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Journal of Constructional Steel Research 61 (2005) 825833

www.elsevier.com/locate/jcsr

Heat transfer model for unprotected steel membersin a standard compartment fire with

participating medium

J.I. Ghojela,, M.B. Wongb

aDepartment of Mechanical Engineering, Monash University, 900 Dandenong Road, Caulfield East,

Victoria 3145, AustraliabDepartment of Civil Engineering, Monash University, Building 60, Clayton, Victoria 3800, Australia

Received 12 March 2004; accepted 19 November 2004

Abstract

A heat transfer model accounting for the radiative properties of combustion products in a

compartment standard fire is presented. The model used is based on a conceptual scheme of a grey gas

mixture exchanging radiative energy with a black enclosure. The proposed model, with a radiation

heat transfer that accounts for the effect of combustion products on the rate of heat transfer from the

fire to the structural elements, is simple to use and the predictions of the temperature response of

unprotected I-columns heated from all sides are superior to those predicted by the classical method

using the StefanBoltzmann radiation equation with constant emissivity.

2005 Elsevier Ltd. All rights reserved.

Keywords:Heat transfer; Model; Steel structures; Gas radiation

1. Introduction

Mathematical modelling of heat transfer in structural members under fire conditions is

an essential part of a standard procedure in fire resistance assessment of these members.

Corresponding author. Tel.: +61 3 9903 2490; fax: +61 3 9903 1084.E-mail address:[email protected] (J.I. Ghojel).

0143-974X/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2004.11.003
http://www.elsevier.com/locate/jcsrhttp://www.elsevier.com/locate/jcsr


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826 J.I. Ghojel, M.B. Wong / Journal of Constructional Steel Research 61 (2005) 825833

The main advantage of using mathematical models is that they are simple to use and can be

easily programmed into spreadsheets. According to Eurocode 3 [1], the steel temperature

increase,Ts during time incrementtin the case of bare unprotected steel elements in a

fire through convection and radiation, is given by

Ts =1

cpss

P

As

(qc + qr)t (1)

where cps specific heat of steel in J/kg K, s density of steel in kg/m3, P/As

massivity or section factor (perimeter/cross-sectional area). The two heat flux components:

convection,qc, and radiation,qr, both in W/m2, are represented as follows:

qc =h c(Tg Ts) (2)

qr =res[(Tg +273)4

(Ts +273)4

] (3)where hc is the heat transfer coefficient (assumed 25 W/m

2 K [24]), Tg and Ts are

the temperatures of the fire and steel, respectively (C), StefanBoltzmann constant

(5.67108 W/m2 K4). The resultant emissivityres is given by

res =1

(1/s)+(1/f)1 (4)

where f ands are the emissivities of the fire and steel member, respectively. A value of

0.85 for f is recommended by SBI [5] for most fire situations. Eq.(4)can be simplified

to [6]res sf (5)

and a value of 0.80 for the emissivity of the fire f is recommended.

Eq.(3)does not include the shape (view) factor because it is based on the assumption

that the fire and the steel member are represented by two infinitely parallel grey planes,

or by a grey body surrounded by a grey enclosure at temperaturesTg andTs , respectively.

The main drawback of this equation is that the fire, which is assumed to be a hot grey

object with constant emissivity res, radiates energy to a structural element through a

perfectly transparent medium with the combustion products playing no role whatsoever

in the radiation heat exchange. Also, it is unlikely that the resultant emissivity will notchange with temperature, and therefore, it should be temperature dependent [7]. For the

current model, it is proposed to use an equivalent temperature dependent emissivity that

is derived from consideration of the effect of the gaseous composition of the combustion

products on the heat transfer by radiation.

2. Models with participating medium

If a grey (partially absorbing and partially emitting) enclosure filled with isothermal

non-grey gas is assumed, the following equation proposed by Edwards and Matavosian [8]

can be used to describe the radiative heat exchange between the gas and a steel member

q = Fg(Tg +273)4 Fs(Ts +273)

4. (6)


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J.I. Ghojel, M.B. Wong / Journal of Constructional Steel Research 61 (2005) 825833 827

Fig. 1. Total emissivity of a mixture of 10% CO 2, 10 H2O and 80% N2.

An exact solution for this model accounting for an infinite number of reflections yields the

following equations for the transfer factors Fg and Fs

Fg =s [g1+ s(g2 g1)+2s(g3 g2)+ ]

Fs =s [g1 + s(g2 g1)+2s(g3 g2)+ ]. (7)

The error in assuming one reflection is only about 4% [8]; therefore the equations in(7)

are reduced, after replacing the reflectivitys of the steel member by (1s), to

Fg =sg1

1(1s)[(g2 g1)/g1]

Fs =

sg1

1(1s)[(g2 g1)/g1] .

(8)

In these equations,s is the emissivity of the inner surface of the grey enclosure, g is the

total emissivity of the gaseous mixture at temperature Tg over the mean beam length of

the enclosure, and g is the total absorptivity of the gaseous mixture for radiation from

a black surface at temperature Ts absorbed over the mean beam length by the gaseous

mixture at temperature Tg. The subscript 1 denotes properties for the mean beam length

of the enclosure and subscript 2 for two mean beam lengths including the effect of one

reflection. The mean beam length is a characteristic dimension of the thickness of the gas

layer transmitting radiative energy and is taken equal to Le = 3.6Vc/Ac. 8V

cand Ac are

the volume and surface area of the compartment, respectively. Thin gas layers transmitmore radiation than thick layers. Figs. 13 show values for g1,g2, g1 and g2 for a

gas mixture comprising 10% CO2, 10% H2O and 80% N2 at a pressure of 1 atm in a


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Fig. 2. Total absorptivity for mean beam length of the enclosure containing a mixture of 10% CO2, 10 H2O and80% N2.

6 m 6 m 6 m enclosure. These properties, total emissivity and total absorptivity, are

determined from a methodology developed in Ref. [9] that is based on the generalised

scaling rules for estimating total properties for homogeneous gases proposed by Edwards

and Matavosian [8]. The total emissivity is first determined from the radiation property

chart at the atmospheric pressure as a function of the average gas temperature, partial

pressure of the water vapour content and the thickness of gas layer, then scaled to any

pressure up to 10 atm using a scaling component chart. The total absorptivity is determinedin two steps by first estimating the total emissivitygs at the enclosure temperatureTs then

calculating the absorptivity from

g =(Tg/Ts)1/2gs .

The special case of a grey gas radiating to a black enclosure (no reflection) can be obtained

by puttings =1.0 in Eqs.(6) and(8)

qr =g(Tg +273)4

g(Ts +273)4. (9)

Ghojel [10], Wong and Ghojel [11] found that using this equation in the radiation

component of heat transfer models resulted in improved correlation between measuredand predicted temperature response of structural members under standard and wood fire

conditions as compared with the Eq.(3).Fig. 4shows a case study taken from [3] for an


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Fig. 3. Total absorptivity for two mean beam lengths including the effect of one reflection of an enclosurecontaining a mixture of 10% CO2, 10 H2O and 80% N2.

I-beam (P/As = 74). The analytical models examined included the classical model (Eq.

(3)) with res =0.6, grey gas surrounded by a grey enclosure (Eqs. (6) a n d (8)) and grey gas

surrounded by a black enclosure (Eq.(9)). It appears that Eq.(9) yields better correlation

with the observed results than the other remaining two in the case of unprotected steel

members heated from all sides.

3. The proposed model

Fig. 4shows that the model of a participating medium with black enclosure (Eq. (9))

gives better correlation with experimental results relative to the other two models. This was

confirmed by modelling similar cases from the UK standard fire tests [12]. Based on these

results a relationship for a variable equivalent emissivity (temperature dependent) that can

be used in the standard StefanBoltzmann equation is deduced by equating Eqs. (3) and

(9), after replacing the resultant emissivity res in Eq.(3) by the equivalent emissivity eq

qr =gT4

g gT4

s = eq[(Tg +273)4 (Ts +273)

4]

from which

eq =g T

4g g T

4s

[(Tg +273)4 (Ts +273)4]. (10)


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Fig. 4. Prediction of temperature response of unprotected steel member by different models. Experimental data

from[3].

Fig. 5. Equivalent emissivity as a function of steel temperature and massivity.

Fig. 5shows the values ofeq calculated from Eq.(10) as a function of steel temperaturefor three I-beams with massivities 50, 200 and 500. When the massivity changes from 50

to 500, the values ofeq changes by eq = 0.54%. Therefore, to simplify the model the


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Fig. 6. Predicted (solid lines) and measured (markers) temperatures, P/As =75.80. Experimental data from [13].

values ofeqfor P/As =200 are taken as the average for all beams within the investigated

range.eq can now be represented by the following fourth degree polynomial fit:

eq =a + bTs +cT2

s +d T3

s +eT4

s (11)

where

a =0.4050437, b = 0.00039097791, c= 1.2346388e06,

d = 2.4208724e09, e =1.3968447e12 and Ts is in C.

Figs. 68show results of calculations using the current model and the standard model

for three cases taken from the UK standard fire tests [12]. The upper range of measured

temperatures represent the mean temperature of the exposed web and the lower range theexposed flanges. These temperature ranges are close during the early stages of the heating

process gradually diverging with time. Generally, the predicted temperature profile is closer

to the mean temperature of the flanges, particularly at the high-temperature end of the

heating process.

The closeness of the predicted temperatures by the standard model depends on the

assumed value of the resultant emissivity res. Wong and Ghojel [13] refer to some of

the recommended values: 0.5 (EC3), 0.64 (UK National Application Document preceding

EC3) and 0.7 (full EN version of EC3). The use of these values in Eq. (3) causes an

overestimation of the temperature profile of structural members, with the results improving

somewhat when res = 0.5. The proposed range of eq = f(T), which is well below0.5, seems to give better overall prediction of the temperature profile of the tested

members.


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Fig. 7. Predicted (solid lines) and measured (markers) temperatures, P/As = 184.97. Experimental data

from[13].

Fig. 8. Predicted (solid lines) and measured (markers) temperatures, P/As = 27.79. Experimental datafrom[13].


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4. Conclusions

The proposed heat transfer model can be used to predict the temperature response of

unprotected steel structural members under standard fire conditions in enclosures withknown temperaturetime histories.

The model yields results that correlate well with observed data for the examples

considered.

The model retains the simplicity of the StefanBoltzmann equation of thermal radiation

while at the same time accounting for real gas behaviour.

Although no attempt has been made here to investigate the effect of the convection

heat transfer component, it is conceivable that better results can be obtained with more

realistic temperature-dependent heat transfer coefficients. It is planned to investigate

this in a future work.

References

[1] EUROCODE 3. ENV 1993-1-2, Design of steel structuresPart 12: General rulesStructural fire design,

CEN; 1995.

[2] Barthelemy B. Heating calculations of structural steel members. Journal of Structural Division, ASCE 1976;

102(8):154958.

[3] Smith CI, Stirland C. Analytical methods and the design of steel framed buildings. In: International seminar

on three decades of structural fire safety. Herts (UK): Fire Research Station; 1983. p. 155200.

[4] Fire Engineering Guidelines. Australian Building Codes Board; 2001.

[5] SBI. Fire engineering design of steel structures, Swedish Institute of Steel Construction, Stockholm; 1976.[6] Purkiss JA. Fire safety engineering design for structures. Butterworth Heinemann; 1996.

[7] Mooney J. Surface radiant-energy balance for structural thermal analysis. Fire and Materials 1992;16:616.

[8] Edwards DK, Matavosian R. Scaling rules for total absorptivity and emissivity of gases. Transactions of

ASME, Journal of Heat Transfer 1984;106:6849.

[9] Mills AF. Heat transfer. Irwin; 1992.

[10] Ghojel JI. A new approach to modelling heat transfer in compartment fires. Fire Safety Journal 1998;31:

22737.

[11] Wong MB, Ghojel JI. Spreadsheet method for temperature calculation of unprotected steelwork subject to

fire. The Structural Design of Tall and Special Buildings 2003;12(2):8392.

[12] Wainman DE, Kirby BR. Compendium of UK standard fire test data unprotected structural steel1, British

Steel Corporation; 1988.

[13] Wong MB, Ghojel JI. Sensitivity analysis of heat transfer formulations for insulated structural steelcomponents. Fire Safety Journal 2003;38:187201.

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