(−t / t_c)

Symbol	Description	Unit
T(t)	Uniform plate temperature at elapsed time t	°C
Tg	Constant gas temperature	°C
Ti	Initial plate temperature	°C
tc	Convective heating time scale	s

Consistent Celsius temperatures can be used in this rearranged expression because it combines temperatures through their difference and the initial reference. This does not authorise Celsius values in a fourth-power radiation law. The displayed model contains convection and an insulated rear face. Adding radiation or rear-face cooling changes the heat balance.

Rockett and Milke discuss a plate that radiates while being heated by hot gas as a separate, more complicated case. Its temperature need not approach the gas temperature in the same way because heat is also lost radiatively. This demonstrates why calling a component thermally thin does not remove the need to identify all relevant surface heat-transfer processes.

Worked illustration for a thin steel plate

A calculated illustration can use a steel thickness of 4 mm with the book's tabulated properties: conductivity 45.8 W/(m K), density 7,850 kg/m³ and heat capacity 460 J/(kg K). Take a convection coefficient of 25 W/(m² K). These values are drawn from Van Rossum's material table and example discussion of steel-roof heating.

For this illustration, impose the SFPE plate model's insulated rear face and one-face convective heating. This is an explicit modelling choice, rather than a reproduction of the book's complete roof calculation. The gas temperature is held at 634 °C and the plate initially has a temperature of 20 °C. Radiation is excluded from this particular illustrative balance.

The Biot number is approximately 0.00218. The internal conduction time is about 1.26 s, while the surface-heating time is about 578 s. Internal redistribution is consequently much faster than the surface-heating process in this model. That separation supports using a nearly uniform plate temperature for the illustrative calculation.

Bi = 25 × 0.004 / 45.8 ≈ 0.00218

t_c = 7,850 × 460 × 0.004 / 25 = 577.76 s

Substituting a duration of 600 s into the exponential solution gives approximately 417 °C. The calculation retains the unrounded time constant until the final result. This is a model output for the specified constant exposure and boundary conditions. It is not a measured steel temperature or a prediction for every 4 mm steel component in a fire.

Calculated heating of the illustrative 4 mm steel plate with its rear face insulated. Based on Rockett and Milke (2002), p.1-33, and Van Rossum (2027), §§5.2 and 5.5.

T(600 s) = 634 + (20 − 634) exp(−600 / 577.76) ≈ 417 °C

The small internal gradient does not mean that the steel is already at the gas temperature. It means that one plate temperature can approximate the temperature through its thickness. The plate can remain considerably cooler than the gas while warming nearly uniformly. The distinction separates the spatial approximation from the elapsed time required to approach the imposed gas temperature.

Worked screening example for a thick concrete wall

Van Rossum gives a separate example of a concrete wall 200 mm thick during the first 15 minutes of heating. The selected diffusivity is 9.88 × 10⁻⁷ m²/s. The duration is therefore 900 s. This example uses the indicative penetration-based criterion, rather than the one-face lumped plate calculation used for the steel illustration.

Twice the characteristic penetration distance is approximately 0.060 m. The actual wall thickness is 0.20 m, substantially larger than that indicative scale. The book consequently describes the wall as thermally thick during the stated early period. The unexposed side remains relatively little affected compared with the exposed surface in that assessment.

The wall is thicker than the approximate screening scale for the stated exposure duration. Based on Van Rossum (2027), §5.2.

L_limit ≈ 2 √(9.88 × 10⁻⁷ × 900) ≈ 0.060 m

The comparison does not provide an exact temperature at the back face. It also does not establish that a wall just below the indicative thickness has an exactly uniform temperature. The source presents a practical rule based on a smooth penetration process. A detailed temperature prediction still requires the material model and boundary conditions.

A steady-state wall calculation assumes that a temperature distribution has developed across the full thickness. Applying that established distribution during the early heating period can overestimate heat flow towards the rear face. Van Rossum uses the concrete example to explain this limitation. The thickness comparison therefore helps identify when a transient treatment is relevant.

Consequences for ignition and exposure geometry

Thermal inertia is particularly relevant to the early surface response of a thick solid. Heat supplied at the exposed face is partly conducted into the cooler interior. The SFPE chapter expresses this response using the product of conductivity, density and heat capacity. A thin-plate balance instead explicitly includes the finite thickness and the energy stored throughout the plate.

This difference affects the interpretation of time to ignition. In a thick fuel, a hot surface can coexist with a much cooler interior. Rockett and Milke discuss how the interior can absorb heat and cool the surface when external heating is removed. Initial ignition and sustained burning therefore need not be equivalent outcomes.

Exposure geometry also matters in structural heating calculations associated with section factor. The Australian Building Codes Board's 2020 data sheets describe a lumped thermal-mass model for protected steel. Its geometrical parameter includes the steel mass per unit length divided by the heated perimeter. This identifies the exposed part of the geometry alongside the thermal mass being heated.

The plate example's thickness represents one specific relationship between heated area and stored material. A section exposed on several sides has a different relationship. The geometry used in a thermal model therefore needs to correspond to the surfaces actually heated. A thin-material description by itself does not supply the exposed perimeter or the details of any protective layer.

Limits when materials and boundary conditions change

The simplified examples use fixed properties. Van Rossum explains that conductivity and heat capacity can change substantially at elevated temperature. Moisture evaporation and material transformation introduce further effects. A classification calculated from one constant diffusivity can therefore become less representative as the material and exposure evolve.

The source identifies melting, charring, material loss and intumescence as changes that can affect the heated construction. These processes may alter thickness, conductivity or the amount of material remaining. They are not captured merely by retaining the original thin or thick label. The appropriate thermal description follows the modelled construction and its assumptions.

Heat loss from the rear face is another possible change. The uniform steel illustration explicitly insulated that face, whereas an actual exposed plate may exchange heat on both sides. Radiation can also become significant as temperature rises. Rockett and Milke treat such changes through the heat balance rather than through a label alone.

When strong temperature gradients, changing properties or complicated geometry must be represented, the SFPE chapter discusses numerical conduction methods. These divide the material into smaller regions and resolve the changing temperature field. Such models still require appropriate material information and boundary conditions. Their spatial detail addresses limitations of a single uniform temperature rather than eliminating those input requirements.

See also

• Opposed-flow flame spread

References

• Australian Building Codes Board. (2020). Fire safety verification method data sheets: Handbook annex (Version 1.1).
• 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.