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The analysis of temperature response in a structural member can be sub-divided into two parts. One is the heat transfer across the boundary from the furnace or fire into the surface of the structural member, which is through the combination of convection and radiation and is usually treated as boundary conditions; the other is the heat transfer within the structural member, which is through conduction and is treated as governing equation expressed by the Fourier equation of heat transfer.

Heat conduction is the transfer of thermal energy from one place to another through a solid or fluid due to the temperature difference between the two places. The transfer of thermal energy occurs at the molecular and atomic levels without net mass motion of the material. The rate equation

describing this heat transfer mode is Fourier law, expressed by

q= −λ∇θ (6.1)

where q is the vector of heat flux per unit area, λ is the thermal con-ductivity tensor and θ is the temperature. For an isotropic solid such as steel, concrete or masonry, λ = λI, where λ is the thermal conductivity that may be a function of the temperature and I is the identity matrix.

The conservation of energy with Fourier’s law requires

ρc∂θ

∂t = −∇ · q + Q (6.2)

where ρ is the density, c is the specific heat, t is the time and Q is the internal heat generation rate per unit volume. The specific heat may be temperature dependent. Substituting Eq. (6.1) into Eq. (6.2) yields

ρc∂θ

∂t = ∇ · (λ∇θ) + Q (6.3)

Equation (6.3) is the heat conduction equation and is solved subject to an initial condition and appropriate boundary conditions. The initial condition consists of specifying the temperature throughout the solid at an initial time. The boundary conditions may take the following several forms.

(1) The fire exposed surface – The surface of the structural member is exposed to a fire on which the heat transfer involves both convec-tion and radiaconvec-tion, although it is generally accepted that the radiaconvec-tion component is the more dominant after the very early stages of the fire. The net heat flux to the surface of the structural member thus is expressed as

˙hnet= ˙hnet,c+ ˙hnet,r (6.4a)

in which,

˙h_net,c= αc(θg− θm)= net convective heat flux per unit surface

˙h_net,r= εmε_fσ[(θg+ 273)⁴− (θm+ 273)⁴] = net radiative heat flux per unit surface

where αcis the coefficient of heat transfer by convection, θgis the gas temperature in the vicinity of the fire exposed surface, θmis the surface temperature of the structural member,  is the configura-tion factor, εm is the surface emissivity of the structural member, ε_f is the emissivity of the fire and σ = 5, 67 × 10⁻⁸W/m²K⁴is the Stephan–Boltzmann constant.

(2) The no heat-flow surface – The surface is a thermal symmetric plane or has a large degree of insulation and thus can be assumed as thermally insulated having no heat flow through it. Therefore, the net heat flux to the surface of the structural member can be simply expressed as

˙hnet= 0 (6.4b)

(3) The ambient exposed surface – The surface is exposed to ambient conditions and thus can be treated similarly to that exposed to a fire but replacing the fire temperature with the ambient temperature, θa, that is,

˙hnet= αc(θa− θm)+ εmε_fσ[(θa+ 273)⁴− (θm+ 273)⁴] (6.4c) (4) The fixed temperature surface – The surface temperature of the struc-tural member is specified to be constant or a function of a boundary coordinate and/or time.

The boundary condition for the surface types (1)–(3) is called Neumann boundary condition which specifies the normal derivative of the temper-ature, that is,

λ∂θ_m

∂n = ˙hnet (6.5a)

where n is the normal of the surface. The boundary condition for the surface type (4) is called Dirichlet boundary condition which specifies the function of the temperature, that is,

θ_m= ¯θ(t) (6.5b)

where ¯θ(t) is the prescribed temperature at the boundary.

Convection is the transfer of thermal energy through a fluid due to motion of the fluid. The energy transfer from one fluid particle to

another occurs by conduction, but thermal energy is transported by the motion of the fluid. However, the convection heat transfer coefficient is not a property of the fluid. It is an experimentally determined param-eter whose value depends on all the variables influencing convection such as the surface geometry, the nature of fluid motion, the proper-ties of the fluid and the bulk fluid velocity. In EN 1991-1-2, it has been suggested that αc= 25 W/m²K for the fire exposed surface when the stan-dard temperature–time curve is used and αc= 9 W/m²K for the ambient exposed surface when assuming it contains the effects of heat transfer by radiation.

Unlike the convection which requires a medium to transfer the heat, the radiation is the transfer of thermal energy between two locations by an electromagnetic wave which requires no medium. The radiation term used here in Eq. (6.4) is the traditional one that has been used in textbooks and also implemented in computer packages (Becker, Bizri and Bresler, 1974; Iding, Bresler and Nizamuddin, 1977a). In litera-ture several different radiation expressions have been suggested. For example, Mooney (1992) proposed a radiation expression based on the concept of the surface radiant energy balance in the fire environment, which uses the representative temperature and representative emissiv-ity, instead of the traditionally used fire temperature and fire emissivity.

It should be noted that, however, whichever expression is used experi-mental data are always required to validate the expression and determine the parameters involved in the expression. To allow for varying radia-tive heat flux levels while keeping the surface and fire emissivities as constants, a configuration factor is introduced in the radiative heat flux expression. A conservative choice for the configuration factor is = 1.

A lower  value may be obtained from the calculation based on the frac-tion of the total radiative heat leaving a given radiating surface that arrives at a given receiving surface, as given in EN 1991-1-2, to take account of so called position and shadow effects. The theory behind the calculation of the configuration factor is given by Drysdale (1998). Figure 6.1 shows a typical example of the variation of the resultant emissivity in the predic-tion of temperatures within a steel column when the configurapredic-tion factor is taken as = 1, i.e. εres= εmε_f = εmε_f (Chitty et al., 1992). It should be noted that the variation of εresshown in Fig. 6.1 can be interpreted as the variation of configuration factor due to the difference in positions while taking εmand εf as constants.

It should be stated here that both the governing, Eq. (6.3) and bound-ary condition, Eq. (6.5) are non-linear. The former is due to the thermal conductivity and specific heat that are temperature dependent, as shown in Chapter 5, and the latter is due to the radiative boundary condition which involves a non-linear term of the temperature. Thus the closed form solution to governing, Eq. (6.3) with boundary conditions, Eq. (6.5) is not

568°C 535°C

589°C 601°C

eres= 0,2 eres= 0,2

eres= 0,2 eres= 0,2

e_res = 0,3

eres = 0,3 eres = 0,5

eres= 0,5 544°C 650°C

664°C

Figure 6.1 Variation of resultant emissivity in the prediction of temperatures within a steel column. The temperatures given correspond to the temperature field calculated at 46 min (temperatures around profile correspond to the centre of the discretized border elements of 10 mm thickness). (Copy with permission from Chitty et al., 1992).

possible for even the simplest geometry. Numerical methods such as finite element methods are usually required to solve this kind of heat transfer problems.