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Concrete In Australia : September 2014
46 Concrete in Australia Vol 40 No 3 FEATURE: CONCRETE PERFORMANCE IN FIRE 2.0 FURNACE OR FIRE? When analysing the heat transfer from the fire to a structural element the problem needs to be formulated in terms of heat fluxes. While temperatures result from the solution to the energy conservation equation, all quantities to be balanced are energies (Incropera & DeWitt, 2002; SFPE, 2009). Heat is transferred from gases to surfaces via radiation and convection resulting in a total heat flux, To t , where (3) and rad is the heat transfer via radiation and con is the heat transferred via convection. For simplicity the problem will be assumed as one dimensional, then the boundary condition for a solid element (structural element) becomes (4) which is a generic version of Equation (2) and where the thermal conductivity (ki) is a property of the solid and the gradient of temperature is taken at the surface. In other words all the heat arriving is conducted into the solid. If there are multiple layers then at each interface the following boundary condition should apply: (5) Where the gradients correspond to each side of the interface and the sub-index “s” is a generic way to represent the next layer of solid. Once the boundary conditions are defined, the energy equation can be solved for each material involved. In the case where two layers of solid are involved (“i” and “s”), then the energy equations take the form (6) (7) The solution to the energy equation will result in the temperature evolution of the material in space and time. Equations (6) and (7) could be repeated for as many layers as necessary. If the geometry is complex, then the problem needs to be resolved in all dimensions. If the properties vary with temperature then, as the temperature increases, these properties need to evolve with the local temperature. Variable properties thus require a numerical solution. If a simple analytical solution is to be obtained then adequate global properties need to be defined. It is important to note that whatever the solution methodology adopted, the temperature of the structure is the result of the resolution of equations (6) and (7) using potential boundary conditions such as those presented in equations (4) and (5). To obtain the numerical solution it is necessary to input material properties for the different layers (“i” and “s”). The material properties required are all a function of temperature and are as follows: ρi , Cpi ,k i ρs , Cps ,k s For some materials such as steel the properties are well known and thus very little difference can be found between the literature (Eurocode 1, 2002). For other materials such as concrete, wood or different thermal insulations the scatter is much greater (Buchanan, 2002). The uncertainty is associated to the presence and migration of water, degradation, crack formation, etc. Furnace data is generally used as a substitute for the uncertainties associated to property definition. In many cases global properties are extracted by fitting temperatures to the furnace data. These properties are then extrapolated and many times used in equations such as (4) to (7) for performance assessment. Nevertheless, this practice also has its unique complexities. First of all the model needs to include all the physical variables necessary, so if physical processes such as the degradation or water are not included in the model, the properties used from the furnace calibration become hybrids that include these physical parameters. Introducing physical phenomena into constants inevitably narrows the range of application, thus most of these calibrated properties can only be used to re-evaluate furnace data. Extrapolation to drastically different scenarios such as a fire becomes doubtful. An important aspect many times overlooked is the need to make sure that the boundary conditions are properly Figure 3: Schematic of the typical temperature distributions for different extreme values of the Biot number. Figure 2: Temperature evolution of the gas phase of a compartment fire and a small cross section unprotected steel beam. Standard temperature time curve per ISO-834 (1975), DFT stands for Dalmarnock Fire Test as per Abecassis-Empis et al. (2008). TH TS TC TC TS≈TH TH TH TS≈TC TC (a) (b) (c) d d d 44-49 - Torero.indd 46 44-49 - Torero.indd 46 26/08/14 10:21 AM 26/08/14 10:21 AM