3.9 Heat Transfer
3.9.1 Conduction
3.8.3 Localized Fires
The discussion of post‐flashover fires in this chapter has been based on the assumption that a fully developed fire occurs and creates the same temperature conditions throughout the fire compartment. In some circumstances, possibly in a large space where there are no nearby combustibles, or in a fire partially controlled by sprinklers, there could be a localized fire which has much less impact on the building structure than a fully developed fire. Tests by Hasemi et al. (1995) have been used by Franssen et al. (1998) to calculate steel temperatures in steel beams above burning cars in car parking buildings. Structural design calculations can be made in such cases if the member temperatures are known, but it is always conservative to assume a fully developed fire. Bailey et al. (1996a) have investigated the structural response of a multi‐bay steel frame to a spreading fire including the effects of cooling during the decay period. For design guidance on localized fires refer to Appendix C of Eurocode 1 Part 1.2 (CEN, 2002b).
unit mass of the material by one degree (with units of J/kgK). Thermal conductivity k repre- sents the amount of heat transferred through a unit thickness of the material per unit tempera- ture difference (with units of W/mK). There are two derived properties which are often needed.
These are the thermal diffusivity given by α = k/ρ cp (with units m2/s) and the thermal inertia kρ cp (with units W2s/m4K2). For a given fire load, rooms lined with materials of low thermal inertia will experience higher temperatures than rooms lined with materials of higher thermal inertia. Thermal properties for common materials are given in Table 3.6. A more extensive list is given in Appendix A of the SFPE Handbook (SFPE, 2008), including temperature dependent thermal properties for metals.
In the steady‐state situation, the transfer of heat by conduction is directly proportional to the temperature gradient between two points, with a constant of proportionality known as the thermal conductivity, k, so that
q kdT dx/ (3.32)
where q is the heat flow per unit area (W/m2), k is the thermal conductivity (W/mK), T is temperature (°C or K) and x is distance in the direction of heat flow (m). The steady‐state calculation does not require consideration of the heat required to change the temperature of the material that is being heated or cooled.
Table 3.6 Thermal properties of some common materialsa
Material Thermal
conductivity, k (W/mK)
Specific heat,
cp (J/kgK) Density, ρ (kg/m3)
Thermal diffusivity, α (m2/s)
Thermal inertia, kρcp (W2s/m4K2)
Copper 387 380 8940 1.14 × 10–4 1.3 × 109
Steel (mild) 45.8 460 7850 1.26 × 10–5 1.6 × 108
Brick (common) 0.69 840 1600 5.2 × 10–7 9.3 × 105
Concrete 0.8–1.4 880 1900–2300 5.7 × 10–7 2 × 106
Glass (plate) 0.76 840 2700 3.3 × 10–7 1.7 × 106
Gypsum plaster 0.48 840 1440 4.1 × 10–7 5.8 × 105
PMMAb 0.19 1420 1190 1.1 × 10–7 3.2 × 105
Oakc 0.17 2380 800 8.9 × 10–8 3.2 × 105
Yellow pinec 0.14 2850 640 8.3 × 10–8 2.5 × 105
Asbestos 0.15 1050 577 2.5 × 10–7 9.1 × 104
Fibre insulating board
0.041 2090 229 8.6 × 10–8 2.0 × 104
Polyurethane foamd
0.034 1400 20 1.2 × 10–6 9.5 × 102
Air 0.026 1040 1.1 2.2 × 10–5 —
Source: Reproduced from Drysdale (2011) by permission of John Wiley & Sons, Ltd.
a From Pitts and Sissom (1977) and others. Most values for 0 or 20 °C. Figures have been rounded off.
b Poly(methyl methacrylate). Values of k, cp and ρ for other plastics are given in Drysdale (2011), Table 1.2.
c Properties measured perpendicular to the grain.
d Typical values only.
heat transfer.
This type of analysis can be extended to two or three dimensions as necessary. There are many methods of solving the heat conduction equation, using analytical, graphical or numerical methods. Some methods are given by Drysdale (2011) and there are many standard textbooks on heat transfer. Calculation of conductive heat transfer can be by simple formulae, the use of design charts or by numerical analysis.
3.9.1.1 Simple Formulae
Standard textbooks on heat transfer contain simple formulae for conductive heat transfer in various materials and geometries. Some of these are available for common materials exposed to fire. A lumped heat capacity formula can be used for a protected or unprotected steel element on the assumption that the internal steel temperatures are constant. This type of formula is most accurate where it is used repeatedly with sequential time steps. Some formulae can take account of the heat required to heat up heavy insulating materials, or the time delay resulting from driving off moisture. Typical formulae are given by Pettersson et al. (1976), Eurocode 3 Part 1.2 (CEN, 2005b) and Milke (2008). A simple spreadsheet formulation is described by Gamble (1989). Examples of these methods are given in Chapter 6.
A semi‐infinite slab analysis can be used in situations where the heat transfer is essentially one‐dimensional, such as with large flat surfaces. This analysis assumes that the heat is absorbed before reaching the unexposed side, so the material has to be relatively thick.
Schaffer (1977) has applied this type of analysis to wood slabs.
3.9.1.2 Numerical Analysis
The most powerful tools for calculating conductive heat transfer are computer‐based numerical methods such as finite element or finite difference formulations. These techniques are well established, but there are not many user‐friendly commercial computer packages customized for fire applications. Special characteristics needed for structural fire applica- tions include internal voids and temperature dependent thermal properties. The most widely used finite element programs for thermal analysis of structural members include SAFIR
(Franssen et al., 2000) and TASEF (Sterner and Wickström, 1990). A review of some of these programs and others that have now been discontinued is given by Sullivan et al. (1994).
Many generic finite element stress analysis programs can calculate heat transfer by conduction. Some widely used commercial programs include ABAQUS (2010) and ANSYS (2009). These are versatile programs which can analyse any three‐dimensional mesh which is input by the user. More information is given in Chapter 11.
Both two‐ and three‐dimensional heat transfer analysis is possible, but two‐dimensional analysis is adequate for almost all fire engineering applications. This is because structural elements are mostly planar or linear, and there will be no temperature gradient along an element if it can be assumed that temperatures are uniform within a post‐flashover fire compartment. This assumption holds for fires in small rooms but is less accurate for fires in large spaces.