Biot number

The Biot number (symbol Bi) is a dimensionless number that compares the resistance to heat flow inside a solid with the resistance to heat transfer at its surface. It equals the surface heat transfer coefficient multiplied by the thickness of the solid and divided by its thermal conductivity. A small Biot number means that conduction inside the solid is fast compared with the supply of heat to its surface, so the whole object stays at nearly one temperature. A large Biot number means that the surface heats up while the interior lags behind.

Fire safety engineers use the number to judge whether a component may be treated as thermally thin, because only then does a simple uniform-temperature calculation apply. This question arises for unprotected steel members, heat detectors, the ignition of solid fuels and the self-heating of stored materials. With room-temperature properties and the convective coefficient usually assumed for a standard fire, a 10 mm steel plate has a Biot number of about 0.0055. A softwood board of the same thickness reaches about 1.8.

Definition

The Biot number is the ratio of two thermal resistances that act in series. The first is the resistance to conduction through the solid, and the second is the resistance of the thin gas layer at its surface. Introduction to Fire Safety Engineering expresses the conduction resistance of a layer as its thickness divided by its thermal conductivity. It expresses the surface resistance as the inverse of the heat transfer coefficient. Dividing the first resistance by the second gives the usual form of the number:

Temperature through a plate heated from the left with an insulated back. Left: small Biot number, the temperature drop sits in the gas layer at the surface. Right: large Biot number, the drop sits inside the solid. Based on Rockett and Milke (2002), p. 1-33, and McGrattan et al. (2026), p. 226.

$$\mathrm{B}\mathrm{i}=\frac{b/k}{1/h}=\frac{hb}{k}$$

Here $h$ is the heat transfer coefficient at the surface, $b$ is the thickness through which heat must be conducted and $k$ is the thermal conductivity of the solid. A thicker layer or a poorer conductor raises the internal resistance and therefore the Biot number. A faster or more turbulent gas flow raises the heat transfer coefficient and also raises the number.

Rockett and Milke derive the same number from two characteristic times of a plate that is heated on one face and insulated on the other. The first time, $t_{\mathrm{c}}$, describes how quickly heat crossing the surface can change the temperature of the whole plate. The second time, $t_0$, describes how quickly heat spreads through the thickness by conduction. Their ratio is the Biot number:

$$t_{\mathrm{c}}=\frac{\rho cb}{h},t_0=\frac{\rho c{b}^2}{k},\frac{t_0}{t_{\mathrm{c}}}=\frac{hb}{k}=\mathrm{B}\mathrm{i}$$

The length in the definition depends on how the body is heated. Rockett and Milke use the thickness of a plate heated on one face with an insulated back. Kanury uses the slab thickness in the same way for ignition problems. For self-heating bodies, Gray defines the number with the smallest physical dimension of the body. The technical guide of the Fire Dynamics Simulator uses the thickness of the solid layer. A Biot number therefore has a clear meaning only together with the length on which it is based.

Symbol	Description	Unit
$\mathrm{B}\mathrm{i}$	Biot number	–
$h$	heat transfer coefficient at the surface	W/(m²·K)
$h_{\mathrm{c}}$, $h_{\mathrm{r}}$	convective and radiative parts of the heat transfer coefficient	W/(m²·K)
$b$	characteristic length, here the thickness of a plate heated on one face	m
$k$	thermal conductivity of the solid	W/(m·K)
$\rho$	density of the solid	kg/m³
$c$	specific heat capacity of the solid	J/(kg·K)
$t_{\mathrm{c}}$	characteristic time for heating the whole plate through its surface	s
$t_0$	characteristic time for conduction through the thickness	s
$t$	time	s
$\bar{T}$	average temperature of the plate	K
$T_0$	initial temperature of the plate	K
$T_{\mathrm{f}}$	temperature of the surrounding hot gas	K
$\mathrm{F}\mathrm{o}$	Fourier number	–
$\alpha$	thermal diffusivity, $k/(\rho c)$	m²/s
$\epsilon$	emissivity of the surface	–
$\sigma$	Stefan-Boltzmann constant, 5.67 × 10−8	W/(m²·K4)
$T_{\mathrm{r}}$	radiative temperature of the surroundings	K
$T_{\mathrm{s}}$	surface temperature; in the steel equation, the steel temperature	K
${\mathrm{B}\mathrm{i}}^{*}$	modified Biot number including radiation	–
$\mathrm{\Delta }{T}_{\mathrm{s}}$	temperature rise of the steel during one time step	K
$c_{\mathrm{s}}$	specific heat capacity of steel	J/(kg·K)
$W$	steel mass per unit length	kg/m
$D$	heated perimeter of the steel section	m
$\mathrm{\Delta }t$	time step	s

Thermally thin and thick behaviour

When the Biot number is small, the temperature of a solid can be represented by its average value. This lumped approach suits thin materials and materials with a high thermal conductivity, where conduction inside is fast compared with the rate at which heat arrives. If the surrounding gas keeps a constant temperature, the average temperature of the plate approaches the gas temperature exponentially:

$$\bar{T}=T_0{e}^{-t/t_{\mathrm{c}}}+T_{\mathrm{f}}(1-e^{-t/t_{\mathrm{c}}})$$

After one time constant the plate has covered about 63 % of the gap to the gas temperature, and after a long time it reaches that temperature. When the Biot number is large, this analysis is inappropriate, because the surface heats much faster than the interior. A single temperature then no longer describes the solid, and the temperature profile through the thickness has to be calculated.

Heating of a thermally thin plate in gas at constant temperature: the temperature rise as a fraction of the gas-plate difference, against time divided by the time constant ρcb/h. Based on Rockett and Milke (2002), p. 1-33.

Thin or thick behaviour also depends on the duration of heating. The Fire Dynamics Simulator guide states that the dimensionless temperature profile inside a solid scales with both the Biot number and the Fourier number, $\mathrm{F}\mathrm{o}=\alpha t/b^2$. The Fourier number compares the rate of heat conduction with the rate of heat storage and acts as a dimensionless time. Introduction to Fire Safety Engineering gives a practical rule based on the same idea. A layer behaves as thermally thick while its thickness exceeds about twice the characteristic penetration depth $\sqrt{\alpha t}$.

By this rule a 200 mm concrete wall is still thermally thick after 15 minutes, because twice the penetration depth is then about 0.060 m. The book adds that a member can be thermally thick early in a fire and behave more and more like a thin one as the exposure continues. It also stresses that the limit is a rule of thumb, not a sharp physical boundary. The two numbers combine in a simple way: their product $\mathrm{F}\mathrm{o}\cdot \mathrm{B}\mathrm{i}=ht/(\rho cb)$ equals the elapsed time divided by the lumped time constant. The Fire Dynamics Simulator uses this product in a method that scales burning curves from cone calorimeter tests to other heat fluxes and thicknesses. In that method the effective heat of gasification of a material was found to collapse onto one curve when plotted against the product of the Fourier number and a modified Biot number.

Heat transfer coefficient and radiation

The heat transfer coefficient is usually the least certain input. It describes how readily heat passes from a gas to a surface, and it increases with flow speed and turbulence. Introduction to Fire Safety Engineering gives guide values of 1 to 5 W/(m²·K) for normal indoor conditions and 3 to 10 W/(m²·K) for a smoke reservoir under a roof. For a fully developed fire it gives 20 to 25 W/(m²·K), for a very hot fire 50 W/(m²·K) and outdoors in a storm up to 200 W/(m²·K). For a standard fire a value of 25 W/(m²·K) is usually assumed.

Other sources give values in the same range. Milke uses 20 to 25 W/(m²·K) for the convective heat transfer to fire-exposed steel. Kanury estimates about 15 W/(m²·K) for a small wooden target in normal air. Gray quotes about 20 W/(m²·K) for smooth surfaces in the rapidly stirred air of a test oven. These values describe heat transfer by convection, the transport of heat by a moving gas.

Radiation complicates the picture. A heated plate loses heat by radiation in proportion to the difference between the fourth powers of its absolute temperature and that of its surroundings. Because of this fourth power, the heating equation can no longer be integrated in closed form. A common approximation linearises the radiation term, which turns it into an extra coefficient that adds to the convective one. For burning solids, the Fire Dynamics Simulator guide notes that external radiation and the radiation of the flame above the surface drive the heating. It therefore replaces the traditional convective Biot number by an alternative number that includes a radiative coefficient:

$${\mathrm{B}\mathrm{i}}^{*}=\frac{(h_{\mathrm{c}}+h_{\mathrm{r}})b}{k},h_{\mathrm{r}}=\epsilon \sigma (T_{\mathrm{r}}+T_{\mathrm{s}})(T_{\mathrm{r}}^2+T_{\mathrm{s}}^2)$$

The radiative coefficient grows quickly with temperature. In the worked example below it is about three times the convective coefficient at the start of heating and about six times at a surface temperature of 500 °C. The Biot number of a component exposed to flames is therefore several times larger than a value based on convection alone.

Typical values

The thermal conductivity of common building materials varies more than a thousandfold, so components of equal thickness can have very different Biot numbers. The table uses the room-temperature properties listed in Introduction to Fire Safety Engineering and a convective coefficient of 25 W/(m²·K). It applies to a 10 mm layer heated on one face.

Material	Thermal conductivity (W/(m·K))	Biot number for 10 mm, 25 W/(m²·K)
Steel	45.8	0.0055
Concrete	2.0	0.125
Gypsum plaster	0.48	0.52
Softwood	0.14	1.8
Polyurethane foam	0.034	7.4

Steel stays far below one, which agrees with the uniform-temperature assumption used for unprotected steel members. Softwood and polyurethane foam exceed one, so their surfaces heat much faster than their interiors. Concrete and gypsum plaster lie in between at this thickness. For such materials the duration of heating also matters, as the penetration-depth rule shows.

Biot number of a 10 mm layer heated on one face with a convective coefficient of 25 W/(m²·K), using room-temperature conductivities. Based on Van Rossum (2027), §5.2 and §5.5.

Worked example

A steel plate and a softwood board, both 10 mm thick, are exposed on one face to hot gas at 900 °C. Introduction to Fire Safety Engineering uses this gas temperature for a fully developed fire in one of its examples. The back faces are insulated, and both start at 20 °C. The thermal conductivities are 45.8 W/(m·K) for steel and 0.14 W/(m·K) for softwood, and the convective coefficient is 25 W/(m²·K).

With convection alone, the steel plate has $\mathrm{B}\mathrm{i}=25\times 0.010/45.8=0.0055$. The softwood board has $\mathrm{B}\mathrm{i}=25\times 0.010/0.14=1.8$. To add radiation, the example uses an emissivity of 0.7, the resultant emissivity that Milke lists for a steel column exposed on all sides. With the Stefan-Boltzmann constant of 5.67 × 10⁻⁸ W/(m²·K⁴) and temperatures of 1,173 K and 293 K, the linearised radiative coefficient is:

$$h_{\mathrm{r}}=0.7\times 5.67\times 10^{-8}\times \frac{1173^4-293^4}{1173-293}\approx 85\mathrm{W}/\mathrm{(}{\mathrm{m}}^{\mathrm{2}}\mathrm{\cdot }\mathrm{K}\mathrm{)}$$

The total coefficient is then about 110 W/(m²·K). This gives a modified Biot number of 0.024 for the steel plate and 7.9 for the softwood board. When the steel surface has reached 500 °C, the radiative coefficient rises to about 153 W/(m²·K) and the steel value to 0.039.

The steel plate therefore remains thermally thin throughout, while the softwood board is clearly thermally thick. For the steel plate, the lumped time constant with convection alone is $t_{\mathrm{c}}=7850\times 460\times 0.010/25\approx 1,440$ s, or 24 minutes. With radiation included and the lumped steel equation stepped in time steps of 10 s, the plate reaches 500 °C after about 3 minutes. For the softwood board the same equation would give misleading results, because its surface runs far ahead of its average temperature.

Applications in fire safety engineering

Heat transfer analyses of steel members estimate the time needed to heat a member to a critical temperature. For unprotected members, the handbook of the Society of Fire Protection Engineers uses a quasi-steady lumped heat capacity analysis, which assumes that the member has one uniform temperature. The temperature rise during a short time step is:

Cross-sections of a wide-flange (I) section and a hollow section. The heated perimeter D is traced in orange; W/D and A/P relate the steel mass or area to that perimeter. Based on Milke (2002), pp. 4-216 and 4-223.

$$\mathrm{\Delta }{T}_{\mathrm{s}}=\frac{h_{\mathrm{c}}+h_{\mathrm{r}}}{c_{\mathrm{s}}(W/D)}(T_{\mathrm{f}}-T_{\mathrm{s}})\mathrm{\Delta }t$$

Here $c_{\mathrm{s}}$ is the specific heat of steel, $W$ the steel mass per unit length and $D$ the heated perimeter. The quasi-steady assumption requires a small time step $\mathrm{\Delta }t$, of the order of 10 s, and the equation is applied step by step over the fire exposure of interest. The geometry of the member enters through a section factor, written W/D for wide-flange sections and A/P for hollow sections. The A/P ratio is the cross-sectional area divided by the heated perimeter, and W/D is essentially the product of steel density and A/P. Written in this way, the steel equation has the same form as the thin-plate equation, with A/P in the place of the plate thickness.

The same uniform-temperature assumption appears in fire detection. Detector models treat a fixed-temperature heat detector as a lumped mass without temperature gradients inside the sensing element. NUREG-1805 considers this reasonable for solder-type detectors, whose operating element has a low mass. For bimetallic detectors the assumption may introduce some error, because heat must reach two slightly different parts.

In the ignition of solids, Kanury describes the Biot number as the characteristic ratio of convective and conductive fluxes. Convective heat losses matter most when the specimen is thin, the irradiance low and the exposure long. In his example, a firwood target 1 cm thick in normal air has a Biot number of about 0.88. A target 0.1 cm thick has about 0.088 and reaches sustained spontaneous ignition after a shorter exposure to the same radiant flux.

In self-heating, the number separates two classical theories. The Semenov theory treats the body as well stirred, with one uniform temperature. The Frank-Kamenetskii theory includes temperature gradients inside the body and often agrees better with experiments on solids of low thermal conductivity. The first corresponds to a zero Biot number and the second to an infinite one, with the surface held at the ambient temperature.

In oven tests with wire-gauze baskets, a Biot number above 30 is effectively infinite. Cellulosic materials such as sawdust conduct about 0.05 W/(m·K), and the surface coefficient in a stirred oven is about 20 W/(m²·K). A test body with a size of 0.1 m then reaches a Biot number of about 40. Inorganic salts conduct 0.2 to 3.0 W/(m·K), which gives only 0.7 to 10. Gray reports that this reduces the critical parameter of the theory by a factor ranging from 0.21 to 0.83.

The critical ambient temperature is the surrounding temperature that separates stable self-heating from runaway to ignition for a body of given size and shape. A finite Biot number makes this temperature sensitive to air movement around the body. For hydrated calcium hypochlorite in a container of 0.175 m radius, this temperature is 60 °C in stirred air but 55 °C in still air. Gray concludes that this raises serious questions about empirical tests used to classify self-heating materials for shipping.

Experiments use the number to choose boundary conditions. In compartment fire tests for the World Trade Center investigation, the National Institute of Standards and Technology computed a Biot number for the unexposed back of the Marinite ceiling boards. Because Marinite conducts heat poorly, the number was much larger than one. The back face could therefore be held at ambient temperature in the heat transfer model.

Limits and uncertainty

A Biot number is only as reliable as its heat transfer coefficient, and that coefficient depends on the flow around the surface. The critical ambient temperature example of Gray shows that a change in air movement alone can shift a measured result by several degrees. Material properties add further uncertainty. The thermal conductivity and specific heat of most building materials vary strongly with temperature, and moisture, charring, melting or intumescence change them further during a fire. A value calculated from room-temperature properties is therefore only an approximation.

The linearised treatment of radiation is itself an approximation. Rockett and Milke note that more accurate answers require higher-order approximations or numerical methods. The simple surface condition behind the self-heating definition also fails when there are heat sources on the boundary, such as incident radiation or frictional heating.

Uneven exposure limits the lumped approach even for steel. A steel beam under a concrete slab is heated on three sides and loses heat to the slab. Its temperature can then vary appreciably from the bottom to the top flange, which stretches the uniform-temperature assumption. For many engineering applications the lumped approach still gives a conservative estimate of the average temperature rise of the beam. When the temperature gradient across the beam matters, another analytical approach is needed.

Related concepts

The Biot number resembles the Nusselt number, but the two describe different things. The Nusselt number divides the product of the heat transfer coefficient and a length by the conductivity of the air. It compares the heat actually transferred with what still air would conduct over the same length. For natural convection in fires it is of the order of 100. The Biot number uses the conductivity of the solid instead, and it compares resistances inside and outside the solid.

Thermal thinness is not a property of thickness alone. In the worked example, a 10 mm steel plate behaves as thin, while a 10 mm softwood board behaves as thick under the same exposure. Introduction to Fire Safety Engineering also stresses that thin and thick behaviour always refer to a particular heating duration. The zero and infinite Biot numbers of the classical self-heating theories are limiting cases, and Gray treats both as special cases of a more general formulation.

See also

• Heat transfer
• Thermal diffusivity
• Thermal inertia
• Thermal radiation
• Emissivity

References

• Gray, B. (2002). Spontaneous combustion and self-heating. 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. 2-211–2-228). National Fire Protection Association; Society of Fire Protection Engineers.
• Hamins, A., Maranghides, A., McGrattan, K. B., Johnsson, E. L., Ohlemiller, T. J., Donnelly, M. K., Yang, J. C., Mulholland, G. W., Prasad, K. R., Kukuck, S. R., Anleitner, R. L., & McAllister, T. P. (2005). Experiments and modeling of structural steel elements exposed to fire (NIST NCSTAR 1-5B). National Institute of Standards and Technology.
• Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
• Kanury, A. M. (2002a). Flaming ignition of solid fuels. 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. 2-229–2-245). 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
• Milke, J. A. (2002a). Analytical methods for determining fire resistance of steel members. 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. 4-209–4-238). National Fire Protection Association; Society of Fire Protection Engineers.
• Rockett, J. A., & Milke, J. A. (2002). Conduction of heat in solids. 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. 1-27–1-43). National Fire Protection Association; Society of Fire Protection Engineers.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.