# Thermal Analysis

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### Thermal Analysis

Figure 1 shows the schematic diagram for the thermal analysis within structural fire engineering design.

Figure 1 Schematic diagram for heat transfer

 Model Design charts / Test data Simple formulae Advanced models Complexity Simple Intermediate Advanced Heat Transfer Modes Conduction Convection Radiation Conduction Analysis ability -Test results -Standard fire conditions -Empirical solutions -Standard fire conditions -Accurate solutions -Any fire conditions Member types Dependent on available test data Mainly steel members Any material & construction methods Input parameters -Construction type -Member geometry -Heat flux or fire curves -Boundary conditions -Member geometry -Material thermal properties Solutions -Cross-sectional temperature charts -Tabulated thermal data Simple cross-sectional temperature profile One to three-dimensional time & space dependent temperature profile Design tools -Fire part of Eurocodes -Test/Research reports -Fire part of Eurocodes -Design guides Finite element package Design charts/tables Spreadsheet Computer models

Heat Transfer

Heat transfer is the science to evaluate the energy transfer that takes place between material bodies as a result of temperature difference. The three modes of heat transfer are conduction, convection and radiation. The thermal analysis on structural fire problems can be divided into two parts:

• The heat transfer by convection and radiation across the boundary from the fire into structural members;

• The heat transfer by conduction within structural members.

Test Data

The thermal analysis in structural members can be extremely complex, especially for materials that retain moisture and have a low thermal conductivity. The simplest method of defining the temperature profile through the cross-section is to use test data presented in tables or charts which are published in codes or design guides. These test data are generally based on standard fire conditions.

Design Formulae

Simple design equations are presented in codes and design guides to predict the temperature development of bare steel. The approach considers both radiative and convective heat transfer and, although a spreadsheet is required to solve the equation over the fire duration, it is simple to use. Similar equations exist for protected steel sections, however the thermal properties of the proposed protection material are needed, which can be difficult to obtain.

Computer Models

It is possible to use simple heat transfer models based on one-dimensional heat flow. However, simple computer programs are needed to solve the heat transfer equations. Alternatively, advance finite-element heat transfer models can be used, but this requires the relevant expertise to ensure the models are applied correctly and used within their limitations.

### Heat Transfer

The thermal analysis can be divided into two parts:

• The heat transfer by convection and radiation across the boundary from the fire into the structural members;

• The heat transfer by conduction within the structural members.

In the following, some guidelines on heat transfer given in codes and design guides will be briefly discussed.

### Heat Transfer across Boundary

The surface of a structural member exposed to a fire is subject to heat transfer by convection and radiation. Generally, the radiation is more dominant than the convection after the very early stages of the fire. The thermal actions can be represented by the net heat flux to the surface of the member.

On the fire exposed surfaces, the net heat flux [W/m2] considering heat transfer by convection and radiation can be determined by:

(1)

where

is the net convective heat flux component as given in Eq.(2) ;

is the net radiative heat flux component as given in Eq.(3) .

For non-fire exposed (unexposed) surfaces of members subjected to partial heating conditions, such as the unexposed side of walls and slabs, heat will transfer from the member to the boundary as the member will have a higher temperature. The heat transfer analysis according to Eq.(1) can be applied, considering different boundary conditions. The following modifications in the heat transfer can be considered (Purkiss 1996):

• For a surface exposed to ambient conditions, the gas temperature is taken to be equal to ambient temperature with both radiation and convection heat transfer considered.

• Alternatively, for a surface exposed to ambient conditions, a fixed temperature equal to ambient temperature can be imposed on the boundary nodes of the surface.

• For a surface with insulation, the boundary can be treated as a no heat-flow condition.

It is worth noting that the heat transfer to the boundary has an important effect on the thermal response of the region within the members close to the unexposed surface. The effect is smaller in the region close to the fire exposed surface.

Convection

The net heat flux [W/m2] due to convection is given by:

(2)

where

is the coefficient of heat transfer by convection as given in Table 1 [W/m2K];

is the gas temperature in the vicinity of the fire exposed member [°C];

is the surface temperature of the member [°C].

 Table 1 Convection coefficients αc according to EN1991-1-2 (2002) Fire model or exposed condition αc[W/m2K] Standard fires 25 External fires 25 Hydrocarbon fires 50 Parametric fires 35 Unexposed side of separating members -withough radiation 4 -with radiation 9

The exact formula for the heat flux due to radiation is complicated, in which the parameters involved depend on the type of surface, the type of flame and the temperature. For simplicity, EN1991-1-2 (2002) provides an approximation of the net heat flux [W/m2] due to radiation as follows:

(3)

where

is the emissivity of the fire (=1.0);

is the surface emissivity of the member (see Table 2);

Φ is the configuration factor (≤ 1.0);

is the effective radiation temperature of the fire environment [°C];

σ is the Stephan Boltzmann constant (=5.67 × 10-8 W/m2K4).

 Table 2 Emissivity of materials according to Eurocodes Material Emissivity εm Reference Carbon steel 0.7 EN1993-1-2 Stainless steel 0.4 EN1993-1-2 Concrete 0.7 EN1992-1-2 Others 0.8 EN1991-1-2

The configuration factor Φ takes into account of varying radiative heat flux levels on the fire exposed surface of the members depending on the position and shallow effects. Annex G (informative) of EN1991-1-2 gives the method for calculating the value of Φ, which will be discussed in the next section. Conservatively, Φ can be taken as 1.0.

Configuration Factor

Annex G (informative) of EN1991-1-2 (2002) provides the simple method for calculating the configuration factor Φ for the determination of the thermal actions for external members.

The configuration factor Φ measures the fraction of the total radiative heat leaving a radiating surface that arrives at a receiving surface, depending on the size of the radiating surface and the distance and the orientation between the two surfaces (see Figure 2).

Figure 2 Radiative heat transfer between two infinitesimal surfaces (EN1991-1-2)

Basically, the value Φ of for a member surface exposed to a fire depends on two effects:

The radiative heat transfer to a convex member surface is determined by the position effect, whereas the transfer to a concave surface is determined by both the position and shadow effects.

The assumptions made in the calculation of Φ for an external member include:

• All radiating surfaces are taken as rectangles in shape, including windows and other openings in fire compartment walls and the equivalent rectangular surfaces of flames.

• The value of Φ is determined for the mid-point P of each face of a rectangular envelope drawn around the cross-section of the member receiving the radiative heat transfer, as shown in Figure 3. This accounts for the shadow effect in an approximate way.

Figure 3 Envelope according to EN1991-1-2

The calculation procedure Φ of is explained in the following steps:

1. Choose a point P on the member.

2. Locate the point X on the radiating surface by drawing a horizontal line perpendicular to the receiving surface from P to the plane containing the radiating surface. The distance s from P to X is the shortest distance from P to the receiving surface.

3. Divide the receiving surface into different zones by drawing a horizontal and a vertical line through X. Normally, it contains four zones if the whole radiating surface is visible from P as shown in Figure 4(a). It will only contain two zones if the radiating surface is only partially visible as shown in Figure 4(b).

4. The Φ of the receiving surface is the sum of the contributions from each visible zone on the radiating surface.

5. If X lies outside the radiating surface, the effective Φ is determined by adding the contributions of the two extending rectangular zones from X to the farther side of the radiating surface, then subtracting the contributions of the two rectangular zones extending from X to the nearer side of the radiating surface.

Figure 4 Calculation of the configuration factor Φ according to EN1991-1-2
Heat Transfer within Members

In the modern physics, the heat transfer by conduction in solids is governed by the Fourier’s equation of heat transfer. Fourier’s equation states that the quantity of heat transferred per unit time across an area A is proportional to the temperature gradient ∂T/∂x as follows:

(4)

where

A is the area across which heat is transferred [m2];

K is the thermal conductivity of the material [W/m K];

Q is the heat transfer rate across the area A [W];

T is the temperature [K];

x is the distance normal to the area A [m].

In the conduction heat transfer of solids made of construction materials, such as steel, concrete and masonry, the solids are generally assumed to be isotropic. Consequently, for a small rectangular block in a Cartesian coordinate system (x, y and z), the three-dimensional heat conduction equation is given by:

(5)

where

ρ is the density of the material [kg/m3];

c is the specific heat of the material [J/kg K];

k is the thermal conductivity of the material [W/m K];

Q is the internal energy generated within the element [W/m3];

T is the temperature [K];

t is the time [sec].

Eq.(2) can be rewritten in the following form:

(6)

where is called the thermal diffusivity of the material [m2/sec]. The larger the value of , the faster heat will diffuse through the material.

To solve the heat conduction equation of a structural member, initial and boundary conditions must be provided, including:

• The initial temperature of the member.

• The temperature or heat flux at the surface of the member for all times.

In steady-state condition where the thermal environment of the member is constant, it is possible to establish simple analytical solutions by using finite difference technique for one and two-dimensional heat transfer of conduction, if the material properties are assumed to be independent of temperature. However, in transient-state condition where the thermal environment is not constant, the solutions can only be carried out by using either finite element or finite difference technique

Heat Transfer - Advanced Models: Computer Packages

The advanced models for heat transfer problems are generally referring to computer models. In structural fire engineering problems, most of the time, the heat transfer is in transient-state condition, coupled with time-dependent boundary conditions and temperature-dependent material properties. Consequently, most advanced models can only be developed based on finite element techniques. The heat transfer analysis can be performed by a two-dimensional (2D) or three-dimensional (3D) models.

The general aspects for the modeling of heat transfer analysis include:
 Meshing The shape and dimensions of the structural model are modeled by a finite element mesh of general flow continuum elements, in form of triangles, quadrilaterals, wedges, or bricks. The boundary elements or interface elements can be line shaped elements for a 2D model, and triangular or quadrilateral elements for a 3D model. Boundary Conditions Heat sources can be represented by either temperature-time functions or heat flux in boundary elements. Convection and/or radiation at boundaries of the structural model can be modeled by the heat transfer coefficient of boundary elements. Material Properties The material can be isotropic, orthotropic or anisotropic. The material thermal properties of conductivity, specific heat and emissivity can be temperature-dependent. Special Features Hydration heat, moisture evaporation and movement, change in contact conditions may be modeled.

The heat transfer analysis of structural fire engineering problems can normally be performed by the commercial computer packages for general finite element modeling, including:

• ABAQUS®

• ANSYS®

• MSC.Marc®

• TNO DIANA®

The special purpose computer programs for structural fire engineering analyses include:

• FIRES-T3 from National Institute of Standards and Technology, USA (Iding, Bresler & Nizamuddin 1996)

• SAFIR from the University of Liège, Belgium (Franssen, Kodur & Mason 2000)

• TASEF-2 from Lund Institute of Technology, Sweden (Wickström 1979)