Contents
- 1Emission from a blackbody and a grey surface
- 2Absolute temperature and the fourth-power dependence
- 3Emitted radiation and net exchange
- 4Geometry and radiation travelling through a fire environment
- 5Worked example: radiation beneath a smoke layer
- 6Surface properties and limits of the grey approximation
- 7Notes
- 8References
The Stefan-Boltzmann law relates the radiant power emitted per unit area of a blackbody to the fourth power of its absolute temperature. A blackbody is an ideal surface that absorbs incident radiation and emits the maximum thermal radiation associated with its temperature. The law is a starting point for calculating thermal radiation from hot surfaces in fire engineering.[1][2]
For a simplified grey surface, the blackbody result is multiplied by an emissivity between zero and one. Emissivity represents the surface's emission relative to the blackbody reference. The calculation gives emitted power per area; determining the radiation received elsewhere additionally requires the geometry, the receiving conditions and any effects of the intervening medium.[1][3]
The fourth-power dependence makes temperature influential. However, the law alone does not specify the heat transfer between every pair of surfaces. Reflection, absorption, surface orientation and radiation from gases can require a more detailed model.[3]
Emission from a blackbody and a grey surface
The blackbody provides an ideal reference for emission and absorption. Van Rossum describes it as a perfect absorber of incident electromagnetic radiation. At a given temperature, it also supplies the reference maximum emission. This idealisation allows a real surface's radiation to be expressed relative to a defined physical limit, rather than described only by its visible appearance.[1]
The Stefan-Boltzmann relationship gives the total emitted power per unit area. It combines emission over the thermal spectrum, rather than describing one selected wavelength. Tien and colleagues present the relationship through integration of the blackbody spectrum. A calculation of total emission therefore differs from a description of the colour or wavelength distribution of that radiation.[2]
The blackbody equation multiplies the Stefan-Boltzmann constant by the fourth power of absolute temperature. The corresponding grey-surface equation adds an emissivity multiplier. In the simplified grey treatment, that multiplier represents the otherwise more complicated variation of emission with wavelength. Setting emissivity to one returns the blackbody result.[1]
| Symbol | Description | Unit |
|---|---|---|
| Eb | Blackbody emissive power per unit area | W/m² |
| E | Grey-surface emissive power per unit area | W/m² |
| ε | Emissivity used in the grey approximation | dimensionless |
| σ | Stefan-Boltzmann constant, here 5.67 × 10−8 | W/(m² K⁴) |
| T | Absolute temperature | K |
The constant used here is the rounded value given by Van Rossum. Its units ensure that multiplying by temperature to the fourth power produces watts per square metre. Tien and colleagues give the more detailed value 5.6696 × 10−8 W/(m² K⁴). Either source representation must be used consistently with absolute temperature and the chosen output units.[1][2]
The Stefan-Boltzmann constant is distinct from the Boltzmann constant used in the spectral radiation equation. The former has the units required for total emitted power per area and temperature to the fourth power. The latter appears in the temperature-dependent distribution of blackbody radiation. Tien and colleagues identify the two separately, so their similar names do not make them interchangeable numerical inputs.[2]
The result is a heat flux, meaning power associated with unit area. It is not the total power emitted by an arbitrarily sized surface. A source-flux calculation and a calculation of the radiation received by a particular target are also separate steps. Van Rossum explicitly distinguishes the source radiation from the flux at the receiving location.[1]
Absolute temperature and the fourth-power dependence
Temperature in this law must be an absolute temperature expressed in kelvin. The fourth power applies to that absolute temperature, not to a Celsius reading. Van Rossum's smoke-layer example uses a layer temperature of 400 °C and enters the rounded absolute temperature 673 K. The receiving surface at 35 °C is represented by 308 K in the same example.[1]
Raising a Celsius value to the fourth power would change the physical result because the Celsius and kelvin scales have different zero points. The source equation identifies kelvin specifically for this reason. A temperature difference used elsewhere in a heat-transfer calculation does not remove the requirement. Each temperature in a fourth-power radiation term remains an absolute temperature.[1]
The mathematical dependence is stronger than a direct proportionality to temperature. If absolute temperature doubles while emissivity remains unchanged, the emitted power per area increases by a factor of sixteen. This follows from the fourth power of two in the source equation. It does not mean that doubling a Celsius reading produces the same factor.[1]
Emissivity has a different role in the same expression. Holding temperature fixed, multiplying emissivity by a given factor multiplies the calculated emission by that factor. Temperature enters through its fourth power, whereas emissivity enters directly. This distinction helps explain why temperature and surface-property assumptions affect a calculation in different ways.[1]
The relationship also concerns radiation beyond visible light. Van Rossum explains that thermal emission relevant to buildings, people and fires lies mainly in the infrared and partly in the visible spectrum. A surface does not need to appear visibly glowing for its thermal radiation to matter. Visible appearance alone therefore does not replace temperature and emissivity in the calculation.[1]
Emitted radiation and net exchange
A receiving surface also emits thermal radiation. Net exchange therefore involves radiation travelling in both directions, rather than only the hot surface's outgoing emission. Van Rossum introduces this by subtracting the colder surface's fourth-power term from the hotter surface's term. The resulting expression represents a simplified exchange calculation with the stated grey-surface assumptions.[1]
| Symbol | Description | Unit |
|---|---|---|
| q″net | Net radiant flux in the simplified exchange calculation | W/m² |
| Th | Hot-side absolute temperature | K |
| Tc | Cold-side absolute temperature | K |
When the two temperatures are equal, the two fourth-power terms cancel. The simplified net exchange is then zero even though both surfaces still emit radiation. This follows directly from the source equation. Zero net exchange therefore has a different meaning from zero emission, and the distinction remains relevant when interpreting a heat balance.[1]
The cold-side term becomes more consequential as its temperature becomes less negligible relative to the source temperature. Omitting it replaces a net-exchange calculation with a source-emission approximation. Whether that approximation is acceptable depends on the temperatures and the intended calculation. Van Rossum specifically includes the receiving temperature when it is not negligible in comparison with the emitting temperature.[1]
The simplified expression is not a general solution for arbitrary reflecting grey surfaces. Tien and colleagues distinguish irradiation, the radiation reaching a surface, from radiosity, the total radiation leaving it. Radiosity includes both emission and reflected incident radiation. Their enclosure treatment accounts for these quantities together with surface properties and geometrical relationships.[4]
This distinction matters where surfaces exchange radiation repeatedly through reflection. A single emissivity multiplier and a temperature difference do not describe every such arrangement. The SFPE treatment uses linked surface relationships to evaluate the exchange. The basic emission law remains part of that calculation, but additional information supplies the enclosure-specific heat transfer.[4]
Geometry and radiation travelling through a fire environment
The view factor describes a geometrical relationship between emitting and receiving surfaces. Tien and colleagues define it as the fraction of radiation leaving one surface that is intercepted by another. Surface area, separation and orientation enter this relationship. Geometry therefore determines how an emitted flux contributes to the exchange between particular surfaces.[3]
Van Rossum applies a receiving-point form of this geometrical treatment to radiation from a finite source surface. The source flux is multiplied by the appropriate view factor for the receiving location. The calculation can represent radiation from a smoke layer towards a point below it. A source temperature alone does not supply that location-dependent result.[1]
The medium between source and receiver can also participate in the exchange. Tien and colleagues discuss absorption and emission by gases and soot particles in fires. Water vapour and carbon dioxide are among the gases relevant to this process. Their involvement means that radiation can change along its path, rather than travelling between surfaces without interaction.[5]
Radiation does not require an intervening material to transport its energy. Van Rossum explains that it can travel through a vacuum, while gases present along the path can absorb and emit radiation. Tien and colleagues consequently distinguish calculations through nonparticipating media from those through participating media. The distinction concerns what happens along the path, rather than a change in the basic blackbody emission law.[1][3]
This is one reason flame radiation cannot be specified by temperature alone. A fire contains gases, soot and spatially varying conditions, whereas the simplest grey-surface calculation uses a single temperature and emissivity. Those simplifications can be useful engineering assumptions. They nevertheless need to represent the source and the path relevant to the particular fire calculation.[5][3]
For surface calculations, uniformity is another modelling assumption. The SFPE radiosity treatment assumes approximately uniform outgoing and incoming radiant flux on each represented surface. Tien and colleagues explain that larger surfaces can require subdivision when those assumptions are not satisfied. Dividing the geometry into appropriate regions preserves the distinction between a calculation element and a physically non-uniform enclosure.[4]
Worked example: radiation beneath a smoke layer
Van Rossum provides an example of radiation reaching a person's head beneath a hot smoke layer. The building plan is 49 m by 49 m and its height is 16 m. The smoke layer is 2 m deep, while the receiving point is 1.75 m above the floor. The example assumes a smoke-layer temperature of 400 °C, a receiver temperature of 35 °C and an emissivity of 0.9.[1]
Subtracting the layer depth and receiver height from the building height gives a separation of 12.25 m. The example divides the source into four equal quadrants relative to the central receiving point. Each quadrant has a tabulated factor of 0.208, giving a total factor of 0.832. These geometry-specific values belong to the stated arrangement, rather than to every smoke layer.[1]
The calculation first determines the simplified net grey-source flux and then applies the view factor. Using the book's rounded absolute temperatures, the hot term uses 673 K and the cold term uses 308 K. The fourth powers are evaluated separately before subtraction. The factor of 1,000 in the final conversion changes watts per square metre to kilowatts per square metre.[1]
Here, F is the dimensionless view factor for the stated receiving point. The calculated grey-source emission is about 10.5 kW/m² before subtracting the receiver term. The net grey-source value is about 10.0 kW/m². Applying the factor of 0.832 gives the final 8.33 kW/m² result, reproducing the book's example after rounding.[1]
The example separates three operations that can otherwise be confused. Emissivity modifies the ideal blackbody source emission. The receiver term changes source emission into the simplified net exchange. The view factor then accounts for the specified source geometry and receiver position. Each operation represents a different physical element of the model.[1]
The receiver term also permits a numerical approximation check. In this example, 308 to the fourth power is about 4.39% of 673 to the fourth power. Omitting that term would overestimate the calculated net result by about 4.59%, with the other factors unchanged. These percentages are derived from the example's temperatures, rather than general error limits for radiation calculations.[1]
The resulting flux describes the example's radiation calculation, not a complete prediction of an exposed person's condition. The calculation assumes the specified source temperature, emissivity, receiving temperature and geometry. It does not independently determine those inputs or resolve all heat-transfer processes at the receiver. Its direct purpose is to calculate the radiant contribution represented by the stated model.[1][3]
Surface properties and limits of the grey approximation
Emissivity depends on more than a material's broad name. Van Rossum's example table gives 0.9 for anodised aluminium but 0.04 for smoothly polished aluminium. It also distinguishes rough concrete from concrete represented under fire conditions. These entries illustrate why a surface condition belongs with an emissivity value used in a calculation.[1]
The grey approximation suppresses additional detail. Actual radiation properties can vary with wavelength, direction and temperature, whereas an engineering calculation may use a single representative value. Tien and colleagues describe the practical reasons for this simplification and the limited availability of detailed surface-property measurements. The resulting overall flux remains tied to the chosen property representation.[3]
A surface's ability to absorb incident radiation also needs care. The SFPE treatment relates spectral absorptivity and emissivity under equilibrium conditions, then discusses the additional assumptions involved in total-property approximations. A total emissivity cannot automatically represent every combination of incident spectrum and material. This distinction becomes relevant when a hot source illuminates a different, cooler surface.[3]
The blackbody law therefore supplies a defined emission relationship within a broader radiation calculation. Its temperature dependence does not remove uncertainty in source temperature, surface state, geometry or participating gases. The worked example remains reproducible because those inputs are stated. More complicated fire environments require the additional radiation treatment described in the SFPE chapter.[1][5][3]
Notes
- ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 Van Rossum (2027), §5.4.
- ↑ 2.0 2.1 2.2 2.3 Tien et al. (2002), chapter 1-4, p. 1-74.
- ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Tien et al. (2002), chapter 1-4, pp. 1-75–1-76.
- ↑ 4.0 4.1 4.2 Tien et al. (2002), chapter 1-4, p. 1-76.
- ↑ 5.0 5.1 5.2 Tien et al. (2002), chapter 1-4, p. 1-73.
References
- Tien, C. L., Lee, K. Y., & Stretton, A. J. (2002). Radiation heat transfer. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., chapter 1-4, pp. 1-73–1-89). National Fire Protection Association.
- Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.