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Lining property: b = P1600 x 0.8 x 1200 = 1239.4 J/(m2 s1/2 K)

0.04 1239.4

The real time at the maximum fire temperature td = 0.587/1.28 = 0.458 h

For the cooling part, Equation 37.9 will be used, giving the time t* necessary to reach the ambient temperature as t* = 1.937 h. The real time is 1.937/1.28 = 1.5136 h.

Figure 37.5 plots the complete fire temperature-time relationship.

Fire exposure time, min

FIGURE 37.5 Temperature-time relationships.

Fire exposure time, min

FIGURE 37.5 Temperature-time relationships.

### 37.3 Introduction to Heat Transfer

Having determined the temperature-time relationship of a fire to which a structural member is exposed, the next step is to calculate the temperatures in the structural member. Building structures are usually treated as either one-dimensional, such as beams and columns, or two-dimensional, such as slabs and walls. For a one-dimensional member, it is common to assume that there is no temperature variation along its length. For a two-dimensional member, temperatures are assumed to change only through its thickness. Calculating temperatures in a structural member exposed to fire involves using heat transfer analysis. Only a brief introduction to heat transfer theory will be given in this chapter, with particular emphasis on applications under fire conditions.

There are three basic mechanisms of heat transfer: conduction, convection, and radiation. In conduction, energy or heat is exchanged in solids on a molecular scale without any movement of macroscopic portions of matter relative to one another. Convection refers to heat transfer at the interface between a fluid and a solid surface. Here, the exchange of heat is due to fluid motion. This motion may be the result of an external force, causing the fluid to flow over the solid surface at speed. This is called forced convection. Convection can also occur due to buoyancy-induced flow when there is a temperature gradient in the fluid, causing a density gradient. This is called natural convection. Radiation is the exchange of energy by electromagnetic waves that, like visible light, can be absorbed, transmitted or reflected at a surface. Unlike conduction and convection, heat transfer by radiation does not require any intervening medium between the heat source and the receiver. Thus, in the context of structural fire safety design, heat conduction describes the heat transfer process inside a structural member and heat convection and radiation describe the thermal boundary condition of the structural member.

37.3.1 Conduction

The basic equation for one-dimensional heat conduction is Fourier's law of heat conduction. It is expressed as dT

dx where, refering to Figure 37.6, dT is the temperature difference across an infinitesimal thickness dx. Q is the rate of heat transfer (heat flux) across the material thickness. The minus sign in Equation 37.10 indicates that heat flows from the higher temperature side to the lower temperature side.

(T ) (T + dT ) FIGURE 37.6 Heat conduction in one dimension.

FIGURE 37.7 Temperature distribution with constant thermal conductivity.

The constant of proportionality k is the thermal conductivity of the material. In many practical applications of fire safety engineering, the material thermal conductivity within the relevant temperature range may be approximated as a constant. Thus, Equation 37.10 may be replaced by its finite difference equivalent

where, refering to Figure 37.7, T1 and T2 are temperatures at the two sides of a material and Dx is the material thickness. DXk expresses the thermal resistance of the material.

### 37.3.2 Convection

Heat convection and radiation are considered at the interface between a structural member and the fire or the ambient temperature air. When applying thermal boundary conditions, it is often assumed that the heat exchange between the fluid and the structural surface is related to the temperature difference at the interface. Therefore, on the fire side from fire to structural surface

On the ambient temperature air side from structural surface to the ambient temperature air

where Tfi and Ta are the fire and air temperatures, respectively and Ts is the structural surface temperature.

Quantities hfi and ha are the overall heat exchange coefficients on the fire and air side, respectively. Depending on the relationship between the fire/air and the structural surface, the heat exchange coefficients (hfi and ha) may only contain the convective component (hc), the radiant component (hr), or both.

The convective heat transfer coefficient depends on many factors. However, for structural fire applications, the structural temperatures are relatively insensitive to its exact values. Eurocode 1 Part 1.2 (CEN 2000a) recommends constant convective heat transfer coefficients as follows: on the fire side, hc = 25 W/m2 and on the air side, hc = 10 W/m2.

### 37.3.3 Radiation

If a structural surface is in direct contact with fire/air, radiant heat transfer between the structural surface and the fire/air may be assumed to occur between two very large parallel plates of area A, whose distance apart is small compared with the size of the plates so that radiation at their edges is negligible. Under this circumstance and assuming graybody radiant heat transfer, the radiant heat exchange coefficient is hr = £rff(T22 + T2)(T2 + T1) (37.14)

where s is the Stefan-Boltzmann coefficient (= 5.876 x 10-8 W/(m K4)) and er is often referred to as the resultant emissivity given by

1 e1e2

in which e1 and e2 are the graybody emissivities of the two surfaces, that is, that of the structural surface and fire/air, respectively.

### 37.3.4 Some Simplified Solutions of Heat Transfer

General heat transfer problems are difficult to solve and will usually require the use of numerical heat transfer procedures. However, for two common cases of unprotected and protected steelworks exposed to fire attack, simple analytical solutions have been derived to enable their temperatures to be calculated quickly. These simple analytical solutions have been derived by using the ''lumped mass method,'' that is, the entire steel mass is given the same temperature. For unprotected steelwork hA

where V and As are the volume and exposed surface area of the steel element, respectively, ps is the density of steel, and Cs is the specific heat of steel. The ratio As/V in Equation 37.16 is often referred to as the section factor of the steel element. Tfi and Ts are the fire and steel temperatures, respectively. h is the total heat transfer coefficient between the fire and the steel surface, including both the convective and radiant components. When using Equation 37.16, a step-by-step approach is necessary and the time increment should be small (At< 5 s).

For protected steelwork

DTs = , n(Tfi ~ Ts)As/\±, At — (ef/10 - 1)DTfi, where f = ^ tp A (37.17)

Additional symbols in Equation 37.17 include tp, the fire protection thickness; kp, thermal conductivity of the fire protection; Cp, specific heat of the fire protection; pp, density of the fire protection; and ATf, the increment in fire temperature during the time interval At. The time increment should not be too large. When using Equation 37.17, the time increment (At) should not exceed 30 s.

Because of the second term in Equation 37.17, it is possible that at the early stage of increasing fire temperature, the increase in steel temperature (ATs) maybe negative. In this case, the steel temperature increase should be taken as zero.

### 37.3.5 Section Factors

Equations 37.16 and 37.17 clearly indicate that the temperature rise in a steel element is directly related to the section factor As/ V, that is, the ratio of the heated surface area to the volume of the steel element. Consider a unit length of a steel element where the end effects are ignored, the section factor may alternatively be expressed as Hp/A, where Hp is the fire exposed perimeter length of the steel cross-section and A is the cross-sectional area of the steel element. Section factors for a few common types of steel sections exposed to fire are given in Table 37.3.

EXAMPLE 37.2 Section factor

Calculate the section factor (Hp/A) for the two cases shown in Figure 37.8.

Calculation results

Case 1, Figure 37.8a

Hp = 2 x 400 + 150 x 3 — 2 x 10 = 1230 mm, A = 2 x 15 x 150 +(400 — 15 x 2)x 10 = 8200 mm2 Hp/A = 0.15 mm-1 = 150 m—1

Case 2, Figure 37.8b

Hp = 2pRo = 300p, A = p(Ro2 — Ri2) = 2900p Hp/A = 0.1034mm—1 = 103.4m—1

### 37.3.6 Thermal Properties of Materials

In order to use Equations 37.16 and 37.17, it is necessary to have available information on the thermal properties (thermal conductivity k, density p, and specific heat C) of steel and insulation materials.

37.3.6.1 Steel

The thermal properties of steel are known with reasonable accuracy and the following values are given in Eurocode 3 Part 1.2 (CEN 2000b):

Density ps = 7850 kg/m3

Thermal conductivity [W/(mK)]

TABLE 37.3 Section Factors of a Steel Element

Fire exposure situation

Unprotected steel section exposed to fire exposure around all sides sè

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