SFPE Fire Exposures...The SFPE Task Group on Fire Exposures to Structural Elements began its work in March 1998. The purpose of this guide is to provide the infor-mation and methodology

The SFPE Task Group on Fire Exposures to Structural Elements

Chairman

James G. Quintiere, Ph.D., FSFPEUniversity of Maryland

Members

Farid Alfawakhiri, Ph.D.American Institute of Steel Construction

Andrew Buchanan, Ph.D.University of Canterbury

Vytenis Babrauskas, Ph.D.Fire Science & Technology Inc.

Jonathan Barnett, Ph.D.,FSFPEWorcester Polytechnic Institute

Thomas Izbicki, P.E.Dallas Fire Department

Stephen Hill, P.E.ATF Fire Research Laboratory

Barbara Lane, Ph.D.ARUP Fire

Sean Hunt, P.E.Hughes Associates, Inc.

Brian Lattimer, Ph.D.Hughes Associates, Inc.

Rodney McPheeCanadian Wood Council

Harold Nelson, P.E., FSFPE

James Mehaffey, Ph.D.Forintek Canada Corp.

Amal Tamim

James Milke, P.E., Ph.D.,FSFPEUniversity of Maryland

Ian Thomas, Ph.D.Victoria University

Christopher Wieczorek, Ph.D.FM Global

Staff

Morgan J. Hurley, P.E.Society of Fire Protection Engineers

Printed in the U.S.A. Copyright ©2004 Society of Fire Protection Engineers. All rights reserved.

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Foreword

The SFPE Task Group on Fire Exposures toStructural Elements began its work in March 1998.The purpose of this guide is to provide the infor-mation and methodology needed to predict thethermal boundary condition for a fire over time. The methods contained herein are based on experi-mental measurements and correlations, and mostlygive global rather than local results. Eventually,“CFD” methods for fire must be subjected to someof the same tests used here and judged accordinglyfor accuracy and application.

On September 11, 2001, the world changed, andthis task took on a new life and significance. Issuesidentified during examination of the collapse of theWorld Trade Center buildings raised questionsregarding the design of fire protection of structures.Indeed, the role of the fire protection engineer(FPE) in structural fire-resistance design maychange and embrace more of these calculations.Presently, the architect is generally responsible forthe fire protection of the structure. An engineereddesign method would involve:

1. A prediction of the fire over time2. Heat transfer analysis of the structural member3. Response of the structural system

Such full calculations will have to be dealt withby the fire protection engineer in conjunction withthe structural engineer. Items 1 and 2 are more inthe domain of the FPE. Note, however, that item 2is not addressed here.

This guide was originally divided into threeareas. The first included fully developed fires incompartments. Since it was an “old” area of studywith many contributors, care was required to sortout the key pieces. The second area was fire plumes,or the exposure of discrete fires to elements. Sinceit was more recent in exposition, this work could beevaluated more easily. A third area intended for thisguide included the effect of window flames on thefaçade and external structural elements. While

this information was not included in this guide, thework of Margaret Law, “Design Guide for FireSafety of Bare Exterior Structural Steel,” TechnicalReports and Designer’s Manual (London, Ove Arup & Partners, 1977), is recommended for suchfire scenarios.

The work in completing this guide was mostlydone voluntarily. All contributions, no matter howsmall, are appreciated and enabled this guide tocome to closure.

This guide is written for those with an under-standing of fire and heat transfer, but should be edu-cational and informative to a structural engineer. Itincludes some theoretical background for orientation,and examples to appreciate the process of calcula-tion. It is the sixth engineering practice guide pub-lished by the Society of Fire Protection Engineers.

I take responsibility for the “theory” on compart-ment fires, and for the general approach of theguide. But the guide could not have been completedwithout the dedicated contribution of MorganHurley, Technical Director of SFPE. He performedthe role of technical editor and personally per-formed the analyses and evaluations of the variousmethods for predicting the temperature–time curvesfor fully developed fires. That comparison hadnever been done before, and it was imperative toconduct in order to make judgment on the methods.In making those comparisons, we decided to use theCIB and Carrington data sets to serve as a bench-mark. While the CIB data are of scales no more that1.5 m in height, the Carrington tests are much morerealistic in scale. However, the theory section shouldoffset any issues of the relevance of small scale.

The section on fire plumes was developed byBrian Lattimer with the assistance of Sean Hunt.That was a significant contribution and had neverbeen assembled before. Christopher Wieczorekorganized the material describing the variousapproaches. Barbara Lane presented a thoroughreview of the time-equivalent method and draftedmaterial on parametric equations for estimating

compartment fire temperatures and durations. Thetime-equivalent method is limited but well known.We included this material to explicitly explain itsbasis and limitations.

Others made noteworthy contributions. JonathanBarnett and his students got us started on theliterature of fully developed fire, and Stephen Hillbrought this to the production point in a presenta-tion for SFPE. James Mehaffey, Ian Thomas, andHarold “Bud” Nelson were early contributors.Others, including Farid Alfawakhiri, Andrew

Buchanan, Thomas Izbicki, Rodney McPhee, AmalTamim, and James Milke, were critical readers, andVytenis “Vyto” Babrauskas continually provideduseful comments and critiques. Readers outside theCommittee included Ulf Wickstrom, TakeyoshiTanaka, Tibor Harmathy, and T.T. Lie, and for thiswe are greatly appreciative.

James G. QuintiereNovember 10, 2003

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The Society of Fire Protection Engineers wishes to acknowledge and thank the American Institute of SteelConstruction, the National Fire Protection Association, the American Forest and Paper Association, and theCanadian Wood Council for their generous support of this project.

Contents

Foreword ...........................................................................................................................................................ii

Executive Summary .......................................................................................................................................xii

Introduction ......................................................................................................................................................1Model Inputs ..................................................................................................................................................1Basis of Fire Resistance.................................................................................................................................2Accounting for Suppression...........................................................................................................................2Heat Transfer Boundary Conditions ..............................................................................................................3Computer Modeling .......................................................................................................................................3

Fully Developed Enclosure Fires ....................................................................................................................4Theory ............................................................................................................................................................5

Theoretical Development ..........................................................................................................................5Wall Heat Transfer.....................................................................................................................................7General Form of Correlations..................................................................................................................12

Methods for Predicting Fire Exposures .......................................................................................................16Eurocode Parametric Fire Exposure Method ..........................................................................................16Lie’s Parametric Method .........................................................................................................................19Tanaka......................................................................................................................................................21Magnusson and Thelandersson Parametric Curves.................................................................................22Harmathy .................................................................................................................................................24Babrauskas...............................................................................................................................................26Ma and Mäkeläinen .................................................................................................................................29CIB...........................................................................................................................................................31Law ..........................................................................................................................................................33Simple Decay Rates.................................................................................................................................34

Recommendations ........................................................................................................................................34

Fire Exposures from Plumes .........................................................................................................................40Axisymmetric Fire Plumes ..........................................................................................................................41Heat Flux Boundary Condition....................................................................................................................44Bounding Heat Flux: Objects Immersed in Flames ....................................................................................45Heat Fluxes for Specific Geometries...........................................................................................................48

Flat Vertical Walls....................................................................................................................................48Fires in a Corner ......................................................................................................................................52Fires Impinging on Unbounded Ceilings ................................................................................................58Fire Impinging on a Horizontal I-Beam Mounted Below a Ceiling .......................................................63

Summary and Recommendations ................................................................................................................68

v

Appendix A – Theoretical Examination of Methods...................................................................................69Results by Harmathy for Wood Cribs..........................................................................................................69Results by Bullen and Thomas for Pool Fires .............................................................................................70CIB Data ......................................................................................................................................................71Eurocode ......................................................................................................................................................71Lie ................................................................................................................................................................71Magnusson, Thelandersson, and Petersson..................................................................................................71Babrauskas ...................................................................................................................................................71Law...............................................................................................................................................................72Ma and Mäkeläinen .....................................................................................................................................72

Appendix B – Comparisons of Enclosure Fire Predictions with Data......................................................73CIB Data ......................................................................................................................................................74Cardington Data ...........................................................................................................................................74Eurocode ......................................................................................................................................................76Lie ................................................................................................................................................................83Tanaka ..........................................................................................................................................................89Magnusson and Thelandersson ....................................................................................................................95Harmathy....................................................................................................................................................101Babrauskas .................................................................................................................................................106Ma and Mäkeläinen....................................................................................................................................113CIB .............................................................................................................................................................118Law.............................................................................................................................................................122

Appendix C – Time-Equivalent Methods ..................................................................................................129Real Structural Response ...........................................................................................................................129Discussion of Methods...............................................................................................................................130

Fire Load Concept .................................................................................................................................130Kawagoe and Sekine .............................................................................................................................131Law ........................................................................................................................................................131Pettersson...............................................................................................................................................132Normalized Heat Load Concept ............................................................................................................133Eurocode Time-Equivalent Method ......................................................................................................133New Zealand Code ................................................................................................................................136

Comparisons...............................................................................................................................................136Limitations and Assumptions.....................................................................................................................137

Appendix D – Examples...............................................................................................................................139

GlossaryNomenclature Used in the Enclosure Fires Section ..................................................................................143Nomenclature Used in the Plumes Section................................................................................................145

References .....................................................................................................................................................147

vi

Illustrations

FIGURE

1 Phases of Fire Development....................................................................................................................42 Model for the Fully Developed Fire .......................................................................................................63 Wall Heat Transfer...................................................................................................................................74 MQH Correlation for Fuel-Controlled Fires .........................................................................................115 Approximate Theoretical Behavior for Fuel Burning Rate ..................................................................156 Approximate Theoretical Behavior of Compartment Temperature ......................................................157 Schematic Illustration of the Heat Balance Equation Terms ................................................................238 Examples of Temperature–Time Curves ...............................................................................................239 Non-Dimensionalized Temperature–Time Curves Developed by Ma and Mäkeläinen .......................29

10 Average Temperature During Fully Developed Burning ......................................................................3111 Normalized Burning Rate During Fully Developed Burning ...............................................................3212 Comparison of CIB Temperature Data to Predictions Using Law’s Method .......................................3513 Comparison of Burning Rate Data to Predictions Using Law’s Method .............................................3514 Comparison of Predictions Using Law’s Modified Method for Cardington Test #1 ...........................3615 Comparison of Predictions Using Law’s Modified Method for Cardington Test #2 ...........................3616 Comparison of Predictions Using Law’s Modified Method for Cardington Test #8 ...........................3717 Comparison of Predictions Using Law’s Modified Method for Cardington Test #9 ...........................3718 Comparison of Predictions from Magnusson and Telandersson’s Method (Type C)

to Data for Cardington Test #3..............................................................................................................3819 Comparison of Predictions from Magnusson and Telandersson’s Method (Type C)

to Data for Cardington Test #4..............................................................................................................3920 Comparison of Predictions from Magnusson and Telandersson’s Method (Type C)

to Data for Cardington Test #5..............................................................................................................3921 Comparison of Predictions from Lie for Cardington Test #6 ...............................................................4022 Axisymmetric Fire Plume .....................................................................................................................4123 Maximum Turbulent Fire Plume Temperatures from Various Sources ................................................4224 Heat Balance at the Material Surface....................................................................................................4425 Magnitude of Surface Temperature Corrections on the Measured Total Heat Flux

Using a Cooled Gauge...........................................................................................................................4526 Averaged Peak Heat Flux as a Function of Angular Position...............................................................4627 Fire Against a Flat Vertical Wall ...........................................................................................................4828 Peak Heat Release Rates Measured in Square Propane Burner Fires Against a Flat Wall ..................4929 Vertical Heat Flux Distribution Along the Centerline of a Square Propane Burner Fire

Adjacent to a Flat Wall..........................................................................................................................5030 Horizontal Heat Flux Distribution (a) Below the Flame Height and

(b) Above the Flame Height with Distance from the Centerline of the Fire ........................................5031 Fire in a Corner Configuration..............................................................................................................5232 Corner with a Ceiling Configuration Showing the Three Regions Where Incident

Heat Flux Correlations Were Developed in the Study of Latimer et al................................................5333 Peak Heat Flux Along the Height of the Walls in the Corner...............................................................5334 Maximum Heat Fluxes to the Walls Near the Corner with Square Burner Sides of ●●-0.17 m,

▲▲-0.30 m, ▼▼-0.30 m (Elevated), and ■■-0.50 m and Fires Sizes from 50 to 300 kW.........................5435 Heat Flux Distribution Horizontally out from the Corner on the Lower Part of the Corner Walls .....55

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36 Maximum Heat Flux Along the Top of the Walls During Corner Fire Tests with Square Burner Sides of ●●-0.17 m, ▲▲-0.30 m, ▼▼-0.30 m (Elevated), and ■■-0.50 m and Fires Sizes from 50 to 300 kW .............................................................................................................56

37 Heat Flux Along the Ceiling Above a Fire in a Corner During Tests with Square Burner Sides of ●●-0.17 m, ▲▲-0.30 m, ▼▼-0.30 m (Elevated), and ■■-0.50 m and Fires Sizes from 50 to 300 kW.......57

38 Unbounded Ceiling Configuration ........................................................................................................5939 Stagnation Point Heat Fluxes on an Unbounded Ceiling with a Fire Impinging on It ........................6040 Heat Fluxes to a Ceiling Due to a Propane Fire Impinging on the Surface .........................................6141 Comparison of the Best Fit Curve Proposed by Wakamatsu and a Bounding Fit to the Data.............6242 I-Beam Mounted Below an Unbounded Ceiling...................................................................................6443 Heat Flux Measured onto the Surfaces of an I-Beam Mounted Below an Unbounded Ceiling

for Fires 95 to 900 kW ..........................................................................................................................6644 Heat Flux Measured on the ●●-Bottom Flange, ■■-Web, and ▲▲-Upper Flange of an I-Beam

Mounted Below and Unbounded Ceiling for Fires 565 to 3,870 kW ..................................................67

A.1 Comparison of Burning Rate Predictions .............................................................................................69A.2 Wood Crib and Liquid Pool Fires .........................................................................................................70

B.1 Histogram of Ratio of Fuel Surface Area to Enclosure Surface Area for the CIB Experiments .........74B.2 Comparison of CIB Temperature Data to Predictions Made Using Eurocode,

Buchanan, and Franssen Methods, qt,d = 100 MJ/m2...........................................................................77

B.3 Comparison of CIB Temperature Data to Predictions Made Using Eurocode, Buchanan, and Franssen Methods, qt,d = 50 MJ/m

2.............................................................................77B.4 Comparison of CIB Burning Rate Data to Predictions Made Using the Eurocode Method ................78B.5 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to

Data from Cardington Test #1...............................................................................................................79B.6 Comparison of Predictions Made Using Eurocode, Buchanan, and Franssen Methods to








Data from Cardington Test #9...............................................................................................................83B.14 Comparison of CIB Temperature Data to Predictions Made Using Lie’s Method...............................84B.15 Comparison of CIB Burning Rate Data to Predictions Made Using Lie’s Method .............................84B.16 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #1 ...................85B.17 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #2 ...................85B.18 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #3 ...................86B.19 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #4 ...................86B.20 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #5 ...................87

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B.21 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #6 ...................87B.22 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #7 ...................88B.23 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #8 ...................88B.24 Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #9 ...................89B.25 Comparison of CIB Temperature Data to Predictions Made Using Tanaka’s Methods .......................90B.26 Comparison of CIB Burning Rate Data to Predictions Made Using Tanaka’s Methods......................90B.27 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #1 ...........91B.28 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #2 ...........91B.29 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #3 ...........92B.30 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #4 ...........92B.31 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #5 ...........93B.32 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #6 ...........93B.33 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #7 ...........94B.34 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #8 ...........94B.35 Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #9 ...........95B.36 Comparison of CIB Temperature Data to Predictions Made Using Magnusson and

Thelandersson’s Method........................................................................................................................96B.37 Comparison of CIB Burning Rate Data to Predictions Made Using Magnusson and

Thelandersson’s Method........................................................................................................................96B.38 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C)

to Data from Cardington Test #1...........................................................................................................97B.39 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C)






to Data from Cardington Test #8.........................................................................................................100B.45 Comparison of Predictions Made Using Magnusson and Thelandersson’s Method (Type C)

to Data from Cardington Test #9.........................................................................................................100B.46 Comparison of CIB Burning Rate Data to Predictions Made Using Harmathy’s Method ................101B.47 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #1 ......102B.48 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #2 ......102B.49 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #3 ......103B.50 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #4 ......103B.51 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #5 ......104B.52 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #6 ......104B.53 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #7 ......105B.54 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #8 ......105B.55 Comparison of Predictions Made Using Harmathy’s Method to Data from Cardington Test #9 ......106B.56 Comparison of CIB Temperature Data to Predictions Made Using Babrauskas’ Method .................107B.57 Comparison of CIB Burning Rate Data to Predictions Made Using Babrauskas’ Method................108B.58 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #1 .....108B.59 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #2 .....109

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B.60 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #3 .....109B.61 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #4......110B.62 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #5......110B.63 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #6......111B.64 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #7......111B.65 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #8......112B.66 Comparison of Predictions Made Using Babrauskas’ Method to Data from Cardington Test #9......112B.67 Comparison of CIB Burning Rate Data to Predictions Made Using Ma and Mäkeläinen’s Method ....113B.68 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from

Cardington Test #1...............................................................................................................................114B.69 Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data from







Cardington Test #9...............................................................................................................................117B.76 Comparison of Cardington and CIB Temperature Data......................................................................118B.77 Comparison of Predictions Made Using the CIB Data to Cardington Test #1...................................119B.78 Comparison of Predictions Made Using the CIB Data to Cardington Test #2...................................119B.79 Comparison of Predictions Made Using the CIB Data to Cardington Test #3...................................120B.80 Comparison of Predictions Made Using the CIB Data to Cardington Test #4...................................120B.81 Comparison of Predictions Made Using the CIB Data to Cardington Test #7...................................121B.82 Comparison of Predictions Made Using the CIB Data to Cardington Test #8...................................121B.83 Comparison of Predictions Made Using the CIB Data to Cardington Test #9...................................122B.84 Comparison of CIB Temperature Data to Predictions Made Using Law’s Method ...........................122B.85 Comparison of CIB Burning Rate Data to Predictions Made Using Law’s Method .........................123B.86 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #1 ...............124B.87 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #2 ...............124B.88 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #3 ...............125B.89 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #4 ...............125B.90 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #5 ...............126B.91 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #6 ...............126B.92 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #7 ...............127B.93 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #8 ...............127B.94 Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #9 ...............128

C.1 Fire Severity Concept..........................................................................................................................130C.2 Law’s Correlation Between Fire Resistance Requirements (tf ) and L/(AwAt )

1/2 ................................137

D.1 Temperature–Time Curve for Burning Duration of 1.5 Hours and Opening Factor of 0.02 m1/2.......141

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TablesTABLE

1 Estimates of Conduction for Common Materials ...................................................................................82 Range of Values for Key Parameters from the 25 Data Sets Used to Develop the Shape Function....303 Rate of Decrease in Temperature ..........................................................................................................344 Selected Heat Fluxes to Objects Immersed in Large Pool Fires ..........................................................47

B.1 Compartment Dimensions of the Cardington Tests ..............................................................................75B.2 Opening Dimensions of the Cardington Tests ......................................................................................75B.3 Properties of Enclosure Materials .........................................................................................................75B.4 Fuel Loading for the Cardington Tests..................................................................................................75

C.1 Fuel Load Density Determined from a Fuel Load Classification of Occupancies.............................134C.2 Safety Factor Taking Account of the Risk of a Fire Starting Due to the Size of Compartment ........134C.3 Safety Factor Taking Account of the Risk of a Fire Starting Due to the Type of Occupancy ...........134C.4 A Factor Taking Account of the Different Active Fire-Fighting Measures ........................................135C.5 Relationship Between kb and the Thermal Property b ........................................................................135C.6 Values for kb Recommended by the New Zealand Fire Engineering Design Guide .........................136

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xii

Executive Summary

Designing fire resistance on a performance basisrequires three steps:

1. Estimating the fire boundary conditions2. Determining the thermal response of the structure3. Determining the structural response

This guide provides information relevant to esti-mating the fire boundary conditions resulting from afully developed fire. Methods are provided for fullydeveloped enclosure fires and for fire plumes. Fullydeveloped enclosure fires can be expected in com-partments with fuel uniformly distributed over theirinteriors. For situations where a fire would not beenclosed or for enclosures with sparse distributionsor concentrated fuel packets, the methods identifiedin the fire plumes section should be used.

Several methods are evaluated for fully developedenclosure fires. Law’s method is recommended forall roughly cubic compartments and in long, narrow

compartments where does not exceed

≈ 18 m–1/2. To ensure that predictions are sufficientlyconservative in design situations, the predictedburning rate should be reduced by a factor of 1.4and the temperature adjustment should not bereduced by Law’s Ψ factor.

Law’s method does not predict temperaturesduring the decay stage. For cases where a prediction

of temperatures during the decay stage is desired, adecay rate of 7ºC/min can be used for fires with apredicted duration of 60 minutes or more, and adecay rate of 10°C/min can be used for fires with apredicted duration of less than 60 minutes.

For long, narrow spaces in which is in

the range of 45 to 85 m–1/2, Magnusson andThelandersson provide reasonable predictions oftemperature and duration. For long, narrow spaces

in which is approximately 345 m–1/2, Lie’s

method is recommended.

For ranges of that fall outside the ranges

identified above, the calculations should be per-formed using the methods identified for the ranges

of that bound the situation of interest, and

the most conservative results should be used.For fire plumes, methods are presented for

conducting a bounding analysis and for specificgeometries. These geometries include flat verticalwalls, corners with a ceiling, unbounded flatceilings, and an I-beam mounted below a ceiling.Additionally, correlations are provided for axisym-metric plumes for those wishing to conduct a heattransfer analysis from first principles.

Fire Exposures to Structural Elements

1

EngineeringGuide

IntroductionAn engineering analysis to evaluate the response

of a structure during a fire must consider both theheat transfer from the fire to the structural membersand the structural response of these members underthe defined threat. The focus of this guide is to definethe heat flux boundary condition due to the fire usedin the heat transfer analysis portion of this problem.Guidance is provided for two potential fire threats:fully developed enclosure fires and local fire plumes.

In fully developed enclosure fires, the conditions(gas temperatures, velocities, and smoke levels) areassumed to be uniform throughout the entire enclo-sure, and all combustible contents are generallyconsidered to be contributing to the fire size andduration. Historically, conditions inside fully devel-oped enclosure fires have been defined by the gastemperatures inside the enclosure, and the enclosurefire section includes a review of the most widelyused methods for predicting gas temperatures.

Local fire plumes may be confined to a singlefuel package in intimate contact with a structuralmember. The thermal exposure from local fires isspatially variable and is dependent on the geometrybeing considered. Though local fires may notexpose as large an area as enclosure fires, the heatfluxes from local fires can be considerable andshould not be neglected in an analysis. Heat fluxesfrom reasonable-size local fires can easily exceed120 kW/m2 and have been measured as high as220 kW/m2 in very large pool fires. Due to thespatially and geometric dependence, the thermalexposure from local fire plumes has historicallybeen measured directly using heat flux gauges.Therefore, the boundary condition for local fireplumes will be provided as a measured heat fluxwith guidance on correcting this measurement basedon the actual structural element temperature.

The methods applicable to fully developed en-closure fires should be used for compartments withfuel uniformly distributed over their interiors. For

situations where a fire would not be enclosed or forenclosures with sparse distributions or concentratedfuel packets, the methods identified in the fireplumes section should be used.

MODEL INPUTS

For fully developed enclosure fires, predictivemethods require as input one or more of the following:

1. Fuel load2. Dimensions of windows, doors, and other similar

horizontal openings3. Wall thermal properties

Thermal properties of walls are generally fixedvery early in the design of a building. They typicallydo not change much during a building’s lifetime.Furthermore, this is the least critical of the threevariables in its effect on the fire temperature–timehistory. Thus, it is generally acceptable to usenormal design values for the thermal properties.

Ventilation is usually handled by simply deter-mining the potential window and door openingsfrom the building’s architectural drawings. Thismay not be a robust strategy since these openingsmay vary as a consequence of alteration of a build-ing. Some serious fire losses have occurred duringconstruction or remodeling. Two examples are theOne Meridian Plaza fire1 and the Broadgate fire.2

During construction or remodeling, the geometricaspects of a building can vary from what they areintended to be during ultimate occupancy. Uncer-tainty in ventilation characteristics can be addressedby a variety of techniques.3 For example, analysescould be conducted using the range of ventilationcharacteristics that could reasonably be expected tooccur. The ventilation characteristics that result inthe most severe exposure could then be used as the basis for design. If uncertainty in ventilationcharacteristics is not addressed during the design,then any change that affects ventilation openings

would require reanalysis to confirm that the build-ing is still within its design basis.

Similarly, fuel loads may vary during the life of abuilding. During construction, periods of work mayexist where the fuel load is great. Such constructionfuel (and debris) may often be much greater thanprojected for the ultimate occupancy. Furthermore,at these times normal fire defense mechanisms—sprinklers, detectors, pull-stations, etc.—are ofteninoperable.

An example may be a building lobby. Duringnormal occupancy, the expected fuel load can betrivial: perhaps a single guard’s desk. Yet duringconstruction or renovation, the lobby may hold thehighest concentration of combustible building andpacking materials. Another example is special events(e.g., school fair exhibits) that are sometimes stagedin lobbies that are generally otherwise fuel free.

Fuel load statistics obtained from building surveysare typically used by designers to derive their inputdata on fuel load. First, these statistics are “typical”values, such as 50% or 80% occurrence values. As“typical” values, these statistics would not providebounding or conservative estimates of fire severity.Additionally, all available fuel load surveys focussolely on normal occupancy characteristics.

Methods of predicting fire exposures from fireplumes also require input values such as heat releaserate or dimension of the fire source. When selectinginput values for these methods, it is recommendedthat bounding or reasonably conservative inputvalues be used.

Whatever input values are used, designers shouldclearly communicate the limits of the design toproject stakeholders such as enforcement officialsand building owners and operators.

BASIS OF FIRE RESISTANCE

Engineered fire protection design is typicallyperformed to meet a set of goals and objectives.These goals and objectives may come from aperformance-based code, from a desire to establishequivalency with a prescriptive code, or from abuilding owner, insurer, or other stakeholder whodesires to have added safety beyond compliancewith a code or standard. Fire resistance might be

used as part of a strategy to achieve life safety,property protection, mission continuity, or environ-mental protection goals.3 More specific objectivescan be developed from these generic goals.

Structural fire resistance has historically beenspecified as ratings for individual structural ele-ments based on a number of building characteristicssuch as occupancy type and building height. Giventhat the fire resistance and permissible materials ofconstruction vary with building use and buildingheight and area, a uniform level of performance doesnot result from compliance with prescriptive codes.

In the case of performance-based codes, the per-formance intended also may vary. The InternationalCode Council Performance Code4 states that somerisk of loss of life may be acceptable, depending uponthe magnitude of the event and performance group ofthe building. Similarly, the serviceability expected ofa building varies with the event size and performancegroup. The National Fire Protection Association’sBuilding Construction and Safety Code5 states thatstructural integrity must be maintained for a suffi-cient time to protect occupants and enable firefighters to perform search and rescue operations.

This guide provides a methodology to estimatethe thermal aspects of a fire as they impact exposedstructural members. Given those heat transfer condi-tions, a structural engineer can compute the effecton the structure.

Prior to designing or analyzing structural fireresistance, it is necessary to determine the objec-tives that the structural fire resistance is intended tomeet. Guidance on determining goals and objectivescan be found in the SFPE Engineering Guide toPerformance-Based Fire Protection Analysis andDesign of Buildings.3

ACCOUNTING FOR SUPPRESSION

Many building codes and design guides permit a reduction in fire resistance when active fire pro-tection systems, such as sprinklers, are used. Forexample, the Eurocode6 contains an approach foraccounting for interventions where the design fireload is reduced by a factor (0.0 to 1.0). This resultsin a design fire load that is less than the actual fire load.

2

The methods presented in this guide for predict-ing fire exposures are based on conditions wherethere is no mitigation of a fully developed fire.Analyses of fire exposures to structures in whichactive mitigation is considered are outside the scopeof this guide.

HEAT TRANSFER BOUNDARYCONDITIONS

Analyzing the thermal response of a structurerequires prediction of the heat flux boundary con-ditions. For fire plumes, methods are provided forestimating the heat flux boundary conditions directly,although basic plume correlations are provided forthose who wish to conduct a heat transfer analysisfrom first principles.

For enclosure fires, most of the predictivemethods contained in this guide provide just thetemperature boundary conditions. Determining theheat flux boundary conditions of a structure requiresprediction of the gas emissivity, the absorbtivity of the element, and the convective heat transfercoefficient. The absorbtivity for a surface in a fullydeveloped enclosure fire can be assumed to be 1.0since the surface will become covered in soot. Thegas emissivity will also approach 1.0 for large fires.*Assuming natural convection, the convective heattransfer coefficient, hc, will generally be approxi-mately 10 W/m2K, although it could be as high as30 W/m2K.* For conservative predictions, a con-vective heat transfer coefficient of 30 W/m2Kshould be used.

For insulated materials, such as concrete or insu-lated steel, a bounding estimate of the heat transferboundary condition would be to assume that thetemperature of the exposed surface is equal to thesurrounding gas temperature.*

COMPUTER MODELING

With one exception,7 all the methods identifiedabove for calculating the temperature–time historyfor a fire in a compartment are relatively simple,closed-form equations. Simple, closed-form equa-tions are possible because of the assumptions madeto solve the fundamental conservation equations, e.g.,

uniform conditions throughout the compartment.Indeed, even the computer model referenced above7

assumes a uniform temperature in the enclosure.Many computer models exist that predict fire

temperatures for user-defined heat release rates. Useof most computer fire models for predicting post-flashover fire boundary conditions requires themodeler to estimate the burning rate in the compart-ment using other methods. Given that the heatrelease rate in a post-flashover compartment fire isa function of the characteristics of the enclosure, itis difficult to apply these models without makingadditional simplifying assumptions. For example,by assuming that burning in the compartment isstoichiometric or ventilation limited, a burning ratecould be estimated as a constant multiplied by theventilation characteristics of the enclosure. Poolfires could be modeled using burning rate correla-tions that were developed for open-air burning;however, these correlations neglect thermal feed-back to the fuel from the enclosure.

Field models such as NIST’s Fire DynamicsSimulator (FDS) allow abandoning the assumptionthat compartment gasses are well stirred.8 Instead ofmodeling the enclosure as one zone, field modelsmodel an enclosure as many rectangular prisms andassume the conditions are uniform throughout eachof these cells.

FDS contains pyrolysis models for solid and liquidfuels. The pyrolysis rate of the fuel is predicted byFDS as a function of the modeled heat transfer tothe fuel, and thermally thick, thermally thin, andliquid fuels can be treated. Combustion is modeledby FDS using a mixture fraction model.

While FDS holds promise in calculating heatrelease rates in fires, it presently must be used withcaution since a number of simplifications are usedas a result of computational, resolution, and knowl-edge limitations. As stated in the FDS User’s Guide,“The various phenomena [associated with modelingcombustion] are still subjects of active research;thus the user ought to be aware of the potentialerrors introduced into the calculation.”9 Any errorsthat are present with pool-like or slab-like fuelswould likely be magnified when considering crib-like fuels such as furniture.

3

____________*See the “Theory” section beginning on page 5 for a derivation of this value.

Fully Developed Enclosure FiresFire in enclosures may be characterized in three

phases. The first phase is fire growth, when a firegrows in size and heat release rate from a smallincipient fire. If there are no actions taken to sup-press the fire, it will eventually grow to a maximumsize, which is a function of the amount of fuel pres-ent or the amount of air available through ventila-tion openings. As all of the fuel is consumed, thefire will decrease in size (decay). These stages offire development can be seen in Figure 1.

The size (magnitude) of the fire and the relativeimportance of these phases (growth, fully devel-oped, and decay) are affected by the size and shapeof the enclosure; the amount, distribution, form, andtype of fuel in the enclosure; the amount, distribu-tion, and form of ventilation of the enclosure; andthe form and type of materials forming the roof (orceiling), walls, and floor of the enclosure.

The significance of each phase of an enclosurefire depends on the fire safety system componentunder consideration. For components such as detec-tors or sprinklers, the fire growth part is likely to bethe most significant because it will have a greatinfluence on the time at whichthey activate. The fire growthstage usually proves no threat tothe structure, but if it can (forexample, if concentrated fuelpackets are located close to an ele-ment), the direct heating by flamesmust be considered in accordancewith the section on fire plumes.The threat of fire to the structureis primarily during the fully devel-oped and decay phases.10,11

There are two methods ofdesign based on fully developedcompartment fires:

1. Methods that predict theboundary conditions to whichthe structure will be exposed,from which a thermal analysisand structural analysis of thestructure may be performed

2. Methods that determine an equivalent exposureto the standard temperature–time relationship

The former is the only true engineering method ofdesigning structural fire resistance. The latter isbased on determining the “equivalent” fire exposureto the “standard” temperature–time relationship,which carries an implicit assumption that the fireresistance requirements contained in prescriptivecodes provide a firm design basis. While the stan-dard temperature–time relationship provides anhourly rating, this rating is only intended to be arelative measure and does not necessarily reflectstructural performance in a fire. Time-equivalentmethods are further discussed only in Appendix C.

With the exception of Babrauskas’ method,which allows for the consideration of pool fires, allthe methods summarized in this guide have theirbasis in fires involving wood cribs. Although manyhydrocarbon-based materials, such as plastics, haveapproximately twice the heat of combustion ofcellulosic materials, such as wood (in other words,burning 1 kg of a plastic can liberate twice theenergy as burning an equal mass of wood), use of

4

Time

Tem

per

atu

re

Dev

elop

men

t

Fla

shov

er

Fully Developed

Cooling Phase

Significant effect on structure

Fire Growth Decay

FIGURE 1. Phases of Fire Development

the methods contained in this guide should be rea-sonable for most design scenarios.

This statement is made for two reasons. First,while real fuels are not wood cribs, cribs mightapproximate structural wood furniture such as desksand chairs. Other furnishings are mostly composedof large flat surfaces that would more easily vapor-ize fuel in a fire. These flat surfaces might be classi-fied as “pools” since they represent a surface fullyexposed to the fire. On the other hand, cribs burnfrom within and feel very little of the surroundingheat of the fire. The heat flux of the fire willincrease vaporization over the ambient level. Thisdepends on the fuel’s heat of gasification (typicallyL = 0.5 to 1 kJ/g for liquids, 2 to 3 for non-charringsolids, and 5 to 10 for charring solids).

Since the fuel volatilization rate is the heat trans-fer to the fuel divided by the heat of gasification ofthe fuel12 and woods tend to have higher heats ofgasification, wood cribs will tend to result in firesof longer duration than other fuels. In ventilation-limited fires involving non-charring fuels, the rateof airflow into the enclosure will govern the heatrelease rate into the enclosure, and fuels that cannotburn inside the enclosure will burn outside oncethey encounter fresh air.

Secondly, the primary fuel in many design oranalysis situations is typically cellulosic in nature(wood, paper, etc.). While many compartments con-tain other fuels, the total mass of non-cellulosicfuels could be a small fraction of the mass of cellu-losic materials. Design or analysis situations inwhich the fuels are not predominantly cellulosic andthe burning is not expected to be ventilation limitedmay require special attention.

Additionally, each of the methods presented inthis guide is subject to the following limitations:

1. The methods are only applicable to compart-ments with fuel uniformly distributed over theirinterior. (Sparse distributions or concentrated fuelpackets should be considered using the methodsidentified in the fire plumes section.)

2. The methods presented in this guide are onlyapplicable to compartments having vents inwalls. (Ceiling and floor vents require a specialformulation, as would underground compart-ments having only roof vents.)

3. Only natural ventilation is considered as wouldoccur through the wall vents. (The effect offorced ventilations and wind and stack-effectflows in tall buildings are not included.)

4. Large fires are considered whose heating effectsare felt uniformly through the compartment.

Concern has been expressed that fires in long,narrow enclosures exhibit different burning behav-ior than fires in other types of enclosures13 and,hence, predictive methods that were developedbased on fires in compartments that are not longand narrow may not accurately predict burningbehavior in long, narrow enclosures. Specifically,these long, narrow compartments with a uniformlydistributed fuel load can exhibit non-uniform heat-ing in ventilation-limited fires. To address this con-cern, the methods presented in this guide have beenevaluated using data from fires in long, narrowenclosures in addition to compartments in which theratio of length to width is nearly one.

THEORY

It would appear that geographical reasons explainthe proliferation of many models for fire resistance.Most of the work on fire resistance took place before1970, when communication and dissemination ofresearch in fire was limited. This might explain theexistence of the different models. However, theirdifferences are superficial for the most part, cloudedby notation or parameters that might appear asdifferent. For that reason, it was felt important todevelop a theoretical base for the models. So doingmight appear to be establishing yet another model.Indeed, the contrary is intended. The purpose of thistheoretical exposition is to present a rationale forthe physics of the models and to show their simi-larities and deficiencies. It is in this context that atheoretical introduction is provided to the modelsthat exist in the literature.

Theoretical Development

The purpose of this theoretical development is to:

1. Present the governing equations2. Explain and justify typical approximations

5

3. Present the equations in dimensionless terms to showa. Their generalityb. Independence of scalec. Relationship to variables used in the

established methods

The common objective of all the models has beento predict the following:

1. Compartment gas temperature2. Burning rate of the fire3. Duration of the fire

The purpose of the studies considered has been topredict the thermal effects of fully developed build-ing fires so that their impact on the structural mem-bers could be assessed. Fully developed fires withconsiderable fuel will tend to produce a fairly uni-form temperature smoke layer that will descend tothe floor. This will particularly occur for a large fireand relatively small vents. The radiation effects ofsuch a fire will further tend to cause uniform heat-ing of the contents. Consequently, the model for thefully developed fire has been an enclosure with uni-form smoke or gas properties. The bounding wallsurfaces are also considered uniform. The structural

elements absorb a small amount of heat relative toheat loss into the wall or ceiling surfaces togetherwith the energy loss out of the vents. These ventsinclude the windows broken by the thermal stress of the impinging flames and heat. The model isdepicted in Figure 2.

The conservation of mass and energy for the con-trol volume (CV), which follows, also applies.

Mass: (Eq. 1)

Energy:

(Eq. 2)

The Equation of State: (Eq. 3)

The volume, V, is constant. The pressure, p, isnearly constant and at the ambient condition forvents that are even very small, e.g., those in theleakage category. Only for abrupt changes in the firewill pressure pulses above or below ambient occur.

The temperature slowly varies during the fullydeveloped fire state. As a consequence, steady-stateconditions can be justified.

6

FIGURE 2. Model for the Fully Developed Fire

(Eq. 4)

The mass flow rate from the vent ( •m) equals theair supply ( •mo) and the fuel gases produced (

•mF ).The energy equation can be written as

(Eq. 5a)

The heat losses ( •q ) consist of the heat transferinto the boundary surfaces and the radiation loss outof the vent. Some simplification can be made since

, so that the second term on theright may be neglected.

(Eq. 5b)

Wall Heat Transfer

The heat transfer into the boundary surface is by convection and radiation from the enclosure,then conduction through the walls. The boundaryelement will be represented as a uniform material of properties:

• Thickness, δ• Thermal conductivity, k• Specific heat, c• Density, ρ

It conducts to a sink at To.The heat transfer can be represented as an

equivalent electric circuit as shown in Figure 3.

The conductances, hi, can be computed as fol-lows from standard heat transfer estimates:

Convection

Convection can be estimated from natural convection.14

It gives hc of about 10 W/m2K. Under some

other flow conditions, it is possible hc might be ashigh as 30 W/m2K.

Conduction

Conduction might be represented as steady orunsteady. The latter is more likely. Only a finitedifference numerical solution can give exact results.Most often the following approximate analysis isused for the unsteady case assuming a semi-infinitewall under a constant heat flux. The exact solutionfor constant heat flux gives:

(Eq. 6a)

or

(Eq. 6b)

This result for hk can be used as an approxima-tion for variable heat flux. For steady conduction,the exact result is

(Eq. 6c)

The steady-state result would be considered to hold for14

7

FIGURE 3. Wall Heat Transfer

Some estimations for commonmaterials are given in Table 1. For a wall 6" thick, δ ≈ 0.15 m, then

Hence, most boundaries might beapproximated as thermally thick sincemost fires would have a duration ofless than 3 hours.

The thermally thick case will predominate undermost fire and construction conditions:

Based on kρc of 103 to 106, it is estimated

Radiation

Radiation heat transfer can be derived from themethod presented in Karlsson and Quintiere15

(p. 170) for enclosures. It can be shown as14

(Eq. 7)

Where:ε = Emissivity of the enclosure gas (flames

and smoke)εw = Emissivity of the boundary surface

Since the boundary surface will become sootcovered in a fully developed fire, εw = 1.

The gas emissivity can be represented as

(Eq. 8)

Where:H = A characteristic dimension of the enclosure,

its height

The absorption coefficient κ, can range fromabout 0.4 to 1.2 m-1 for typical flames (see Karlsson and Quintiere,15 p. 167). Experimentalfires might use H ≈ 1 m, while buildings generallyhave H ≈ 3 m. For the smoke conditions in fullydeveloped fires, κ =1 m-1 is reasonable in the least.Hence, ε ranges from about 0.6 for a small experi-mental enclosure to 0.95 for realistic fires.

It follows that:

(Eq. 9)

where ε is generally nearly 1. It can be estimatedfor ε = 1, and T = Tw, that

hr = 104 – 725 W/m2K

for T = 500 to 1200°C.From the circuit in Figure 3, the equivalent con-

ductance, h, allows

(Eq. 10a)

Where:

(Eq. 10b)

It follows from the estimates that h ≈ hk , whichimplies Tw ≈ T for fully developed fires. This resultapplies to structural elements that are insulated,including unprotected concrete elements. Hence,predicting the fire temperature provides a simpleboundary condition for the corresponding computa-tion for the structural element. Its surface tempera-ture can be taken as the fire temperature.

This result is very important and helps to explainwhy most of the methods only present the fire tem-perature without any detailed consideration of the

8

Approximate Properties

Concrete/Brick Gypsum Mineral Wool

k (W/mK) 1 0.5 0.05

kρc (W2s/m4K2) 106 105 103

k/ρc (m2/s) 5 × 10-7 4 × 10-7 5 × 10-7

TABLE 1. Estimates of Conduction for Common Materials

t (min) hk (W/m2k)

10 0.8-26

30 0.3-10

120 0.2-5

heat transfer in representing the fully developedfire. From the estimates made here, the gas phaseradiation and convection heat transfer have negligi-ble thermal resistance compared to conduction intothe boundary. As a consequence, the fire tempera-ture is approximately the surface temperature. Thisboundary condition is “conservative” in that it givesthe maximum possible heat transfer from the fire.

Radiation Loss from the Vent

From Karlson and Quintiere15 (p.170), an analy-sis of an enclosure with blackbody surfaces (εw = 1)gives the radiation heat transfer rate out of the ventof area Ao as

(Eq. 11)

Since ε is also near 1 and Tw ≈ T, it follows that

(Eq. 12)

This blackbody behavior for the vents has beenverified.16

The total heat losses can be written as

(Eq. 13)

Vent Mass Flow Rate Air

The mass flow rate of air can be approximatedfor small ventilation as (Karlsson and Quintiere,15

p.100)

or in general

(Eq. 14)

where ko = 0.145 (for ρ0 = 1.1 kg/m3). This result is

prevalent in all analyses, and the parameter ( )shows up in many experimental correlations.

The Fire—Firepower and Burning Rate

To complete the energy equation in order to solvefor the temperature, the fire must be described. Theheat of the flames and smoke causes the fuel tovaporize, supplying a mass flow rate, •mF . While allthe fuel may eventually burn, it may not necessarilyburn completely in the compartment. This dependson the air supply rate. Either all the fuel is burned,or all the oxygen in the incoming air is burned.What burns inside gives the firepower within the enclosure.

Thus,

(Eq. 15)

The equivalence ratio, φ, determines if the com-bustion is fuel-lean (1).

(Eq. 16)

Where:s = Stoichiometric air-to-fuel ratio∆Hc = Heat of combustion (chemical heats of

combustion according to Tewarson17)∆Hair = Heat of combustion per unit mass of

air ≈ 3kJ/g, which holds for most fuels

Note:

(Eq. 17)

The mass supply rate of the fuel, •mF , depends onthe fuel properties, its configuration, and the heattransfer. Most studies have been done using woodcribs. These are composed of ordered layers ofsquare sticks of side b. Gross18 and Heskestad19

have developed correlations to describe how theyburn. For cribs that have sufficient air supply, their

burning rate per unit area is found as

(Eq. 18)

where C depends on the wood (approximately1 mg/cm1.5s).

9

For a range of crib experiments in compartments,Harmathy20 gives

while Tewarson17 gives 11 g/m2s. These values givean approximation for wood, but it should be notedthat, in general, it depends on the stick size.

Real fuels are not wood cribs, although cribsmight approximate structural wood furniture such asdesks and chairs. Other furnishings are mostly com-posed of large flat surfaces that would more easilyvaporize fuel in a fire. These flat surfaces might beclassified as “pools” since they represent a surfacefully exposed to the fire. On the other hand, cribsburn from within and feel very little of the sur-rounding heat of the fire.

In general, the mass flux of fuel produced in afire can be represented as

(Eq. 19)

The fire “free”-burning flux is how the fuelwould burn in ambient air. In a fire, this would bemodified by the oxygen concentration the fuel expe-riences. Also, the heat flux of the fire willincrease vaporization over the ambient level. Thisdepends on the fuel’s heat of gasification (typicallyL = 0.5 to 1 kJ/g for liquids, 2 to 3 for non-charringsolids, and 5 to 10 for charring solids). It is knownthat large fires, burning in air, reach an asymptoticburning flux as their flames reach an emissivityof 1. Such values are tabulated (see Tewarson17 orBabrauskas21). Since the radiant heat transfer domi-nates, the fuel mass loss rate in typical buildingcompartments, where the fire is large, can beapproximated as

(Eq. 20)

Here, it is assumed that for φ < 1, the “fuel-controlled” fire, the fire burns as a large fire withsufficient air. Such “large” fires need only achieve aburning diameter of greater than about 1 to 2 m. Inthe “ventilation-controlled” fire, φ > 1, the fuelmass loss rate is composed of all that burns insidewith the available airflow plus what is vaporized by

radiant heating. The radiation geometric view factorF is, in the limits, 0 and 1, respectively, for crib-likeand pool-like fuels. This expression is the governingequation for the mass loss rate. Together with theenergy equation, there are two equations and twounknowns: T and •mF

Development of a Solution andDimensionless Groups

The equations will be examined to achieveinsight into the form of a solution. They are notdifficult to solve by iteration using a computer.However, analytical approximations can be ofvalue. A dimensionless form of the equations willbe presented to demonstrate the important variables.These variables will be used to explain the theoreti-cal and experimental results presented in this guidein terms of the methods available in the literature.

Compartment Temperature

Substituting for the heat loss rate fromEquation 13 into the energy equation (5b) yields:

(Eq. 21a)

Dividing the numerator and denominator byand representing

gives

(Eq. 21b)

By substituting for , thefollowing dimensionless groups emerge. Thedimensionless variables are presented in terms of afrequently used Q* factor.

(Eq. 22)

10

(Eq. 23)

(Eq. 24)

(Eq. 25)

(Eq. 26a)

or .

(Eq. 26b)

The correlation by McCaffrey, Quintiere, andHarkleroad (MQH)22 is

(Eq. 27)

This result has only been developed from datawhere φ < 1. But Tanaka, Sato, and Wakamatsu23

have applied it for φ > 1 with some success.

Maximum Possible Temperature

Examine the limit of the stoichiometric adiabaticstate that would yield the maximum temperature.Here

Qw* = Qr

* = 0

And from Equations 15 and 22

With φ = 1, the adiabatic stoichiometric firetemperature is

(Eq. 28)

The experimental results for an adia-batic turbulent fire plume24 suggest (T – To)ad ≈ 1500°C at most. This mightrepresent as well the maximum possibletemperatures attainable in a compart-ment fire. The plume adibaticity occursdue to smoke preventing the radiationloss. This occurs as the diameter of thefire becomes large. Large compartmentfires can act similarly as the floor areabecomes large, and only smoke is seenfrom the windows, particularly in anover-ventilated state, φ < 1.

11

FIGURE 4. MQH Correlation for Fuel-Controlled Fires.X1 ≡ Q*, X2 = Qw*

700

600

500

400

300

200

100

00 0.3 0.6 0.9 1.2 1.5 1.8

Tem

per

atu

re R

ise

Un

der

Cei

ling

(T

– T

0)

(K)

X1N X2

M

Burning Rate

The form of Equation 26 suggests a correspond-ing dimensionless form for Equation 20:

(Eq. 29)

12

The last term suggests another dimensionlessgroup governing compartment feedback.

Define

(Eq. 30)

Significant Relationships

Now examine the values of the dimensionlessvariables. Estimating values are as follows:

For typical building compartments, the geometric

compartment parameter is ≈ 1 m–1/2

for full windows, ≈ 10 m–1/2 for typical windows,and ≈ 100 m–1/2 for very small vents.

Since the fuel surface area is similar and related

to the room area, has a similar range.

The burning rate term can be estimated as

≈ 10-3 – 1 for wood and ≈ 10-2 – 10

for liquid fuels from very large to very small vents,respectively.

The heating terms can be estimated as follows:

Qw* ≈ 3 × 10-5 – 90 for large to small vents,

from estimates of hk

Qr* ≈ 1 × 10-4 – 2 × 10-4 for Ho ≈ 3 m

QF* ≈ 1.3 × 10-4 × for wood,

1.3 × 10-3 × for liquid fuels

Therefore, all terms can be significant undersome circumstances.

General Form of Correlations

The dimensionless variables developed here canbe used to explain the methods presented in thisguide. From Equations 26 and 29, the approximatefollowing solutions, in general, can be derived:

(Eq. 31a)

(Eq. 31b)

(Eq. 31c)

A functional form of these equations is givenfrom the theoretical approximation given here, butcomplete analytical solutions cannot be determined.Only limiting analytical solutions are possible, but these still depend on empirical factors, e.g.,

, etc. Some limiting cases are as follows:

Large Ventilation

Large ventilation,

In this case, ko is not a constant (Equation 14),

but depends on due to the effect of

temperature difference on the buoyancy velocity,

i.e., and .

For the case of large vents (φ < 1), Equation 26acan be rewritten as

This suggests that

(Eq. 32)

This is consistent with the MQH correlation forφ < 1 given by Equation 27.

The mass loss rate for large ventilation (φ < 1) isgiven directly by Equation 31a.

(Eq. 33a)

or alternatively

(Eq. 33b)

Both forms of are used in the experimentalcorrelations; however, the ratio has notgenerally been included in their results. It should be recalled that, for well-ventilated wood cribs,

, where b is the stick thickness.

The temperature, from Equation 27, can bewritten as

(Eq. 34)

Small Ventilation

Small ventilation,

From Equation 31b, it can be estimated for woodcribs and for large pool fires where the radiationfeedback is small:

(Eq. 35)

The radiation feedback is negligible for cribsbecause of the stick blockage and for large poolfires because of obscuration by smoke. For small-scale pool fires in compartments, there can be aconsiderable enhancement in the burning rate due toradiation feedback.

The corresponding temperature can be estimatedas follows, neglecting the vent radiation, since thevent is small.

(Eq. 36)

But Q* depends on the airflow, so, by Equation 31c,

or

(Eq. 37)

For small-scale pool fires in compartments, theeffect of heat feedback from the compartment islarge and cannot be neglected as above.

13

14

Summary

The theory suggests that the correlations be ofthe following form:

• Large ventilation,

(Eq. 38a)

(Eq. 38b)

• Small ventilation,

(Eq. 39a)

(Eq. 39b)

Usual forms of the correlations have been

for wood and liquid pool fires. This would lead toresults as shown in Figure 5.

A typical form for temperature is

From Equation 38a, it follows that

This results in the following trends, as shown inFigure 6.

In the theoretical development, the dimensionlessvariables that should show up in the literature corre-lations have been identified. The dimensionless

variables contain the scaling factorsthat allow for the extrapolation ofresults over geometric scales. In addi-tion, the dimensionless groups exhibitthe proper combination of other vari-ables including time and material

properties. The theoretical results give the followingfunctional behavior:

These dimensionless variables are not usuallyrepresented in the literature correlations in the same manner. They have equivalent surrogates. For example:

• , Maximum Gas Temperature, is usuallygiven as T only.

• , Burning Rate/Vent Flow, isusually given as .

• Q*, Fire Power or heat release rate; usually onlyventilation-limited fire states are considered, and,consequently, this variable does not explicitlyshow up; however, in general,

Note that in the latter case (φ > 1) Q* is constant. The former, or fuel-controlled, state contains the effect of fuel.

• ,

Wall Heat Loss, is usually repre-sented as a scaling factor for timethat allows for the temperature to be represented over dimensionlesstime,

• ,

Vent Radiation Loss, usually doesnot appear in the correlations since

likely has a small variationover the range of data considered.

• ,

Enhanced Fuel Vaporiza-tion; for wood cribs this term is small, but for otherforms of fuel in the form of flat surfaces it can beconsiderable. Compared towood cribs, it will reducethe duration of the fire,making the wood cribmodel conservative indesign since it would give a longer duration.

FIGURE 6. Approximate Theoretical Behavior of CompartmentTemperature

15

FIGURE 5. Approximate Theoretical Behavior for Fuel Burning Rate

0

200

400

600

800

1000

1200

0 10 20 30 40 50

Tem

per

atu

re °

C

A/AoHo1/2 m–1/2

Fuel leanWell-ventilated

Fuel richVentilation-limited

Increases asfuel mass flux, heat of combustion,fuel areaincrease

Increases asheat loss to wallsdecreases

Φ < 1 Φ > 1

METHODS FOR PREDICTING FIRE EXPOSURES

Several methods are available for predictingtemperatures and duration of fire exposure in acompartment. These methods are presented in anarbitrary order.

Eurocode Parametric Fire Exposure Method

The Eurocode 1, Part 2.2,6 provides three “stan-dard” fire curves and a parametric fire exposure.The standard fire curves include the ISO 834 curve,an external fire curve, and a hydrocarbon fire curve;these standard curves are not addressed further inthis guide. The parametric fire exposure in the Euro-code was originally developed by Wickstrom.25

Wickstrom stated25 that this method assumes thatthe fire is ventilation controlled and all fuel burnswithin the compartment.

Wickstrom modified an approximation of theISO 834 standard fire curve by altering the timescale based on the ventilation characteristics andenclosure thermal properties. The modified time scale compares the enclosure of interest toMagnusson and Thelandersson’s “type A” enclosurewith an opening factor of 0.04 m1/2. Wickstromfound that the resulting curve approximated theISO 834 standard fire curve.

The Eurocode states that this parametric exposuremay be used for fire compartments up to 100 m2

only, without openings in the roof, and for a maxi-mum compartment height of 4 m. The Eurocodedoes not provide any basis for these limits.

The Eurocode provides the following tempera-ture–time curve for a natural fire (also known as aparametric curve):

Where:T = Temperature (°C)t* = tΓ (hours)t = Time (hours)

Where:The opening factor has limits of

Ao = Area of vertical openings (m2)

Ho = Height of vertical openings (m)A = Total area of enclosures (walls, ceilings,

and floor including openings) (m2)b = (J/m2 s1/2 K) and has the limits

1000 ≤ b ≤ 2000k = Thermal conductivity of enclosure lining

(W/m-K)ρ = Density of enclosure lining (kg/m3)c = Specific heat of enclosure lining (J/kg-K)

For enclosures with different layers of material, b = is calculated as follows:

b = (J/m2 s1/2 K)

Where:δi = Thickness of layer i (m)ci = Specific heat of layer i (J/kg K)ki = Thermal conductivity of layer i (W/m K)bi = (J/m

2 s1/2 K)

To account for different materials in walls,ceiling, and floor, b = should be calculated as follows:

Where:Atj = Area of enclosure including openings with

the thermal property bj (m2)

16

The temperature–time curves in the coolingphase are given by:

Where:Tmax = Maximum temperature (°C) in the

heating phase for t* = td*

td* = (hours)

with:qt,d = Design value of fuel load

density related to surface area Aof the enclosure whereby qt,d = qf,d Afloor/A (MJ/m

2). The limits 50 ≤ qt,d ≤ 1000 (MJ/m

2) should be observed.

qf,d = Design value of the fuel loaddensity related to the surface areaAfloor of the floor (MJ/m

2).

By making simple substitutions, td* can also be

expressed as:

Where:E = Total energy content of the fuel in the

compartment, expressed by

Buchanan10 suggested that the temperatures inthe Eurocode are often too low and that it would bemore accurate to scale based on a reference of 1900 J/m2 s1/2 K. This would result in thefollowing modified equation for Γ :

Franssen26 noted two shortcomings of theEurocode procedure for accounting for layers ofdifferent materials:

1. The Eurocode procedure does notdistinguish which material is on the sideexposed to a fire.

2. The contribution of each material to the b factor is weighted by thickness, sothe adjusted b factor for an enclosure witha nominal thickness of an insulatingmaterial over a much thicker, heaviermaterial will be biased towards the bfactor of the thicker, heavier material.

Franssen therefore suggests the followingalternative method of accounting for layers of dif-ferent materials:

1. If a heavy material is insulated by a lightermaterial, the b factor for the lighter materialshould be used.

2. If a light material is covered by a heavier material,for example in a sandwich panel, then a limitthickness should be calculated according to:

where the subscript 1 indicates the properties ofthe material on the side exposed to the fire and tis the duration of the heating phase of the fire inseconds, which can be calculated as

If δ1 > δlim, then the b factor for the heaviermaterial should be used; otherwise,

Franssen observed26 that, as the ratio between thefuel load and the ventilation factor decreases, theEurocode predicts unrealistically short burningdurations. Therefore, Franssen suggests that if

17

is less than 20 minutes, then the following proce-dure should be used:

1. The opening factor should be set

equal to , Γ should be set equal

to , and td* should be

set equal to ,

where 0.33 is 20 minutes, expressed in hours.

2. If > 0.04 m1/2 (calculated based

on actual compartment geometry, not as modified above) and qt,d < 75 MJ/m

2 and b <1160 J/m2 s1/2 K, then Γ should be set equal to

where is calculated based on actualcompartment geometry.

Data Requirements

1. Enclosure thermal properties, k, ρ, and c. If thelining is not the same over the entire surface, thepercentage of the enclosure area composed ofeach material is required. If multiple layers ofmaterial are present in the enclosure, thethickness of each layer is required. For thermallythick enclosure materials, it should be sufficientto account only for the innermost layer.

2. The fuel load density present in the enclosure,qf,d.

3. The area and height of the enclosure opening(s),Ao and Ho.

4. The interior surface total area of the enclosure,including the floor and openings, A.

Data Sources

1. Thermal properties: SFPE Handbook of FireProtection Engineering27 or manufacturer’s data.

2. Several surveys have been published of mass ofcombustible materials per unit area for differentoccupancies.28,29,30,31 Given that fire loading can vary significantly over the life of a building,uncertainty should be carefully considered. Heats of combustion are available in the SFPEHandbook of Fire Protection Engineering32,33

or other sources. To determine qf,d, sum theproducts of the heat of combustion and the totalmass of each material and divide this sum by thetotal floor surface area. Given the uncertaintythat is expected in estimating the mass ofmaterials, 40 MJ/kg is a reasonable estimate ofthe heat of combustion of plastics and otherhydrocarbon-based materials, and 15 MJ/kg is areasonable estimate of the heat of combustion ofwood and other cellulosic materials.

3.

Documents

SFPE Fire Exposures...The SFPE Task Group on Fire Exposures to Structural Elements began its work in March 1998. The purpose of this guide is to provide the infor-mation and methodology