Adiabatic surface temperature

The adiabatic surface temperature (AST) is the temperature that a perfectly insulated surface would reach in a fire. The documents used for this article name the quantity but do not define it: a 2016 validation study lists the abbreviation and reports a series of Swedish experiments named after it. The definition used here follows from the adiabatic boundary condition of the Fire Dynamics Simulator, under which no heat is conducted into an insulated face, applied to the face exposed to the fire. At that temperature the heat the surface absorbs from radiation and hot gases equals the heat it loses again. As the heat balance below shows, the quantity thus combines the two ways in which a fire heats a surface, thermal radiation and convection, into a single temperature. That temperature can serve as a boundary condition when the heating of a structural member is calculated, for instance with finite element analysis.

Experiments on the adiabatic surface temperature at the SP Technical Research Institute of Sweden in 2008 and 2011 used plate thermometers. Their data are part of a 2016 validation study of fire models for nuclear power plants in the United States. As the balance and the worked example show, the adiabatic surface temperature belongs to a surface in a given exposure rather than to the gas. It depends on the emissivity of the surface and on the convective heat transfer coefficient.

Heat transfer to a surface

A fire transfers heat to a surface by convection and by radiation. In an open fire, about 70% of the heat leaves the fire by convection and about 30% by radiation. Convection is heat transfer by moving gas. Its heat flux to a surface equals a heat transfer coefficient times the difference between the gas temperature and the surface temperature. The coefficient summarises the effects of flow speed and turbulence in the thin boundary layer at the surface.

The value of the coefficient has to be chosen. NIST assumed a value of 25 W/(m²·K), citing the Eurocode, for the convective flux to structural members in the hot layer of the World Trade Center fires. Introduction to Fire Safety Engineering uses the same value in a worked example.

Radiation does not need a medium. A surface emits radiation in proportion to its emissivity and to the fourth power of its absolute temperature, with the Stefan–Boltzmann constant of 5.67 × 10⁻⁸ W/(m²·K⁴). The emissivity is a material property between 0 and 1. Rough concrete has 0.91, concrete in a fire 0.7 and painted surfaces 0.9. The Fire Dynamics Simulator, a computational fluid dynamics model, writes the net radiative flux to a solid as absorbed incoming radiation minus emitted radiation. The surface absorbs the incoming radiation in proportion to its emissivity and emits εσT⁴.

At the surface, the convective and radiative fluxes together are conducted into the solid. When the back of a layer is perfectly insulated, the Fire Dynamics Simulator applies an adiabatic condition: the temperature gradient at that face, and hence the heat conduction, is zero. An adiabatic surface is the same idea applied to the exposed face.

The heat balance

Combining the convective and radiative terms gives the net heat flux into a surface with temperature T_s. Here ε is the emissivity, q̇″_inc the incident radiation, σ the Stefan–Boltzmann constant, h the convective heat transfer coefficient and T_g the gas temperature near the surface: $${\dot{q}}^{\prime }{\text{'}}_{net}=\epsilon {\dot{q}}^{\prime }{\text{'}}_{inc}-\epsilon \sigma T_s^4+h(T_g-T_s)$$ An adiabatic surface conducts no heat into the material, so its net heat flux is zero.

The adiabatic surface temperature T_AST is therefore the root of the following balance, which follows from setting the net heat flux to zero: $$\epsilon \sigma T_{AST}^4+h{T}_{AST}=\epsilon {\dot{q}}^{\prime }{\text{'}}_{inc}+h{T}_g$$ In this balance, the radiation term pulls the result towards the temperature at which a black surface would emit the incident radiation, while the convective term pulls it towards the gas temperature. The adiabatic surface temperature therefore lies between these two temperatures. When incident radiation and gas temperature correspond to the same temperature, the adiabatic surface temperature equals it, whatever the emissivity. This mirrors a result for enclosures: an adiabatic, re-radiating wall has a temperature that does not depend on its emissivity.

Subtracting the balance from the net heat flux gives a useful form of the heat flux into a real surface: $${\dot{q}}^{\prime }{\text{'}}_{net}=\epsilon \sigma (T_{AST}^4-T_s^4)+h(T_{AST}-T_s)$$ One temperature thus replaces the two exposure quantities, incident radiation and gas temperature. This allows a measured or calculated adiabatic surface temperature to act as the boundary condition for a heat conduction calculation.

Symbol	Description	Unit
${\dot{q}}^{\prime }{\text{'}}_{net}$	net heat flux into the surface	W/m2
${\dot{q}}^{\prime }{\text{'}}_{inc}$	incident radiation on the surface	W/m2
$\epsilon$	emissivity of the surface	dimensionless
$\sigma$	Stefan–Boltzmann constant, 5.67 × 10−8	W/(m2·K4)
$h$	convective heat transfer coefficient	W/(m2·K)
$T_g$	gas temperature near the surface	K
$T_s$	surface temperature	K
$T_{AST}$	adiabatic surface temperature	K

Worked example

Introduction to Fire Safety Engineering states that a hot smoke layer above 500 °C radiates about 20 kW/m² to objects on the floor. Consider a painted object on the floor, with an emissivity of 0.9, in air that is still at 20 °C. For the convective coefficient, the book gives 20 to 25 W/(m²·K) for a fully developed fire and usually uses 25 for a standard fire. That value is assumed here, and the balance becomes 0.9 × 5.67 × 10⁻⁸ × T⁴ + 25 T = 0.9 × 20,000 + 25 × 293.15. Solving it numerically gives T_AST = 649.6 K, or 376.5 °C, rounded up to 377 °C.

With this coefficient, a painted object in that position cannot become hotter than this as long as it does not burn and the exposure stays the same. The result depends strongly on the assumption: with the book's range of 1 to 5 W/(m²·K) for ordinary indoor conditions, the adiabatic surface temperature would be between about 470 and 490 °C. For concrete in a fire, with an emissivity of 0.7, the value falls to 346 °C, because the surface absorbs less and convection cools it relatively more.

The picture changes for an object inside the smoke layer. At 500 °C a black surface would emit about 20,300 W/m², close to the 20 kW/m² received. For an object immersed in gas at 500 °C and exposed to 20 kW/m², the adiabatic surface temperature is 498 °C for an emissivity of both 0.9 and 0.7. Radiation and convection now point to almost the same temperature, so the emissivity hardly matters.

The third equation then gives the heat flux into the floor object while it heats up. At a surface temperature of 20 °C it receives 17,600 W/m², at 200 °C about 10,900 W/m² and at 300 °C about 5,500 W/m². These values agree exactly with the direct calculation from incident radiation and gas temperature. The heating slows as the surface approaches its adiabatic surface temperature, where the net flux reaches zero.

Use with fire models and structural analysis

Design fires for structural elements are often expressed as an imposed temperature or heat flux over time. From that exposure, the temperatures inside a cross-section can be calculated. Computational fluid dynamics can give such thermal loads for any type of structure in much more detail than a zone model. A finite element model can then calculate how the heat penetrates into the structure.

Simpler methods describe the exposure by a gas temperature alone. In the 2016 validation study, the empirical correlations took the hot gas layer temperature of the MQH correlation, or the plume temperature of McCaffrey's correlation, as the exposing fire temperature for the SP experiments. The adiabatic surface temperature is one way to pass the fire exposure to the structural model as a single temperature for each surface. Fire models can also calculate it: in the 2016 validation study, the Fire Dynamics Simulator predicted the adiabatic surface temperatures measured in the SP experiments, as well as the plate thermometer and steel temperatures.

The investigation of the World Trade Center collapse needed a similar coupling. NIST developed a method that computed heat transfer to structural elements with a simple radiative transport model. It divided the compartment locally into a hot, sooty upper layer and a cooler, clearer lower layer, with properties taken from time averages of the fire simulations. Finite element models of the steel members and their fireproofing then gave the three-dimensional, time-dependent thermal response in a form suited to the structural models. The model predictions compared favourably with large-scale experiments.

A fire resistance test provides data under one standardised heating regime. The Australian Data Sheets note that the limitations of such a single exposure must be considered when the results are applied to design scenarios with varying fire exposures. In such tests the furnace heats the element according to a standard temperature–time relation.

Measurement and validation

In 2008, SP carried out three compartment experiments sponsored by Brandforsk, the Swedish Fire Research Board. They were to demonstrate how plate thermometer measurements near a simple steel beam can supply the boundary conditions for a multidimensional heat conduction calculation of the beam. The compartment measured 3.6 m by 2.4 m by 2.4 m, with a door of 0.8 m by 2.0 m and walls of lightweight concrete blocks. A propane burner of 450 kW heated a steel beam suspended 20 cm below the ceiling, with measurement stations at three positions along it.

A second series followed in 2011. A vertical steel column, 6 m long and 0.2 m in diameter, stood in the centre of diesel pool fires of 1.1 m and 1.9 m and of a heptane pool fire of 1.1 m. Gas, plate thermometer and surface temperatures were measured at heights from 1 m to 5 m, with the column partly engulfed in flames.

The 2016 supplement to the United States validation study for nuclear power plant fire models includes these data among its target temperature experiments. Each model receives a bias factor, which shows how far it over- or under-predicts on average, and a relative standard deviation for the scatter. These statistics cover all target experiments in the study, not only the SP tests. For the rise in target temperature, the Fire Dynamics Simulator had a bias factor of 0.99 and a relative standard deviation of 0.17, against an experimental uncertainty of 0.07. The zone models showed higher bias and uncertainty for target temperature, and their assumption of uniform layers led to some under-prediction near the fire.

Limits and common misconceptions

The adiabatic surface temperature is not the temperature of the real element. A real element conducts heat into its interior, so while it heats up its surface stays cooler than the adiabatic surface temperature, which it approaches only when its interior is hot or well insulated. The adiabatic surface temperature is therefore an upper bound only while the exposure is maintained. In the first SP test the measured adiabatic surface temperature fell sharply after 30 minutes, and ten minutes later the steel was hotter than it. Fire models calculate the temperature of a target as a function of time and take the heat conducted into the target material into account.

The adiabatic surface temperature also does not describe the fire alone. With the assumed coefficient, the same exposure in the worked example gives 377 °C for a painted surface and 346 °C for concrete. The convective coefficient is also uncertain. NIST assumed 25 W/(m²·K), and in a large eddy simulation the Fire Dynamics Simulator takes it from empirical natural and forced convection correlations. In compartment fire models, an adiabatic wall temperature is sometimes introduced, because the local boundary-layer temperature, rather than the bulk gas temperature, drives the convective heat transfer.

The simple heat balance also assumes that radiation is absorbed in a thin layer at the surface. The Fire Dynamics Simulator notes that many common materials are not opaque to thermal radiation, so radiation can penetrate some distance into them. The exposure itself must be predicted well, too. Zone models, which assume uniform gas layers, tend to under-predict temperature and heat flux for targets near the fire.

See also

• Steel in fire
• Critical steel temperature
• Section factor
• Standard fire curve

References

• Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
• Gann, R. G., Hamins, A., McGrattan, K. B., Mulholland, G. W., Nelson, H. E., Ohlemiller, T. J., Pitts, W. M., & Prasad, K. R. (2005). Reconstruction of the fires in the World Trade Center towers (NIST NCSTAR 1-5). National Institute of Standards and Technology.
• Lie, T. T. (2002). Fire temperature-time relations. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 4-201–4-208). National Fire Protection Association; Society of Fire Protection Engineers.
• McGrattan, K., Hostikka, S., Floyd, J., McDermott, R., Vanella, M., Mueller, E., & Paul, C. (2026). Fire Dynamics Simulator technical reference guide: Vol. 1. Mathematical model (6th ed., NIST Special Publication 1018-1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.SP.1018
• Prasad, K., & Baum, H. (2005). Fire Structure Interface and thermal response of the World Trade Center towers (NIST NCSTAR 1-5G). National Institute of Standards and Technology.
• Quintiere, J. G. (2002a). Compartment fire modeling. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-162–3-170). National Fire Protection Association; Society of Fire Protection Engineers.
• Stroup, D., Lindeman, A., McGrattan, K., Peacock, R., Overholt, K., Joglar, F., LeStrange, S., & Montanez, S. (2016). Verification and validation of selected fire models for nuclear power plant applications: Supplement 1 (NUREG-1824, Supplement 1; EPRI 3002002182). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Tien, C. L., Lee, K. Y., & Stretton, A. J. (2002). Radiation heat transfer. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 1-73–1-89). National Fire Protection Association; Society of Fire Protection Engineers.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.