fire exposures to structural elements

Society of Fire Protection Engineers7315 Wisconsin Avenue, Suite 620 E

Bethesda, MD 20814

Engineering Guide

Fire Exposures toStructural Elements

May 2004

35438_COVER 10/15/04 3:43 PM

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

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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...........................................................................77B.3 Comparison of CIB Temperature Data to Predictions Made Using Eurocode,

Buchanan, and Franssen Methods, qt,d = 50 MJ/m2.............................................................................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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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 someother 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 isprevalent 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), or fuel-rich (>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/m2 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/m2). The limits 50 ≤ qt,d ≤ 1000 (MJ/m2) should be observed.

qf,d = Design value of the fuel loaddensity related to the surface areaAfloor of the floor (MJ/m2).

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/m2 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. Building characteristics can be obtained fromsurveys of existingbuildings or architecturalplans of new buildings.

Validation andLimitations

See Appendix B forcomparisons of predictions with test data.

The Eurocode method, without modifications,bounds all CIB temperature data for qt,d = 50 MJ/m2

and most data for qt,d = 100 MJ/m2. The Eurocode,without modification, overpredicted the burning rateof all the CIB data and, hence, underpredicted theburning duration. In Cardington tests #1, 2, 8, and 9

, the Eurocode, without

modifications, bounds average temperatures, butunderpredicted burning duration. In tests #3, 4, 5,

and 6 , the Eurocode,

without modifications, reasonably predicted theburning duration but underpredicted temperature. Intest #7, which was square in plan view, the Eurocode,without modification, underpredicted temperaturebut predicted the burning duration; however, afaster decay was predicted than was observed.

18

Predictions for CIB data using the Buchananmodification bound all temperature data, more so that the Eurocode method without modifica-tion, for qt,d = 50 MJ/m2 and qt,d = 100 MJ/m2. In Cardington tests #1, 2, 8, and 9

, Buchanan’s modi-

fication bounds peak temperature and under-predicts burning duration. In tests #3, 4, 5, and 6

, Buchanan’s modifi-

cation reasonably predicted average temperaturesand the burning duration; however, peak tempera-tures were underpredicted. In test #7, Buchanan’smodification underpredicted temperature butpredicted the duration of peak burning; however,Buchanan’s modification predicted a faster decaythan was observed.

The Franssen modification fell within the scatter

of temperature data for values of between

0 m–1/2 and approximately 15 m–1/2 for

qt,d = 50 MJ/m2 and for values of

between 0 m–1/2 and approximately 20 m–1/2

for qt,d = 100 MJ/m2. For values of

between 20 and 50 m–1/2, Franssen’s modificationbounds all temperature data. Franssen’s modifica-tion reasonably predicts peak temperatures andunderpredicted the burning duration in Cardington

tests #1, 2, 8, and 9 . In

tests #3 and 4 , Franssen

reasonably predicts average temperatures and burn-ing duration; however, Franssen’s modification pre-dicts a faster decay than was observed in test #4(where the fire load was 40 kg/m2). In tests #5 and 6

Franssen’s modification

slightly underpredicted average temperatures.Franssen’s modification reasonably predicted burn-ing duration in tests #5 and 6. In test #7, Franssen’smodification reasonably predicted burning durationbut underpredicted temperature data.

Lie’s Parametric Method

Lie suggested that, if the objective is to develop amethod of calculating fire resistance requirements,then it is necessary only to find a fire temperature–time curve “whose effect, with reasonable proba-bility, will not be exceeded during the use of thebuilding.”34 Lie developed an expression based onthe series of temperature–time curves computed byKawagoe and Sekine35 for ventilation-controlledfires, which he proposed could be used as anapproximation for the most severe fire that is likelyto occur in a particular compartment.36

He describes the opening factor

Where:Ao = Area of vertical openings (m2)Ho = Height of vertical openings (m)A = Total area of enclosures (walls, ceilings,

and floor including openings) (m2)

The rate of burning of the combustible materialsin the enclosures is given by:

Where:= Mass burning rate of fuel

Thus, if is the fuel load per unit area of the

surfaces bounding the enclosure, the duration of thefire, τ, is:

Where:τ = Duration of fire (hours)

For given thermal properties of the materialbounding the enclosure, the heat balance can besolved for the temperature as a function of theopening factor F. Besides depending on F, the tem-perature course is also a function of the thermalproperties of the material bounding the enclosure.

Lie derived a series of temperature–time curvesfor ventilation-controlled fires in two types ofenclosures: “dominantly heavy materials” and“dominantly light materials.”

A

HAF oo=

19

He found these curves could be reasonablydescribed by the expression

Where:T = Time in hoursC = Constant taking into account influence of

the properties of the boundary material onthe temperature:C = 0 for heavy material with a density

ρ ≥ 1600kg/m2

C = 1 for light materials ρ < 1600kg/m2

Lie states that the expression is valid for

If t > (0.08/F) + 1 a value of t = (0.08/F) + 1should be used.

If F > 0.15 a value of F = 0.15 should be used.Lie also derived an expression to define the tem-

perature course in the decay period, over time:

with the condition T = 20 if T < 20°C.

Where:Tτ = Temperature at time τ (°C)

Data Requirements

1. Enclosure density, ρ2. The mass of fuel in the enclosure, mf3. The area and height of the enclosure opening(s),

Ao and Ho4. The interior surface total area of the enclosure,

including the floor and openings, A

Data Sources

1. Density: SFPE Handbook of Fire ProtectionEngineering27 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 canvary significantly over the life of a building,uncertainty should be carefully considered.

3. Building characteristics can be obtained fromsurveys of existing buildings or architecturalplans of new buildings.

Validation and Limitations

See Appendix B for comparisons of predictionswith test data.

Lie’s method bounded almost all the CIB temper-ature data. Lie’s method generally overpredictedburning rate and underpredicted burning duration

for . For

predictions using Lie’s method fell within thescatter of points. The data in the ventilation-

controlled regime can be

bounded by multiplying and dividing Lie’s burningrate prediction by a factor of 1.8.

In Cardington tests #1, 2, 8, and 9

, Lie’s method predicted or

slightly underpredicted average temperatures andunderpredicted peak temperatures. The burningduration was underpredicted in these experiments.

In test #7 , Lie underpredicted

temperature and duration. Lie’s method underpre-dicted temperatures in tests #3, 4, and 5

; however, predictions

improved as increased. Lie’s method

reasonably predicted burning duration in these

experiments. In test #6, ,

Lie’s method reasonably predicted both temperatureand duration.

20

Tanaka

Tanaka extended the equation for pre-flashoverroom fire temperature developed by McCaffrey et al.22 to obtain equations for ventilation-controlledfire temperatures of the room of origin and thecorridor connected to the room. 37 The temperaturerise in a compartment can be predicted by the fol-lowing equation according to McCaffrey et al.

where the effective heat transfer coefficient defined as

Substituting hk and the values of g, c0 , ρ0, and T∞,the equation reduces to

Where:g = Gravity, 9.81 m/s2

c0 = Specific heat of air, 1.15 kJ/kg Kρ0 = Density of air, 1.2 kg/m3

= Heat release rate (kW)T = Temperature (K)T∞ = 300 KA0 = Area of opening (m2)H0 = Height of opening (m)A = Total surface area of room, excluding

opening (m2)t = Time (s)k = Thermal conductivity of enclosure lining

(kW/m K)ρ = Density of enclosure lining (kg/m3)c = Specific heat of enclosure lining (kJ/kg K)

Tanaka studied the effect of an opening betweenthe corridor and the outdoors when the corridor wasconnected to the room of origin. His equations canbe reduced where there is no opening between the

room of origin and the connected corridor and can be used for predicting the temperature of asingle fire room. In this case becomes

and substituting

Tanaka’s method performs allcalculations in Kelvin; the equationfor temperature in degrees Celsiusfollows.

Tanaka uses Kawagoe and Sekine’s method ofpredicting the mass burning rate as follows:

Where:= Mass burning rate of fuel

Upon comparison of the results of the simpleequations to results of a more detailed computermodel, Tanaka refined the equations to improveaccuracy. Tanaka defined the parameter as

and the equations for

temperature of the fire room are

or

Where:

and KF reduces to 1. can be simplified to

21

The equation for temperature must be re-dimen-sionalized and converted to degrees Celsius in thesame manner as before.

Data Requirements

1. Enclosure thermal properties, k, ρ, and c2. The height and area of the enclosure opening(s),

Ao and Ho3. The interior total surface area of the enclosure,

including the floor, but excluding the opening(s), A

4. The mass of fuel in the enclosure, mf

Data Sources






Both of Tanaka’s methods bounded all the CIBtemperature data; however, the refined method moreclosely approximates the values. Both Tanaka’ssimple and refined methods use the same correlationfor burning rate. Tanaka’s methods overpredictedburning rate and underpredicted burning duration

for . For

Tanaka’s methods fell within the scatter of points. Burning rate for those tests in the ventilation-

controlled regime can be

bounded by multiplying Tanaka’s prediction by afactor of 1.6 and dividing by a factor of 1.9.

Tanaka’s simple and refined methods overpredicttemperatures but underpredict duration for Carding-

ton tests #1, 2, 8, and 9 .

The simple method overpredicts temperature andreasonably predicts duration for test #7

, while the refined method

reasonably predicts both values. The simple methodgreatly overpredicts temperature, and the refinedmethod reasonably predicts average temperature for

tests #3, 4, and 5 , while

both underpredict duration. For test #6, Tanaka’ssimple method overpredicts temperature, and therefined method underpredicts temperature, yet bothreasonably predict duration. The quality of tempera-ture predictions using Tanaka’s refined method

decreases as increases.

Magnusson and ThelanderssonParametric Curves

Magnusson and Thelandersson38 studied thevariations in the development of energy, the effectsof air supply, and the resulting evolution of gaseswith time in the course of a fire. They determinedthe temperature of the combustion gases from woodfuel fires, in an enclosed space as a function oftime, under different conditions.

Magnusson and Thelandersson made adjustmentsto Kawagoe’s work to accommodate the effect of a cooling phase since Kawagoe and Sekine’s workis more applicable to the flame phase process of fire development.

They used the equation of energy balance derivedby Kawagoe and Sekine35:

Where:= Rate of heat energy released per unit time

during combustion= Rate of heat energy withdrawn per unit

time from the enclosed space owing toreplacement of hot gases by cold air

= Rate of heat energy withdrawn per unittime from enclosed space through thewall floor or ceiling and roof structures

= Rate of heat energy withdrawn per unittime from the enclosed space by radiationthrough the openings in the enclosed space

22

= Rate of the heat energy stored per unittime in the gas volume that is containedin the enclosed space

Magnusson and Thelandersson also use the opening factor,

Where:Ao = Area of opening (m2)Ho = Height of opening (m)A = Total surface area of room,

excluding opening (m2)

Magnusson and Thelanderssonevaluated eight specific types of enclo-sures and developed temperature–timecurves for each, assuming wood fuel.The opening factor and the fuel loadwere varied for each of the eight typesof enclosures, and temperature as afunction of time was presented in bothgraphic and tabular formats. Figure 8shows examples of temperature–timecurves developed by Magnusson andThelandersson.

For practical design, they suggest that the designerchoose the type of enclosed space most similar toone of the eight types with respect to the thermalproperties of the bounding structure. The designershould then determine the opening factor and thefuel load for his/her case, and finally interpolatelinearly, if necessary.

Alternatively, the designer can choose a curvethat is determined without interpolation so as to beon the safe side; the designer chooses the nexthigher value of opening factor and fuel load.

23

FIGURE 7. Schematic Illustration of the Heat BalanceEquation Terms38

FIGURE 8. Examples of Temperature–Time Curves

Data Requirements

1. Construction materials of the enclosure2. The fuel load density (related to the surface area

of the enclosure), q3. The area and height of the enclosure opening(s),


including the floor and openings, A

Data Sources

1. Several surveys have been published of mass ofcombustible materials per unit area for differentoccupancies.28,29,30,31 Given that fire loading canvary significantly over the life of a building,uncertainty should be carefully considered. Heats of combustion are available in the SFPEHandbook of Fire Protection Engineering27,33

and other sources. (Note that values expressed inMJ/kg must be converted to Mcal/kg by multi-plying by 0.239.) To determine q, sum theproducts of the heat of combustion and the total mass of each material and divide this sumby the total enclosure surface area. Given theuncertainty that is expected in estimating themass of materials, 40 MJ/kg (10 Mcal/kg) is areasonable estimate of the heat of combustion ofplastics and other hydrocarbon-based materials,and 15 MJ/kg (4 Mcal/kg) is a reasonableestimate of the heat of combustion of wood and other cellulosic materials.




For values of for which Magnusson and

Thelandersson provide predictions, Magnusson andThelandersson’s predictions bounded the tempera-ture data from the CIB tests. Magnusson andThelandersson’s predictions overpredicted burningrate and underpredicted burning duration for

. However, for

Magnusson and Thelandersson’s predictions fellwithin the scatter of points. Those tests in the

ventilation-controlled regime

can be bounded by multiplying Magnusson andThelandersson’s prediction by a factor of 1.3 and dividing by a factor of 2.3.

Magnusson and Thelandersson’s method predicts peak temperatures, but underpredictsduration, for Cardington tests #1, 2, 8, and 9

. Magnusson and

Thelandersson reasonably predict average tempera-tures and duration for Cardington tests #3 and 4

. For test #5 ,

Magnusson and Thelandersson reasonably predictduration but slightly underpredict temperature. In

Cardington Test #7 , which was

square in plan view, predictions made usingMagnusson and Thelandersson’s method almostcoincided with the data.

Harmathy

Harmathy published a method for predictingburning rates and heat fluxes in compartment fireswith cellulosic fuels.39,40 Harmathy’s method isbased on theory, with a number of simplificationsand comparisons of data to define constants. Themethods that Harmathy presented are applicable tofully developed fires in compartments that areventilation limited or fuel bed controlled.

Harmathy developed a method for calculating theburning rate as follows:

Where:= Mass burning rate of fuel (kg/s)

ρ0 = Density of air (kg/m3)g = Gravitational constant (9.81 m/s2)Ao = Area of ventilation opening (m2)Ho = Height of ventilation opening (m)Af = Surface area of fuel (m2)

24

Harmathy notes that a “critical regime” existswhere the burning rate is poorly predicted using theabove equations. This regime is the range

Harmathy established the duration of the fullydeveloped burning period as the time that the com-bustible mass remaining in the compartment is 80%or more of the initial mass. Using this definition,Harmathy established the following expressions forthe duration of the fully developed fire exposure:

Where:τ = Time of primary (fully developed) burning (s)

Harmathy provides a method of computing theeffective heat flux from the compartment fire toobjects within the compartment as follows:

Where:

To apply the temperature for heat flux, it isnecessary to determine T. To do this, it is firstnecessary to determine the surface temperature of boundary elements in the compartment.Harmathy recommends the following equation to determine the surface temperature of boundary elements:

Where:Tw = Surface temperature of boundary

elements (K)κ = k /ρck = Thermal conductivity of enclosure lining

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

Harmathy states that, where boundary materialsare not homogeneous, a weighted average can beused. Also, Harmathy suggests that, where lining

materials are layered,the properties of theinner layer may beused.

Where:σ = Stefan-Boltzmann constant,

5.67 x 10–8 W/m2 T4

η = Factor (-) (0.9)τ = Burning duration (s)

This results in two equations and twounknowns. Harmathy suggests selecting avalue for T and inserting it into the equa-tion for determining the effective heatflux. The calculated value for can thenbe substituted into the equation for deter-mining T, which can be substituted backinto the equation for determining . Thisprocess of iteration can be repeated untilthe changes in calculated values are small.

25

Decay

Harmathy suggests that during the decay periodthe temperature can be calculated as follows:

Data Requirements

1. Enclosure thermal properties, k, ρ and c2. The density and specific heat of air, ρ0 and c03. The total mass of fuel, mf4. The total free surface area of the fuel, Af5. The area and height of the enclosure opening(s),


including the floor but not including openings, A,and the height of the interior of the enclosure, H

7. Heat of combustion of the volatiles and char,∆Hv and ∆Hc

Data Sources


2. Density and specific heat of air: 1.2 kg/m3 and1150 J/kg-K, respectively.

3. The surface area-to-mass ratio of the fuel typi-cally varies between 0.1 and 0.4 m2/kg for largerwood cribs and conventional furniture, and moreoften varies between 0.12 and 0.18 m2/kg.40

4. For wood products, the heat of combustion ofvolatiles can be assumed to be 16.7 × 106 J/kg,and the heat of combustion of char can be takenas 33.4 × 106 J/kg.39





Due to the iterative nature of Harmathy’s method,it was not possible to compare predictions to the CIBtemperature data. For ventilation-limited fires, pre-dictions made using Harmathy’s method fell withinthe scatter of the test points. The burning rate data canbe bounded by multiplying and dividing predictionsmade using Harmathy’s method by a factor of 1.8.

In the CIB tests, for fuel-controlled fires,

fell within a range of approximately

0.003 to 0.012 . Since occurs

in the denominator of both terms, ranged fromapproximately 0.003 to 0.012 A. In the CIB tests,the average value of AF /A was approximately 0.75. Substituting, ranged from approximately0.002 to 0.009 (kg/m2s) AF. Therefore, multiplyingHarmathy’s burning rate prediction for fuel-controlledfires by 1.5 and dividing it by 2.8 bounds most of the data.

Harmathy’s method underpredicted temperatureand duration in Cardington tests #1, 2, 8, and 9

, and in tests #3, 4, and 5

overpredicted tempera-

tures but underpredicted duration. Harmathyreasonably predicted duration in test #6

but overpredicted temperature.

In test #7 , which was square in

plan view, Harmathy’s method predicted durationwell but overpredicted temperature.

Babrauskas

The software program COMPF was completedand released to the public in 1975.7 The documenta-tion of the program comprised a user’s guide and acomplete source code listing of the program. Acomprehensive presentation of the theory was thenpresented as part of Babrauskas’ Ph.D. disserta-tion.41 The portions of the dissertation pertinent toCOMPF theory were subsequently made availableas a pair of journal articles.42,43

26

The original COMPF program treated only woodcrib fuels, or else arbitrary fuels for which burningrate data were known and could be inputted. Asecond version, COMPF2,44 allowed treatment ofliquid and thermoplastic pools.

During the development of COMPF, it was real-ized that not all the input data that might be desiredwould necessarily be available to the designer.Thus, the idea of “pessimization” was introduced.In addition to running in a purely deterministicmode, two other modes of computation were avail-able. In one case, the fuel mass loss rate would becomputed as usual, but window ventilation wouldnot be set to the maximum open area. Instead, theinstantaneous open area was computed by the pro-gram to always be a value that would lead to thehighest room temperature (up to the maximum full-opening size). In a second pessimization mode, thewindow ventilation would have a fixed value, butthe fuel mass loss rate would be instantaneouslyadjusted to give the highest room temperature.

Babrauskas used COMPF2 to create a series ofclosed-form algebraic equations that can be used toestimate temperatures resulting from fully devel-oped fires. According to Babrauskas, estimationsmade using the closed-form equations are accurateto within 3% to 5% of COMPF2 predictions, typi-cally closer to 3%.45 The general equation follows:

Where:T = Temperature in compartment (°C)To = Ambient temperature (°C)T* = Constant = 1452°C

The first variable, θ1, known as the burning rate stoichiometry, is found for two separateregimes using:

Where:Ao = Area of ventilation opening (m2)Ho = Height of ventilation opening (m)

= Mass burning rate of fuel (kg/s)= Mass burning rate of fuel at

stoichiometry (kg/s)φ = Equivalence ratio (-)s = Ratio such that

1 kg fuel + s kg air = (1 + s) kg products∆Hc = Heat of combustion (MJ/kg)σ = Stefan-Boltzmann constant

(5.67 × 10–11 kW/m–2-K–4)

For pool fires,

Where:Tb = Fuel boiling point (K)Af = Surface area of fuel (m2)∆Hp = Heat of vaporization of liquid (kJ/kg)

Additionally, the heat release rate may be used inplace of the mass loss rate according to the follow-ing equation:

Where:= Heat release rate (kW)

The second variable, θ2, accounts for wallsteady-state losses and is determined using thefollowing equation:

Where:A = Interior surface area of the enclosure,

excluding the floor and openingsδ = Thickness of wall surface (m)k = Thermal conductivity of enclosure lining

(W/m-K)

27

Transient wall losses are incorporated into θ3as follows:

Where:t = Time (hours)c = Specific heat of enclosure lining (J/kg-K)ρ = Density of enclosure lining (kg/m3)

If only steady-state temperatures need to beevaluated, θ3 = 1.

The variable θ4 accounts for the effect that theheight of a vent in relation to the total vent size canhave on a compartment’s radiative losses and isgiven as follows:

The final variable, θ5, describes the effect ofcombustion efficiency on the compartment tempera-ture. This variable takes into account the fact thatthe gases in the compartment may not be complete-ly mixed, and is found using:

Where:bp = Maximum combustion efficiency

(ranges from 0.5 to 0.9)

Data Requirements

1. Mass pyrolysis rate of fuel, , or heat releaserate,

2. The ratio s where 1 kg fuel + s kg air = (1+s) kgproducts or the chemical formula of the fuel

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

4. The interior surface total area of the enclosure, A,not including the floor or openings

5. For liquid fuels, the heat of vaporization of theliquid, ∆Hp, the fuel boiling point, Tb, and thepool area, Af

6. Enclosure thermal properties, k, ρ, and c, and thethickness of the enclosure, δ

7. The combustion efficiency, bp

Data Sources

1. For ventilation-controlled fires, the masspyrolysis rate of fuel can be calculated from

.44 For fuel-controlled fires,Harmathy39 suggests , where Af isthe free surface area of the fuel. The surface area-to-mass ratio of the fuel typically varies between0.1 and 0.4 m2/kg for larger wood cribs andconventional furniture, and more often variesbetween 0.12 and 0.18 m2/kg.40

2. For hydrocarbon-based fuels, s can be calculatedas follows:

ep051 03

where

and .

Babrauskas46 suggests that for wood fuels s = 5.7. Harmathy39 notes that a typical woodwould have the chemical formulaCH1.455O0.645•0.233H2O, which would result ina value of s of 6.0.


4. Properties of liquid fuels: SFPE Handbook ofFire Protection Engineering.32



7. For design purposes, a value of 0.9 should beassumed for bp since this would result in themost conservative prediction of T. θ5 is onlyrelevant if the theoretical heat of combustion isused. If an effective heat of combustion is used,e.g., “chemical” heats of combustion fromTewarson,33 θ5 = 1.0.

28



Predictions using Babrauskas’ method boundedthe average temperatures measured in the CIB testsfor ventilation-controlled fires but underpredictedaverage temperatures for fuel-controlled fires. Forventilation-controlled fires, Babrauskas’ burningrate prediction falls in the scatter of points. Theburning rate data can be bounded by multiplyingBabrauskas’ prediction by a factor of 1.3 and bydividing by a factor of 2.3.

Babrauskas’ method reasonably predicted peaktemperatures but underpredicted burning duration inall of the Cardington tests; however, predictions ofburning rate improved as increased.

Ma and Mäkeläinen

Ma and Mäkeläinen developed a parametric tem-perature–time curve for compartments that are smallor medium in size (floor area < 100 m2). The methodwas developed for use mainly with cellulosic fires.Their aims were to develop a simple calculation pro-cedure that would reasonably estimate the tempera-ture, with time, of a fully developed compartment fire.

Ma and Mäkeläinen noted that fires generallyonly impact the structures during the fully devel-oped and decay stages. Theydeveloped a general shapefunction to define the tempera-ture history of a compartmentfire that is a function of fuelloading, ventilation conditions,and geometry and materialproperties of the compartment.

The general shape functionwas developed by non-dimen-sionalizing temperature–timedata from 25 different data setsand was based on the maxi-mum gas temperature, Tgm,and the time to reach the maxi-mum temperature, tm. Thenon-dimensionalized data col-lapses to the general shapeshown in Figure 9.

The shape of the curve is determined using thefollowing equation, and an appropriate value for theshape constant, δ. The recommended values for theshape constant are 0.5 for the ascending phase and1.0 for the decay phase. These values produce acurve that encompasses a majority of the experimen-tal data. It is reported, however, that values for theshape factor of 0.8 for the ascending phase and 1.6for the descending phase provided a best-fit curve tothe data.47 Both curves are shown in Figure 9.

Where:T = Temperature in compartment (°C)To = Ambient temperature (°C)Tgm = Maximum temperature in compartment (°C)t = Time (min)tm = Time corresponding to maximum gas

temperature (min)

tm =

mf = Mass of fuel (kg)= Mass burning rate of fuel (kg/min)

δ = Appropriate shape constant of thetemperature–time curve discussed above

29

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Tem

per

atu

re R

atio

(T

g/T

gm

) δ = 0.5, 1.0δ = 0.8, 1.6

Time Ratio (t / tm)

FIGURE 9. Non-Dimensionalized Temperature–Time CurvesDeveloped by Ma and Mäkeläinen47

Where:A = Surface area of interior of enclosure (m2)Ao = Area of ventilation opening (m2)Ho = Height of ventilation opening (m)D = Depth of compartment (m)W = Width of wall containing ventilation

opening (m)

For fuel-controlled fires, Ma and Mäkeläinenuse Harmathy’s correlation for the burning rate offuel-controlled fires:

Where:Af = Surface area of fuel (m2)

For furniture, the value for Af /mf is generallybetween 0.1 and 0.4 m2/kg; however, the mostcommon value is between 0.12 and 0.18 m2/kg,and 0.131 represents the value obtained from aseries of Japanese tests.47 The maximum gas tem-perature is determined using

with the maximum fire temperature in the criticalregion, Tgmcr, determined by

and the value of ηcr determined using

Where:Af = Surface area of fuel (m2)g = Gravitational constant (9.81 m/s2)

= Ratio of floor area to the totalcompartment surface area

m"f = Mass of fuel per unit area (kg/m2)ρ0 = Density of air (kg/m3)

The shape function is based on 25 experimentaldata sets whose key parameters, fuel load density,ventilation factor, thermal boundary properties, androom dimensions varied between experimentalstudies. The ranges for each of these parameters arelisted in Table 2.

Data Requirements

1. Ratio of floor area to total surface area2. The mass of fuel per unit area, m"f3. The area and height of the enclosure opening(s),


including the floor and openings, A, and thewidth, W, and depth, D, of the enclosure

5. The surface area-to-mass ratio of the fuel, Af /mf

Data Sources

1. The surface area-to-mass ratio of the fuel typi-cally varies between 0.1 and 0.4 m2/kg for largerwood cribs and conventional furniture, and moreoften varies between 0.12 and 0.18 m2/kg.40

30

Property Range Units

Fuel load density, m"f 10 – 40 kg/m2

Ventilation Factor, 5 – 16 m5/2

555 – 1800 J/m2 s1/2 K

Compartment floor < 100 m2

area, Afloor

Maximum height, H < 4.5 m

Shape of 0.5 – 2.0compartment, (W/D)

TABLE 2. Range of Values for Key Parametersfrom the 25 Data Sets Used to Develop theShape Function

For ventilation-controlled fires, Ma and Mäkeläinen use Law’s correlation to describe theduration of the fully developed stage:

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.




Predictions made using Ma and Mäkeläinken’smethod’s maximum temperature predictions boundedthe average temperatures measured in the CIB testsfor ventilation-limited fires but underpredicted aver-age temperatures for fuel-limited fires. Given thatmaximum temperature predictions using Ma andMäkeläinken’s method were compared to the CIBdata, which represented the average temperaturesmeasured during the fully developed stage, andpredictions of average temperature would be lowerthan average temperatures, Ma and Mäkeläinken’smethod would underpredict much of the CIBtemperature data.

Ma and Mäkeläinken’s method reasonablypredicted average temperatures and duration in Cardington tests #1, 2, 8, and 9

; however, as

increased, predictions increasingly deviated from

the test data. See the conclusions regardingHarmathy’s and Law’s methods for an evaluation of burning rate predictions.

CIB

In 1958, under the auspices of CIB W014, labora-tories from several countries agreed to investigate thefactors that influence the development of enclosurefires.48 Compartments with dimension ratios of 211,121, 221, and 441 (where the first number denotescompartment width, the second number denotes com-partment depth, and the last number denotes com-partment height) with length scales of 0.5 m, 1.0 m,and 1.5 m were analyzed. A total of 321 experimentswere conducted in still air conditions. The fuel load-ing (m"f ) in the compartments ranged from 10 to 40 kg/m2 of wood cribs with stick spacing to stickwidth ratios of 1/3, 1, and 3. Test data was modifiedthrough statistical analysis to account for systematicdifferences between test laboratories.

Average temperature and normalized burning rate

were presented as a function of in graphical

form (A was defined to exclude the area of theventilation opening and the floor area). Separategraphs were presented for cribs with 20 mm thickwood sticks spaced 20 mm apart, and for cribs with20 mm wide sticks spaced 60 mm apart, or with10 mm wide sticks spaced 30 mm apart. Becausethe cribs with 20 mm thick wood sticks spaced 20 mm apart resulted in higher compartment tempera-

tures and lower normalizedburning rates (and, hence,longer predicted burning dura-tions), these graphs are recom-mended for design analysisand are presented here.

Figure 10 shows the averagecompartment temperature dur-ing the fully developed burningstage, where “fully developedburning” was defined as theperiod where the mass of fuelwas between 80% and 30% ofthe original, unburned fuelmass. The line represents abest-fit through the data.

31

FIGURE 10. Average Temperature During Fully Developed Burning

200

400

600

800

1000

1200

0 10 20 30 40 50

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

CIB DataCIB Curve

0

Figure 11 shows theburning rate, , duringthe fully developed burn-ing stage, normalized bythe ventilation factor

and the squareroot of the ratio of com-partment depth to width(where the width is thedimension of the wallcontaining the ventilationopening).

To apply these graphs in a design context, first calculate the factor

and use

Figure 10 to determine theaverage gas temperature.Then use Figure 11 to determine the normalizedburning rate. This normalized burning rate can bere-dimensionalized by multiplying by the ventila-tion factor and dividing by the square rootof the ratio of compartment width to depth. Theduration of burning can be determined by dividingthe total mass of fuel, mf , by the burning rate.

Data Requirements

1. The total mass of fuel, mf2. The area and height of the enclosure opening(s),


excluding the floor and openings, A, and thewidth, W, and depth, D, of the enclosure

Data Sources





The averaged Cardington temperature data fallsin the same range as the CIB temperature data for

values of less than 30. However, once the

opening factor exceeds 30 m–1/2 the CIB tempera-ture graph underpredicts temperature, and the CIBdata has much lower values than the Cardington data.

As a curve fit through data, the CIB temperaturegraph reasonably predicted the aggregate of all CIBtemperature and burning rate data but underpredictedsome experiments and overpredicted others.

Using the CIB graphs resulted in reasonablepredictions of average temperature and duration inCardington tests #1, 2, 8, and 9, and reasonable pre-diction of duration but underprediction of tempera-ture in Cardington tests #3 and 4. In test #7, usingthe CIB graphs resulted in reasonable predictions ofduration but underprediction of temperatures. Due to

the large values of in Cardington tests #5

and 6, predictions were not possible using the CIB graphs.

32

A/AoH

o1/2 (m–1/2)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 10 20 30 40 50 60

. mf/A

oH

o1/

2 (D

/W)1/

2 (k

g/s

– m

5/2 )

121221211441Curve Fit

FIGURE 11. Normalized Burning Rate During Fully Developed Burning

33

Law

Law derived a method of predicting compartmenttemperatures resulting from fully developed firesbased on data from tests conducted under theauspices of CIB. Law’s method takes into accountthe geometry of the compartment. The area of thecompartment’s lining surface through which heat islost is expressed by subtracting the vent area fromthe total interior compartment surface area (A – Ao);the temperature in the compartment is thereforedependent on A, as well as variables incorporated inthe ventilation factor, Ao, and H.

Law derived the following equation to determinethe maximum temperature of the compartment withnatural ventilation49:

Where:Tgm = Maximum compartment temperature (°C)A = Surface area of interior of enclosure (m2)Ao = Area of ventilation opening (m2)Ho = Height of ventilation opening (m)

This equation does not account for the effects oncompartment temperature due to fuel loading. Itsimply represents the maximum temperatureachieved in a compartment for a given geometryand ventilation. The following equation incorporatesthe effect of fuel loading on the temperature and isvalid for wood-based fuels:

Where:

The mass loss rate is correlated as

Where:= Mass burning rate of fuel (kg/s)

W = Length of wall containing ventilationopening (m)

D = Depth of compartment (m)

The duration of burning can be calculated bydividing the total mass of combustibles by the burn-ing rate as follows:

Where:τ = Burning duration (s)

Data Requirements

1. The total mass of fuel, mf2. The area and height of the enclosure opening(s),


including the floor and openings, A, and thewidth, W, and depth, D, of the enclosure

Data Sources





Without applying the adjustment factor Ψ, Law’stemperature predictions bounded all the CIB data.Law reasonably predicted the CIB burning rate data,and, if the burning rate was adjusted by a factor of 1.4, Law bounded all the CIB burning rate data.

34

Law reasonably predicted average temperatureand duration for Cardington tests #1, 2, 8, and 9

.

For Cardington tests #3, 4, 5, and 6

,

Law reasonably predicted duration but under-predicted temperature. In test #7

,

which was square in plan view, Law’s methodunderpredicted temperatures but predicted theburning duration.

Simple Decay Rates

Many of the methods cited previously do not contain amethod of estimating thecompartment temperatureduring the decay stage. Forthese methods, a number ofsimple decay rates can beapplied if the engineer wishesto account for heating thatoccurs during the decay phase.

Decay cannot be modeled by basic physics because the“decay rate” is actually theheat transfer from the compart-ment and the heat release rateof combustibles that have charred and collapsed onto the floor, with poor access ofoxygen and therefore limited heat release rate. Allmethods are wholly empirical.

The simplest way to determine the temperature–time profile during the decay phase is to use a fixedrate of temperature decay. Originally, the tempera-ture decay during the cooling phase was selectedarbitrarily. Kawagoe first suggested that the rate oftemperature decrease during the cooling period wasa function of the fire duration, reporting values of7°C/min for fire durations greater than 60 minutesand 10°C/min for fire durations less than 60 min-utes.50 The pioneering work of Magnusson andThelandersson indicated that the rate of decrease in

temperature was a function of the fire duration andopening factor.51,52,53 Magnusson and Thelanderssonreported that for shorter duration fires the rate oftemperature decrease was higher than 10°C/min,while for longer duration fires the rate of tempera-ture decrease was lower than 10°C/min.38 Based ona series of short-duration fires, Harmathy reporteddecay rates between 15 and 20°C/min, which areconsistent with the results presented by Magnussonand Thelandersson. Typical rates found in the litera-ture are listed in Table 3.

In the absence of better information, it would beappropriate to select a decay rate of 7°C/min forfires with a predicted duration of 60 minutes ormore, and a decay rate of 10°C/min for fires with apredicted duration of less than 60 minutes, sincethese rates would result in the slowest decay ratesaccording to the above.

RECOMMENDATIONS

Based on comparison of predictions to the datafrom the CIB and Cardington tests, Law’s method isrecommended for use in all roughly cubic compart-ments (compartment width to depth ratio within therange of 0.5 to 2.0) and in long, narrow compart-

ments where does not exceed ≈ 18 m–1/2.

To ensure that predictions are sufficiently conserva-tive in design situations, the predicted burning rateshould be reduced by a factor of 1.4, and the tem-perature adjustment should not be reduced by thefactor Ψ. See Figures 12 through 17, which show

TemperatureDecay

(°C/min) Restrictions Reference

10 τ < 60 min Kawagoe

7 τ > 60 min Kawagoe

>10 τ < 60 min Magnusson and Thelandersson

<10 τ > 60 min Magnusson and Thelandersson

10 No restrictions Swedish Building Regulations

15 – 20 Short-duration fires Harmathy

TABLE 3. Rate of Decrease in Temperature

35

comparisons made using Law’s method to the CIB data and to data for Cardington tests

#1 , #2 ,

#8 , and #9 .

Law’s method does not predict temperatures dur-ing the decay stage. For cases where a prediction oftemperatures during the decay stage is desired, adecay rate of 7°C/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.

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

CIB DataLaw (max)

A/AoH

o1/2 (m–1/2)

Tem

per

atu

re (

°C)

FIGURE 12. Comparison of CIB Temperature Data to Predictions UsingLaw’s Method

A/AoHo1/2 (m–1/2)

. mf/A

oH

o1/

2 (D

/W)1/

2 (k

g/s

– m

5/2 )

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60

121221211441Law X 1.4

FIGURE 13. Comparison of CIB Burning Rate Data to Predictions Using Law’s Method

36

Cardington Test #1

Time (h)

Tem

per

atu

re (

°C)

MeasuredLaw Adjusted

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

FIGURE 14. Comparison of Predictions Using Law’s Modified Method for Cardington Test #1

Tem

per

atu

re (

°C)


Cardington Test #2

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


37

Tem

per

atu

re (

°C)


Time (h)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)


Time (h)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


For long, narrow spaces in which

is in the range of 45 to 85 m–1/2, Magnusson andThelandersson provide reasonable predictions of temperature and duration. See Figures 18 through 20, which show comparisons made usingMagnusson and Thelandersson’s method to data

for Cardington tests #3 ,

#4 , and

#5 .

38

Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

Measured

Magnusson (Type C)

FIGURE 18. Comparison of Predictions from Magnusson and Thelandersson’s Method (Type C) toData for Cardington Test #3

39

Time (h)

Tem

per

atu

re (

°C)

MeasuredMagnusson (Type C)

0

200

400

600

800

1000

1200

1400

0 1 2 3 4


Time (h)

Tem

per

atu

re (

°C)

Measured

Magnusson (Type C)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3


For long, narrow spaces in which is

approximately 345 m–1/2, Lie’s method is recom-

mended. Note that this value of is outside

Lie’s stated range of applicability. However, based

on comparison of predictions to the Cardington data,its use is still recommended. See Figure 21, whichshows comparisons made using Lie’s method to

data for Cardington test #6 .

40

Measured

Lie

Tem

per

atu

re (

°C)

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8

Time (h)

FIGURE 21. Comparison of Predictions from Lie for Cardington Test #6

Fire Exposures from PlumesThis section of the guide focuses on predicting

the heat transfer from area exposure fire plumes toadjacent surfaces. Area exposure fires are burningobjects or fuel located adjacent to or near the sur-face being heated. For certain scenarios, the localfire exposure may produce a more extreme expo-sure than the hot gas layer that develops in the areaof consideration. Some examples are open parkinggarages, large warehouses, and bridges and over-passes. To analyze these scenarios, one needs tohave knowledge of the incident heat flux levels pro-duced by local fire plumes.

The boundary condition between the fire plumeand the structural element needs to be properlydefined in order to predict the temperature of thestructural element with time. This part of the guide

develops heat transfer boundary conditions for twodifferent types of exposure:

1. Bounding, or elements immersed in a fire plume2. Specific geometries, or specific element shapes

and orientations

Detailed modeling of the fire from first principlescan also be conducted to predict the boundary condi-tion; however, this type of analysis is not addressedin this guide. If detailed modeling is conducted, the model should be verified with existing data forsimilar configurations to validate predicted heatfluxes. Some additional data on gas temperaturesand velocities generated by a fire plume are alsoincluded to aid in this type of modeling effort.

The fire exposure recommended for a boundinganalysis will consist of a constant fire exposure. If a

more refined analysis is required, guidance is pro-vided on how to predict the boundary conditionwith the fire in specific geometries. These geome-tries include the following:

• Flat vertical walls• Corners with a ceiling• Unbounded flat ceilings• I-beam mounted below a ceiling

The boundary condition in these configurations isbased on experimental data and may be limited tothe conditions tested in the study.

Studies have also been conducted to measure the heat flux boundary condition with fires in otherconfigurations. Lattimer54 provides a review ofexisting incident heat flux data and correlations forexposure fires and burning surfaces in a variety ofconfigurations including flat walls, corners, cornerswith a ceiling, parallel flat walls, walls above a win-dow containing a fire plume, unbounded ceiling,and an I-beam under a ceiling.

AXISYMMETRIC FIRE PLUMES

The simplest fire plume is the unconfinedaxisymmetric fire plume, shown inFigure 22. Correlations for velocity andtemperature produced by an axisymmetricplume are provided in this section to aidthose in modeling the heat flux to elementsfrom first principles. Unconfined axisym-metric fire plumes are typically approxi-mated as point heat sources when estimatingthe local velocity and temperature profile.This section describes how to estimate thelocation of the virtual point source relative tothe base of the fire, the flame height, and thevelocity and temperature distribution withinthe fire plume.

Velocity Profile

The velocity profile of the fire plume is afunction of the elevation above the virtualorigin and the distance from the plume cen-terline. The velocity at the plume centerline

can be calculated using the following equation forregions above the average flame height, Lf:

55

(Eq. 40)

Where:Um,c(Z) = Centerline plume velocity (m/s)χr = Fraction of energy released as

radiation in the fire= Fire heat release rate (kW)

Z = Target elevation above the base of the fire (m)

zo = Elevation of the virtual origin relativeto the base of the fire (m)

The centerline plume velocity for regions belowthe average flame height may be determined usingEquation 4156 where all terms have been defined.

(Eq. 41)

41

r

buPlumeRegion

FlameRegion

Lf

D

Zo Z

FIGURE 22. Axisymmetric Fire Plume

The virtual origin may be calculated using Equa-tion 42 where D is the effective fire diameter (m)57:

(Eq. 42)

For noncircular fuel packages with a length to widthratio of near one, the equivalent diameter of the fuelpackage can be estimated using the surface area, A,of the noncircular fuel package:

(Eq. 43)

Where:A = Surface area of the fuel package (m2)

The average flame height can be calculated usingthe relation developed by Heskestad58:

(Eq. 44)

Where:= Heat release rate of the fire (kW)

D = Diameter of the fuel package (m)

The velocity distribution within a fire plume has been found to fit a Gaussian profile, though notheoretical grounds exist for this.55,59 The followingequation may be used to determine the velocity as a function of the distance from the plumecenterline55:

(Eq. 45)

Where:U(r) = Velocity in plume at a distance r (m)

from the centerline (m/s)bu = Plume width parameter (m)

The plume width parameter is found viaEquation 46 where all terms have been defined.55

(Eq. 46)

Temperature Profile

The temperature profile is also a function of theelevation above the plume virtual origin and the

42

800

900

1000

1100

1200

1300

1400

1500

1600

0 0.1 0.2 0.3 0.4 0.5

Nat. Gas 0.3 mHeptane 1.7, 6Methanol 1.7Kerosene 30JP-4 15

Max

. Tu

rbu

len

t F

lam

e T

emp

erat

ure

(°C

)

Fire Plume Radiation Fraction, Xr

FIGURE 23. Maximum Turbulent Fire Plume Temperatures from VariousSources61,62,63,64

distance from the plume centerline. The centerlinetemperature may calculated using Equation 47 forelevations above the average flame height55:

(Eq. 47)

Where:Tm,c(Z) = Centerline plume temperature (K)T∞ = Ambient temperature (K)

The centerline plume temperature for elevationsbelow the average flame height may be determinedusing the following where all terms have beendefined56:

(Eq. 48)

(Eq. 49)

These temperatures represent average tempera-tures in the flaming and plume regions, and theywill tend to be higher when the radiative fraction,χr, of the fire is decreased. For turbulent fireplumes, having a radiative loss fraction χr, theturbulent flame (centerline) temperature follows therelationship60

(Eq. 50)

From the best available data,61,62,63 the turbulentmixing parameter, kT is found to be about 0.5 for cp = 1 kJ/kg-K. As the fire diameter increases, theradiative fraction falls due to soot blockage.64

Figure 23 shows flame temperature data for turbu-lent plumes as a function of χr. The extrapolatedadiabatic temperature is about 1500°C.

Temperatures have been measured to be as lowas 820°C for flames produced by fuels with a radia-tive fraction of χr ~ 0.20.65 Thus, Equations 48 and49 correspond to fires of χr ≈ 0.3.

As the pool fire diameter is increased, flamesproduce more soot, reducing the flame radiationbeing emitted to the surroundings. From the SFPEEngineering Guide on Assessing Flame Radiation toExternal Targets from Pool Fires66 and Beyler,67

radiative fraction will decrease linearly from anaverage radiative fraction of 0.22 for a small- (~0 m) diameter pool fire to approximately 0.04 fora 50 m diameter pool fire. Baum and McCaffrey68

clearly showed the dependence of gas temperatureon diameter, with measured gas temperatures ashigh as 1000°C for 6 m diameter fires and 1250°Cfor 30 m diameter fires. These data are represented

in Figure 23.The temperature dis-

tribution as a function ofthe distance from theplume centerline also fits a Gaussian profile.55

Equation 51 can be used todetermine the temperatureat any distance r (m) fromthe plume centerline55:

(Eq. 51)

Where:bt = Thermal plume width parameter (m)

The thermal plume width parameter may becalculated using Equation 52 where all terms havebeen defined55:

(Eq. 52)

Data Requirements

1. Source fire heat release rate, (kW)2. Radiative fraction, χr3. Elevation above source fire, Z (m)4. Radial (horizontal) separation from centerline of

source fire, r (m)

43

Data Sources

1. Heat release rate data may be obtained fromBabrauskas,69 Hoglander and Sundstrum,70 orMudan and Croce.71

2. Radiant fraction data may be obtained fromTewarson.72

Assumptions

1. The axisymmetric fire plume may beapproximated as a point heat source. Thisassumption is valid for many types of firesincluding pool fires, but may yield poor resultsfor three-dimensional burning objects (i.e., sofa),momentum-driven plumes (jets), or regions nearthe base of the fire.

2. The effect of a hot smoke layer formation in acompartment on the temperature and velocityprofiles in a fire plume is ignored. Refer toEvans73 and Cooper74 for a discussion of hotlayer–plume interactions.

3. There is no air movement (wind, vent flows) inthe vicinity of the plume. Such air motions maycause a plume to deflect.

Validation

There have been numerous experiments on thecenterline temperature and velocity in fire plumes.The form of the correlations is generally identical;however, there is some variation among the corre-lated constants.55,75 Those presented in this sectiontend to be conservative in terms of predicting thegreatest velocity and centerline temperature for agiven heat release rate and target elevation.

Limitations

The fire plume equations in this section arelimited to open, axisymmetric thermal plumes in aquiescent environment. The source fire should havea relatively square plan area, though fuel packages orsource fires with aspect ratios on the order of two orthree may be acceptable. Larger aspect ratios couldresult in a line fire. Refer to Quintiere and Grove76

for a discussion of line fire thermal plumes.

HEAT FLUX BOUNDARY CONDITION

The governing boundary condition for a fire heat-ing an adjacent surface is determined using the heatbalance shown in Figure 24 to be

(Eq. 53)

assuming negligible heating from the surroundingenvironment (i.e., no hot gas layer heating). Toapply this relation directly, the local gas tempera-ture, Tf , local heat transfer coefficient, h, and theemissivity of the gases, εf , must be known. The sur-face absorbtivity, αs, and emissivity, εs, must alsobe known, but approach 1.0 as they become sootcovered. All these parameters are scenario depend-ent, and all are not readily known or predicted. As aresult, several research efforts have been conductedto measure the total incident heat flux to a surfacein a variety of configurations. This is typically doneusing cooled total heat flux gauges. These gaugesare cooled so that their surface temperature remainsnear ambient and are coated with a high-emissivitypaint to maximize the absorbed radiation. By setting the surface temperature to the ambient inEquation 53, the boundary condition at the totalheat flux gauge is represented by Equation 54:

(Eq. 54)

44

FIGURE 24. Heat Balance at theMaterial Surface

Cooling the gauge surface maximizes the convec-tive heat transfer and minimizes the radiative losses;thus, the cooled heat flux gauges measure the maxi-mum total incident heat flux. Assuming that thesurface absorbtivity and emissivity are identical,and the emissivity of the heat flux gauge is similarto that of the material surface (εs,hfg ≈ εs), the totalincident heat flux measured using the heat fluxgauge, Equation 54, is related to the actual heat fluxthrough the following relation,

(Eq. 55)or

(Eq. 56)

Therefore, measuring the heat flux has removedthe need to predict both the gas temperature and theemissivity of the gases. To get the actual net heatflux into the surface from the measured heat flux, asurface temperature correction needs to be appliedas done in Equation 56.

A conservative estimate of thenet heat flux into the structuralelement can be determined byeither not applying any surfacetemperature correction or onlyapplying the radiative correction.A closer estimate of the actualnet heat flux into the surfacewould include both radiative andconvective corrections. Applyinga convective correction involvesestimating the local heat transfercoefficient, h, which is depend-ent on the local velocity and gas temperature.

Local heat transfer coeffi-cients may range from 0.015 to0.030 kW/(m K) for hot gas flowup a wall or along a ceiling. Atpoints where hot gases impingeon a surface, this value may behigher. Based on data fromKokkala77,78 and You andFaeth,79,80 the local convectiveheat transfer coefficient where a

diffusion flame impinges on a ceiling is on the orderof 0.050 kW/(m-K). Figure 25 contains plots of theradiative correction for different element surfacetemperatures along with convective correction forconvective heat transfer coefficients of 0.015 and0.050 kW/(m K). From Equation 56 and Figure 25,overestimating the convective correction will resultin a non-conservative boundary condition. There-

fore, a convective heat transfer correctionis only recommended in simple configura-tions where local heat transfer coefficientscan be calculated (e.g., flat walls).

BOUNDING HEAT FLUX:OBJECTS IMMERSED IN FLAMES

The simplest and most conservative way to treatthe fire exposure boundary condition would be toapply a constant, bounding heat flux to all structuralelements in the area of interest. The bounding heatflux boundary condition was developed from dataon objects immersed in large hydrocarbon poolfires. The heat flux data for objects immersed in

45

Surface Temperature (°C)

0 100 200 300 400 500 600 700 800 900 1000

Hea

t F

lux

[kW

/m

0

10

20

30

40

50

60200 400 600 800 1000 1200 1400 1600 1800

Surface Temperature (°F)

FIGURE 25. Magnitude of the Surface Temperature Correctionson the Measured Total Heat Flux Using a Cooled Gauge (seeEquation 56). Radiation (—), Convection with h = 0.015 kW/(m K) (– . .–), and Convection with h = 0.050 kW/(m K) (– – –).

Hea

t F

lux

(kW

/m2)

fires are presented in this section and used to deter-mine the magnitude of the bounding heat flux. Theinformation in this section is derived primarily fromdirect or indirect measurements of heat flux taken inopen hydrocarbon pool fires with optically thickflames. There is insufficient data available at thistime to adequately address the impact of a boundarysuch as a wall or ceiling on the heat flux conditionsto an immersed object. It is expected that the dataobtained from optically thick flames in unconfinedpool fires is bounding.

Test Data

A series of 30-minute, 9.1 m by 18.3 m hydrocar-bon pool fires (JP-4) conducted by Gregory, Mata,and Keltner81 provided useful temperature and heatflux data at various elevations above the base of thefire. Steel cylinders filled or lined with insulation(referred to as small or large calorimeters, respec-tively) at several locations were used to indirectlymeasure the net heat flux for objects immersed inthe fire. The temperature inside the cylinder wasrecorded, and the net heat flux was extracted usingthe inside temperature as a boundary condition. The

measurements were taken at various elevations andangular positions on the calorimeters. The cold-wall(i.e., peak) heat fluxes to the large calorimeter variedbetween 100 kW/m2 and 160 kW/m2 at any one loca-tion, with the largest peak heat fluxes observed onthe underside and the lowest on the top. Figure 26shows the average peak heat flux at various angularpositions as a function of the external surface tem-perature of the large calorimeter, which increases asa function of time, and the angular position.

The cold-wall fluxes to the small calorimetervaried between 150 kW/m2 and 220 kW/m2. As withthe large calorimeter data, the maximum heat fluxeswere observed on the bottom of the calorimeter andthe minimum were observed on the top. There wasno decrease in the cold-wall heat flux detected overthe elevation range (1 to 11 m) sampled.

Russell and Canfield82 immersed a steel cylinderin a 2.4 m by 4.9 m JP-5 pool fire in windy condi-tions. The inside surface temperature of the cylinderwas directly measured, and the exposure heat fluxwas determined in the same manner as Gregory,Mata, and Keltner.81 The peak heat fluxes to thesurface of the cylinder were measured at variousangular positions. The peak heat fluxes ranged from

46

External Surface Temperature of Large Calorimeter (K)

400 600 800 1000

Ave

rag

e H

eat

Flu

x (k

W/m

2 )

0

20

40

60

80

100

120

140

BottomTopLeft SideRight Side

FIGURE 26. Averaged Peak Heat Flux as a Function of Angular Position

18 kW/m2 on the windward side to 144 kW/m2 onthe leeward side. The heat fluxes on the top andbottom of the cylinder were 48 kW/m2 and103 kW/m2, respectively.

Cowley83 summarized the peak heat fluxesmeasured directly or indirectly to objects immersedin various large-scale pool fires. The values rangebetween 80 kW/m2 and 270 kW/m2. Table 4 sum-marizes some of this information. Cowley speculatesdifferences between low- and high-volatile fuelswith heat fluxes as high as 300 kw/m2 are possiblein the latter.

Most of the heat flux test data suggest a bound-ing cold-wall heat flux between 150 kW/m2 and170 kW/m2. Although some data (small calorimeter)indicate that the peak may be as high as 220 kW/m2,these appear to be exceptional.

The heat flux in a flame increases with firediameter and where the object or flame impinge-ment is located. The upper bound of heat flux canbe calculated as follows:

(Eq. 56a)

Data Requirements

The flame temperature is needed to performthis calculation.

Data Sources

For pool fires, the radiative fraction can be deter-mined as a function of pool diameter from the SFPEEngineering Guide to Assessing Flame Radiation toExternal Targets from Pool Fires. This radiativefraction can be substituted into Figure 23 to esti-mate the flame temperature. For noncircular poolswith a length-to-width ratio of near one, the equiva-lent diameter of the pool can be estimated using thesurface area, A, of the noncircular pool:

(Eq. 56b)


Assumptions

1. The flame emissivity and surface absorbtivity areequal to 1.0.

2. The impact of a compartment on the heat fluxesat the surface of an immersed object can beignored.

3. Reduction in net heat flux due to heating of thetarget is not considered.

Validation

Equation 56a is based on firstprinciples. Heat fluxes calculatedusing Equation 56a are much largerthan measured heat fluxes. Forexample, Baum and McCaffrey68

reported gas temperatures as highas 1250°C in 30 m diameter poolfires. Assuming that the gases are optically thick, emissivity of 1.0, the cold-wall heat flux is305 kW/m2. As seen in Table 4,measured values are less than thisvalue, indicating that the assumedemissivity may be significantlyless than 1.0 or the effective gastemperatures providing the radia-tion are lower than measured orreported temperatures.

47

PeakHeat Flux

Test Pool Size Fuel (kW/m2)

AEA Winfrith84 0.49 x 9.4 m Kerosene 150

US DOT84 Not listed Kerosene 138

USCG84 Not listed Kerosene 110-142

US DOT84 Not listed Kerosene 136-159

Sandia84 Not listed Kerosene 113-150

HSE Buxton84 Not listed Kerosene 130

Shell Research84 4.0 x 7.0 m Kerosene 94-112

Large cylinder82 9 x 18 m JP-4 100-150

Large cylinder82 9 x 18 m JP-4 150-220

Russell and Canfield83 2.4 x 4.9 m JP-5 144

TABLE 4. Selected Heat Fluxes to Objects Immersed in Large Pool Fires83

Limitations

The results of this section are limited to Class A(plastic or wood-based) combustible material firesor hydrocarbon pool fires. Gaseous jet flames arebeyond the scope of this section because they mayproduce larger cold-wall (200 to 270 kW/m2) heatfluxes to immersed objects.83

The results are also not applicable to objects thatare located near (collocated), but not in, the burningregion. Methods of estimating the incident heat fluxto collocated objects are available in anotherEngineering Guide.66

HEAT FLUXES FOR SPECIFIC GEOMETRIES

The incident heat flux from a fire plume to asurface is dependent on:

• Geometry• Dimensions of the fire• Fire heat release rate• Effective radiative path length• Soot production rate

Research has been conducted to evaluate the effectsof each of these variables on the incident heat fluxfrom a fire. However, a general engineeringapproach has not been developed for predicting the incident heat flux from a fire to an adjacent sur-face. This section provides empirical correlationsfor estimating the heat flux boundary condition insome specific geometries. These correlations weredeveloped over a specific range of fire source size,heat release rate, and geometry, which limits theirgeneral applicability.

The heat transfer from a flame to an adjacent sur-face or object has historically been characterizedwith respect to the flame length. Many of the heatflux correlations developed in the literature arebased on flame length data taken in a particularstudy. Measured flame lengths can vary dependingon the measurement technique, definition, and sur-rounding geometry. For the studies considered inthis section, the data were nondimensionalized witheither the average (50% intermittent) flame length

or the flame tip length. Therefore, heat flux correla-tions should be applied using either the flame lengthcorrelation developed in the study or with one thathas been demonstrated to predict the flame length inthat study.

Flat Vertical Walls

The simplest geometry is with the fire directlyagainst a flat wall as shown in Figure 27. Correla-tions are developed in this section to estimate thevertical and horizontal variation in the heat flux tothe wall due to a fire in this configuration.

Correlations to estimate the incident heat fluxfrom an exposure fire against a flat wall have beendeveloped through an experimental study performedby Back et al.84 In this study, fires were generatedusing square propane sand burners with edge lengthsof 0.28, 0.37, 0.48, 0.57, and 0.70 m. Heat flux fieldswere measured for fires ranging from 50 to 520 kW.The flame height to burner diameter aspect ratioranged from approximately 1 to 3 in these tests.

48

Wall

Lf

A

Z

FIGURE 27. Fire Against a Flat Vertical Wall

The average flame length of fires against a flatwall was determined to be equal to the averageflame length of unconfined fire plumes. Flamelengths can be calculated using the relation devel-oped by Heskestad58:

(m) (Eq. 57)

Where:= Heat release rate of the fire (kW)


Flame lengths are taken relative to the base of thefire. For noncircular fuel packages with a length towidth ratio of near one, the equivalent diameter ofthe fuel package can be estimated using the surface area, A, of the noncircular fuel package:

(Eq. 58)


A plot of the peak heat fluxes measured for eachof the fires considered in the study is shown inFigure 28. Peak heat fluxes for the different firesevaluated were determined to be a function of thefire heat release rate. This dependence was attributedto the larger size fires resulting in thicker boundarylayers on the wall, thus increasing the radiation pathlength. Based on gray-gas radiation theory, theauthors found the following relation adequatelyrepresented the data:

(Eq. 59)

These peak heat fluxes were measured in thelower part of the fire (z/Lf ≤ 0.4) along the center-line. Above this region, the heat fluxes were

49

Heat Release Rate (kW)

0 100 200 300 400 500 600

.P

eak

Hea

t F

lux,

q" p

eak

(kW

/m2 )

0

20

40

60

80

100

120

140

Aspect Ratio ~1Aspect Ratio ~2Aspect Ratio ~3

FIGURE 28. Peak Heat Release Rates Measured in Square PropaneBurner Fires Against a Flat Wall84

measured to decrease with distance above the fire,z. The heat flux data measured along the centerline

are shown in Figure 29. Lines in this plot are ageneral correlation of the centerline data:

50

(Eq. 60a)

(Eq. 60b)

(Eq. 60c)

z/Lf

0.01 0.1 1 10

.C

ente

rlin

e H

eat

Flu

x, q

" cl (

kW/m

2 )

1

10

100

1000

Q ≈ 59 kWQ ≈ 121 kWQ ≈ 212 kWQ = 313 kWQ = 523 kWCorrelation for Q = 59 kWCorrelation for Q = 523 kW

FIGURE 29. Vertical Heat Flux Distribution Along the Centerline of a SquarePropane Burner Fire Adjacent to a Flat Wall84

FIGURE 30. Horizontal Heat Flux Distribution (a) Below the Flame Height and(b) Above the Flame Height with Distance from the Centerline of the Fire84

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Hea

t F

lux/

CL

Hea

t F

lux

Hea

t F

lux/

CL

Hea

t F

lux

Distance/Burner Half Length(a)

Distance/Burner Half Length(b)

0 0.5 1.0 1.5 2.0 2.0 3.0 3.5 0 0.5 1.0 1.5 2.0 2.0 3.0 3.5

Heat fluxes were measured by Back et al.84 todecrease with horizontal distance from the center-line as shown in Figure 30. Significant heat fluxeswere measured as far as twice the burner radiusfrom the centerline. Conservatively, it can beassumed that the heat flux is equal to the centerlineheat flux at distances as far as twice the fire radiusfrom the centerline.

Data Requirements

1. Diameter of the fuel package, D. For noncircularfuel packages, the equivalent diameter may becalculated using Equation 58 and the surface areaof the fuel package.

2. Heat release rate of the fire, .3. Elevation along the flame length, z.

Data Sources


Assumptions

This analysis assumes that the fire is attached tothe wall and that the wall is vertical. Walls that arenot vertical may result in different total incidentheat flux levels due to the flame’s becoming sepa-rated from the wall or the difference in entrainmentinto the plume.

Validation

Some studies have made measurements of incidentheat fluxes from various burning objects to walls, butthe data is sparse. Incident heat fluxes at the rim ofwastebasket fires were reported by Gross and Fang.85

At the rim, heat fluxes as high as 50 kW/m2 weremeasured; however, the authors noted that peak heatfluxes for these fires occurred approximately 0.22 mabove the rim. Mizuno and Kawagoe86 performedexperiments with upholstered chair fires against aflat wall. In these tests, Mizuno and Kawagoe meas-ured heat fluxes to the wall of 40 to 100 kW/m2 overthe continuous flaming region (~z/Lf < 0.4). Allthese tests were performed using foam-padded chairs.These data do provide evidence that the magnitude

of the incident heat flux levels measured in thepropane burner experiments is consistent with fireproduced by burning items. In tests with propane gasburners against a non-combustible boundary, similarheat flux levels have been measured by other inves-tigators for limited conditions.87,88

Limitations

Correlations for incident heat fluxes were devel-oped using luminous flames in an open environmentwith the fire directly against a flat vertical wall.Using these relations inherently assumes:

• There is negligible heating from a hot gas layerin the surroundings.

• The fire is against the wall.• The flames are luminous.• The wall is vertical.

The experimental study considered fire diameters aslarge as 0.70 m and heat release rates as large as520 kW. No data was available to validate the corre-lations against fires with larger diameters or higherheat release rates. The presence of a hot gas layermay increase the total incident flux onto the wall,and if significant in the area of interest adding thiscontribution to the total incident heat flux from thefire plume may be warranted.89 Moving the fireaway from the wall will eventually cause the inci-dent heat fluxes to become lower, largely becausethe flame becomes detached from the wall.90 Thus,the use of correlations in this section for fires thatmay be slightly spaced from the wall will yieldconservative results. Flames less luminous thanthose produced by the propane fires (i.e., naturalgas) may transmit lower total incident heat fluxes tothe wall because the radiative heat flux to the wallwill be lower.87,88 Propane fuel fires used to developthe heat flux data presented in this section producea moderate amount of soot; therefore, heat fluxlevels presented in this section should be consideredto be average but not bounding for all differentfuels. Propane burners are also used extensively instandard fire tests as an exposure fire that is repre-sentative of real fires. Therefore, the incident heatfluxes from these flames are considered to be repre-sentative of those produced by most fires.

51

Fires in a Corner

Fires located in a corner geometry as shown inFigure 31 produce a more complicated flow field,particularly when a ceiling is present. As indicatedin Figure 31, fires in a corner rise vertically in thecorner until the gases impinge on the ceiling, atwhich point the fire will be redirected along theceiling and the top of the walls. Near the top of thewalls, flaming vortices will flow out from the cor-ner resulting in elevated heat fluxes along the top ofthe wall as much as twice the ceiling jet thickness.

Incident heat flux correlations in a corner with aceiling were developed by Lattimer et al.91 Thestudy was conducted using a 2.4 m high open cor-ner constructed of two walls and a ceiling. Fireswere produced using square propane burners havingsingle side lengths of 0.17, 0.30, and 0.50 m. Fires

were placed directly in the corner. The studyincluded fires with heat release rates ranging from50 to 300 kW.

Correlations were developed for the three regionsin the corner shown in Figure 32. The regions werethe corner walls on the lower part of the walls, thetop portion of the walls near the ceiling, and alongthe ceiling. The corner walls region extended fromthe fire to approximately 1.8 m above the floor.Above this region, the incident heat flux onto thewalls was measured to be affected by the hot gasesflowing along the ceiling. The distance of 1.8 m is approximately twice the ceiling jet thicknessbelow the ceiling or H – 2δ where H = 2.2 m andδ = 0.1H.92 Correlations for the top part of the walls,which are heated by the ceiling jet, were developedusing data from 1.8 m to 2.2 m or H – 2δ < z < H.

The flame length in the corner with a ceiling wastaken to be the flame length in the corner plus anyflame extension along the ceiling. The followingrelation can be used to calculate the flame tip lengthwith the fire in the corner:

(Eq. 61)

Where:

(Eq. 62)

= Heat release rate of the fire (kW)D = Diameter of the fuel package (m)ρ0 = Density of air at initial ambient conditions

(1.2 kg/m3)cp = Specific heat capacity of air at initial

ambient conditions [1.0 kJ/(kg K)]To = Temperature at initial ambient conditions

(293 K)g = Gravitational acceleration (9.81 m/s2)

Flame lengths are taken relative to the base of thefire. This correlation can be used to estimate flamelengths in a corner with or without a ceiling. Fornoncircular fuel packages with a length to widthratio of near one, the equivalent diameter of the fuelpackage can be estimated using the surface area, A,of the noncircular fuel package:

(Eq. 63)

52

Wall

Lf,w

A

Z

Wall

Ceiling

H

Lf,c r

X

FIGURE 31. Fire in a Corner Configuration


Walls at Corner

Correlations in this section can be usedto estimate the incident heat flux in thecorner with the fire. These correlations canbe used to estimate the incident heat fluxin a corner configuration with or without a ceiling. When a ceiling is present, thecorrelations are valid up to an elevation ofz = H – 2δ, where δ = 0.1H.92 Along theheight of the walls in the corner, the peakheat fluxes were typically measured nearthe base of the fire. These peak heat fluxeswere measured to be a function of the firediameter as shown in Figure 33. The curvein Figure 33 is a correlation to the dataand is expressed using Equation 64:

(Eq. 64)

Where:= Peak heat flux in the corner (kW/m2)


53

CornerWallsRegion

2DH

Z

D

Exposure Fire

Top of WallsRegion

X

Ceiling Above Corner

Cornerwith Fire

CeilingRegion

r

FIGURE 32. Corner with a Ceiling Configuration Showing the Three RegionsWhere Incident Heat Flux Correlations Were Developed in the Study ofLattimer et al.91

Length of Area Burner Side, D (m)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

.P

eak

Hea

t F

lux,

q" p

eak

(kW

/m2 )

0

10

20

30

40

50

60

70

80

90

100

110

120

FIGURE 33. Peak Heat Flux Along the Height of theWalls in the Corner. Data from Lattimer et al.91

The vertical distribution in the maximum heatflux along the walls near the corner is shown inFigure 34 plotted with the elevation above the fire, z, normalized with respect to the flame tip

length. Peak heat flux levels were measured in thelower part of the flame (z/Lf,tip ≤ 0.4) and decreasedwith distance above z/Lf,tip = 0.4. A general correla-tion to represent this behavior is as follows:

54

z/Lf,tip

0.01 0.1 1 101

10

100

1000

.

Max

imu

m H

eat

Flu

x, q

" pea

k (k

W/m

2 )

FIGURE 34. Maximum Heat Fluxes to the Walls Near the Corner withSquare Burner Sides of ●●-0.17 m, ▲▲-0.30 m, ▼▼-0.30 m (Elevated), and■■-0.50 m and Fire Sizes from 50 to 300 kW. Data from Lattimer et al.91

(Eq. 65a)

(Eq. 65b)

(Eq. 65c)

Where:= Maximum heat flux at a particular

elevation in the corner (kW/m2)= Peak heat flux in the corner (kW/m2)

z = Elevation along the flame height in thecorner (m)

Lf,tip = Flame tip length calculated usingEquations 61 and 62 (m)

Heat fluxes will decay with distance away fromthe corner as shown in Figure 35. Significant heatfluxes can exist as far as two fire diameters horizon-tally out from the corner. For a conservative analy-sis, the maximum vertical heat fluxdistribution measured in the cornershould be assumed from the cornerto two fire diameters horizontallyout from the corner.

Top of Walls

This section provides correlations to estimateheat fluxes along the top of the walls in a cornerconfiguration with a ceiling. These incident heatflux correlations apply to the top of the wallsapproximately twice the ceiling jet thickness belowthe ceiling or H – 2δ < z < H where δ = 0.1H.92

Along the top part of the wall, the maximum heatfluxes were measured at locations less than 0.15 m

below the ceiling. The maximum heat fluxes areshown in Figure 36 plotted against the dimensionlessdistance along the flame, (x + H)/Lf,tip. These heatfluxes can be estimated using the following relations:

Where:x = Distance horizontally out from the

corner (m)H = Distance between the base of the fire and

the ceiling (m)Lf,tip = Flame tip length calculated using

Equations 61 and 62 (m)

55

x/D

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

.

.

q"/

q"m

ax

0.00

0.25

0.50

0.75

1.00

1.25

D = 0.17 mD = 0.30 mD = 0.50 mCorrelation

FIGURE 35. Heat Flux Distribution Horizontally out from the Corneron the Lower Part of the Corner Walls

(Eq. 66a)

(Eq. 66b)

The assumed plateau in thecorrelation was based on themaximum heat flux levelsmeasured in larger fire tests with burning boundaries.91

Heat fluxes will decrease withdistance below the ceiling.Conservatively, it can beassumed that incident flux along the top of the walls isconstant with distance below the ceiling and is equal to themaximum incident flux pre-dicted through Equation 66.

Ceiling Above a Corner

Correlations in this sectioncan be used to predict the inci-dent heat flux distribution radi-ally out from a corner along theceiling. The heat fluxes to theceiling were determined to be afunction of dimensionless dis-tance along the flame length,(r + H)/Lf,tip. A plot of the heatfluxes measured along the ceil-ing out from the corner is shownin Figure 37. A correlation to predict the heatflux distribution along the ceiling is as follows:

Where:r = Radial distance from the corner (m)H = Distance between the base of the fire and

the ceiling (m)Lf,tip = Flame tip length calculated using

Equations 61 and 62 (m)

This correlation is similar to the one developedfor predicting the maximum heat flux along the topof the walls, Equation 66, except the length scalehere is r instead of x. Again, the assumed plateau inthe correlation was based upon the maximum heatflux levels measured in larger fire tests with burning

boundaries.91 The heat flux at the impingement pointcan be estimated using Equation 67 with r = 0.

Data Requirements


2. Heat release rate of the fire, .3. Distance between the base of the fire and the

ceiling, H.

56

(x+H) / Lf,tip

0.1 1 101

10

100

1000

.

Max

imu

m H

eat

Flu

x, q

" max

(kW

/m2 )

FIGURE 36. Maximum Heat Flux Along the Top of the WallsDuring Corner Fire Tests with Square Burner Sides of ●●-0.17 m,▲▲-0.30 m, ▼▼-0.30 m (Elevated), and ■■-0.50 m and Fire Sizes from50 to 300 kW. Data from Lattimer et al.91

(Eq. 67a)

(Eq. 67b)

4. Location along the surface where incident heatflux level is needed. This could be the elevationalong the height of the corner, z, horizontaldistance from the corner along the top of thewalls, x, or radially out from the corner along the ceiling, r.

Data Sources


Assumptions

This analysis assumes that the fire is attached tothe corner walls, the corner walls are vertical and ata 90° angle, and the ceiling is horizontal and at a90° angle with the corner walls. Walls that are not

vertical may result in different total incident heatflux levels as a result of the flame’s becoming sepa-rated from the wall or the difference in entrainmentinto the plume. Incident heat fluxes to the cornerwalls across the width of the fire are constant and areequal to the maximum vertical heat flux distributionin the corner. Heat fluxes along the top of the wallsare constant and equal to the maximum horizontalheat flux distribution along the top of the walls.

Validation

Other studies have been conducted with propanefires in a corner configuration with and without aceiling. Corner heat flux data with no ceiling93

agree well with the heat flux data in Figure 34 whenconsidered relative to the flame tip. In a study withfires in a corner and a ceiling, Hasemi et al.94

measured incident heat flux levels on the walls and

57

(r+H) / L f,tip

0.1 1 101

10

100

1000

Hea

t F

lux

to C

eilin

g (

kW/m

2 )

FIGURE 37. Heat Flux Along the Ceiling Above a Fire in a CornerDuring Tests with Square Burner Sides of ●●-0.17 m, ▲▲-0.30 m,▼▼-0.30 m (Elevated), and ■■-0.50 m and Fire Sizes from 50 to 300 kW.Data from Lattimer et al.91

ceiling from both exposure fires and simulatedburning boundaries. Trends in incident heat fluxlevels measured by Hasemi et al.94 along the top ofthe walls and the ceiling agree well with the data inFigures 36 and 37 when using the dimensionlessdistances used in these figures. In tests with propanegas burners against a non-combustible boundary,similar heat flux levels have been measured byother investigators for limited conditions.87,88

Ohlemiller, Cleary, and Shields95 measured peakheat fluxes approximately 10% to 20% higher usingsimilar size propane square burners.

Lattimer et al.91 also demonstrated that the corre-lations for incident heat fluxes in the three regionsof the corner configuration also hold when theboundary is combustible and burning. For this case,a modified length scale is required to correctly pre-dict flame length.

Limitations

Correlations for incident heat fluxes were devel-oped using luminous flames in an open environmentwith the fire directly in the corner. Using theserelations inherently assumes:

• There is negligible heating from a hot gas layerin the surroundings.

• The fire is against the wall.• The flames are luminous.• The corner walls are vertical and at a 90° angle.• The ceiling is horizontal and at a 90° angle

with the corner walls.

The experimental study considered fire diameters aslarge as 0.50 m and heat release rates as large as300 kW. No data were available to validate the cor-relations against fires with larger diameters or high-er heat release rates. The presence of a hot gas layermay increase the total incident flux onto the wall,and if significant in the area of interest adding thiscontribution to the total incident heat flux from thefire plume may be warranted.89 Moving the fireaway from the corner will eventually cause the inci-dent heat fluxes to become lower, largely becausethe flame becomes detached from the wall.90 Thus,the use of correlations in this section for fires that

may be slightly spaced from the corner will yieldconservative results. Flames less luminous thanthose produced by the propane fires (i.e., naturalgas) may transmit lower total incident heat fluxes tothe surfaces because the radiative heat flux to thewall will be lower.87,88,96 The propane fuel firesused to develop the heat flux data presented in thissection produce a moderate amount of soot; there-fore, heat flux levels presented in this section shouldbe considered to be average but not bounding for alldifferent fuels. Propane burners are also used exten-sively in standard fire tests as an exposure fire thatis representative of real fires. Therefore, the incidentheat fluxes from these flames are considered to berepresentative of those produced by most fires.

Fires Impinging on Unbounded Ceilings

Fires that impinge onto an unbounded ceiling asshown in Figure 38 have flames that are redirectedradially out from the impingement point. Thehighest heat fluxes onto the ceiling will be at theimpingement or stagnation point. Heat fluxes willtend to decrease with radial distance away from thestagnation point. Correlations are provided in thissection to estimate the heat fluxes from such a fireto the ceiling.

The incident heat flux due to a fire impingingonto an unbounded flat ceiling has been experimen-tally characterized by Hasemi et al.97 In this study,Hasemi et al.97 conducted a series of fire tests usingpropane gas burners located at different distancesbeneath a non-combustible unbounded ceiling. Thetest configuration is shown in Figure 38 along withimportant variables. Fires as large as approximately400 kW were considered in the study. Heat fluxgauges were used to measure the incident heat fluxalong the ceiling both directly above the centerlineof the fire (i.e., stagnation point) and radially outfrom the stagnation point.

A plot of the heat flux levels at the stagnationpoint is shown in Figure 39. Heat fluxes at thestagnation point are shown in this figure to plateauat approximately 90 kW/m2. In order to collapse the data, the unconfined flame tip length wasnormalized with respect the distance between theceiling and fire, H, plus the virtual source origin

58

correction, z'. The unconfined fire flame tip lengthwas calculated using the following relation:

(Eq. 68)

Where:n = 2/5 for Q*

D > 1.0n = 2/3 for Q*

D < 1.0

(Eq. 69)

= Heat release rate of the fire (kW)D = Diameter of the fuel package (m)ρ0 = Density of air at initial ambient conditions


ambient conditions [1.0 kJ/(kg K)]T0 = Temperature at initial ambient conditions


59

H

Lf(Unconfined

FlameLength)

Z

Stagnation Point

Ceiling

r

LH

Exposure Fire

Virtual PointSource Correction

Z'

D

FIGURE 38. Unbounded Ceiling Configuration

For noncircular fuel packages with a length towidth ratio of near one, the equivalent diameter ofthe fuel package can be estimated using the surfacearea, A, of the noncircular fuel package:

(Eq. 70)


The virtual point source correction for this geom-etry was determined using the following relations:

(Eq. 71a)

(Eq. 71b)

Where:Q*

D = Dimensionless quantity defined inEquation 69


The radial distribution in the incident heat fluxdecays with distance from the stagnation point asshown in Figure 40. The length of the flame used tocorrelate this data was the measured flame extensionplus a virtual origin correction. The measured flameextension was defined as the distance between thefire and the ceiling, H, plus the radial extension of the flame out from the center of the fire, LH. The location of the flame tip in this geometry was found to correlate with Q*

H, which is defined thesame as in Equation 69 except the length scale is Hinstead of D. The flame tip correlation was deter-mined to be

(Eq. 72)

60

D = 1.0m H = 1.0m

H = 1.2m

H = 0.8m

H = 0.6m

H = 0.4m

D = 1.0m H = 0.64m

H = 0.8m

H = 1.0m

D = 0.3m H = 1.0m

H = 0.8m

Lf /(H + z') (-)

100

80

60

40

20

0 0 1 2 3 4 5 6 7 8 9 10

. qs"

(kW

/m2 )

FIGURE 39. Stagnation Point Heat Fluxes on an Unbounded Ceiling with a Fire Impinging on It.Data from Hasemi et al.97

Where:

(Eq. 73)

LH = Flame extension along ceiling from thestagnation point to the flame tip (m)

H = Distance between the base of the fire andthe ceiling (m)

= Heat release rate of the fire (kW)ρ0 = Density of air at initial ambient conditions




The radial heat flux distribution along the ceilingat w > 0.45 can be estimated using the correlationrecommended by Wakamatsu98:

(Eq. 74a)

Where:

w = (-) (Eq. 74b)

r = Radial distance along the ceiling from thestagnation point (m)

H = Distance between the base of the fire andthe ceiling (m)

z' = Virtual source origin correction (m)LH = Flame extension along ceiling from the

stagnation point to the flame tip (m)

61

D = 0.5m H = 1.0m

H = 1.2m

H = 0.8m

H = 0.6m

H = 0.4m

D = 1.0m H = 0.64m

H = 0.8m

H = 1.0m

D = 0.3m H = 1.0m

H = 0.8m

(r + H + z')/(LH + H + z') (-)

100

10

00.1

. q"

(kW

/m2 )

1.0 10.0

FIGURE 40. Heat Fluxes to a Ceiling Due to a Propane Fire Impinging on the Surface. Data fromHasemi et al.97

Figure 41 contains a plot of Equation 74 (dashedline) along with a representation of the data ofHasemi et al.97 for a flat unbounded ceiling. Asnoted in Equation 74, this correlation adequatelyestimates the data when w > 0.45, but significantlyoverestimates heat flux levels for smaller values of w. Based on the data from Hasemi et al.97 andother data from fires impinging on I-beams mountedto a ceiling,98 a correlation was developed topredict the bounding heat flux levels where w isdefined in Equation 74b:

(Eq. 75a)

(Eq. 75b)

This correlation is shown in Figure 41 as the solidline. The peak heat flux of 120 kW/m2 at w ≤ 0.5bounds nearly all the heat flux measurements madein this range for the studies of Hasemi et al.97 andMyllymaki and Kokkala.98

Data Requirements



ceiling, H.4. Radial location out from the centerline of the

fire, r, where the incident heat flux level is needed.

Data Sources

1. Heat release rate data may be obtained fromBabrauskas,69 Hoglander and Sundstrum,70

or Mudan and Croce.70

62

w = (r + H + z')/(LH + H + z')

0.1 1 10

.

Hea

t F

lux,

q"

(kW

/m2 )

1

10

100

1000

FIGURE 41. Comparison of the Best Fit Curve Proposed by Wakamatsu (– –) and aBounding Fit to the Data (—). The unbounded ceiling data of Hasemi et al.97 is representedas the outlined area.

Assumptions

The fire is assumed to be impinging on ahorizontal, flat ceiling far from walls or any other obstructions.

Validation

Several experimental and theoretical studies havebeen performed on fires impinging on an unboundedceiling.77,78,79,80,97,99,100,101 Total heat fluxes fromfires and fire plumes impinging on the ceiling weremeasured by Hasemi et al.97, You and Faeth,79,80

and Kokkala.77,78 Due to the fuel type and size offires evaluated, heat flux levels measured by Hasemiet al.97 were higher than those measured in otherstudies. Therefore, the correlations developed usingthe data of Hasemi et al. are considered conservative.

Limitations

Correlations for incident heat fluxes were devel-oped using luminous flames in an open environmentwith the fire beneath an unbounded flat ceiling.Using these relations inherently assumes negligibleheating from a hot gas layer in the surroundings, theflames are luminous, and the ceiling is horizontal.The presence of a hot gas layer may increase thetotal incident flux onto the wall, and if significant inthe area of interest adding this contribution to thetotal incident heat flux from the fire plume may bewarranted. Flames less luminous than those pro-duced by the propane fires (i.e., natural gas) maytransmit lower total incident heat fluxes to the wallbecause the radiative heat flux to the wall will belower. Propane flames do not have the highest sootproduction of any fuel, and, therefore, incident heatfluxes may not be bounding. However, propaneburners are used extensively in standard fire tests asan exposure fire that is representative of real fires.Therefore, the incident heat fluxes from theseflames are considered to be representative of thoseproduced by most fires.

Fire Impinging on a Horizontal I-BeamMounted Below a Ceiling

The final geometry considered is an I-beam thatis mounted below a ceiling as shown in Figure 42,with the fire impinging on the lower flange of theI-beam. The focus here is the heat fluxes from thefire onto the I-beam. This case turns out to be quitesimilar to a fire impinging onto an unbounded ceiling.

Two separate studies have been conducted toevaluate the heat flux incident onto an I-beammounted below a ceiling with an exposure fireimpinging upon the beam (Hasemi et al.,97

Wakamatsu et al.,102 and Myllymaki andKokkala98). In these studies, the heat flux wasmeasured along the four surfaces of the I-beamnoted in Figure 42:

1. Downward face of the lower flange2. Upward face of the lower flange3. The web4. Downward face of the upper flange

The I-beam evaluated in these studies was 3.6 mlong, a web 150 mm high and 5 mm thick, andflanges 75 mm wide and 6 mm thick. For each ofthese surfaces, heat fluxes were measured from thestagnation point of the fire (centerline of the fire)along the length of the I-beam.

Results from these studies have demonstratedthat the incident heat flux onto all surfaces of thebeam will be equal to or less than the heat fluxlevels measured with a fire impinging onto a flatunbounded ceiling. Wakamatsu et al.102 measuredthis for fires up to 900 kW. Flame lengths wereobserved to be different along the lower flange,upper flange, and center of the web of the I-beam.Correlations to predict these flame lengths weredeveloped for the lower flange,102

(Eq. 76)

Where:

(Eq. 77)

63

the upper flange,102

(Eq. 78)

Where:

(Eq. 79)

and for the center of the web,98

(Eq. 80)

Where:

(Eq. 81)

64

HB

Ceiling

LB

Exposure Fire

D

HWHC

I-Beam

Ceiling

Upward Face ofLower Flange

Stagnation Point

LW

LC

Web

Downward Face ofLower Flange

Downward Face ofUpper Flange

FIGURE 42. I-Beam Mounted Below an Unbounded Ceiling

LB = Flame extension along lower flange fromthe stagnation point to the flame tip (m)

LC = Flame extension along upper flange fromthe stagnation point to the flame tip (m)

LW = Flame extension along the web center fromthe stagnation point to the flame tip (m)

HB = Distance between the base of the fire andbottom of the lower flange (m)

HC = Distance between the base of the fire andthe ceiling (m)

HW = Distance between the base of the fire andthe center of the web (m)

= Heat release rate of the fire (kW)ρ0 = Density of air at initial ambient conditions




The form of these correlations is similar to thatfor the unbounded ceiling flame length correlationgiven in Equation 72. The dimensionless distancealong the flame beneath the downward face of thelower flange was taken to be

(Eq. 82)

Where:r = Radial distance along the I-beam from the

stagnation point (m)HB = Distance between the base of the fire and

the lower flange (m)z' = Virtual source origin correction (m)LB = Flame extension along lower flange from

the stagnation point to the flame tip (m)

The dimensionless distance for the upper flangeon the I-beam was taken to be

(Eq. 83)


stagnation point (m)HC = Distance between the base of the fire and

the upper flange (m)

z' = Virtual source origin correction (m)LC = Flame extension along upper flange from


The dimensionless distance for the web on theI-beam was taken to be

(Eq. 84)


stagnation point (m)HW = Distance between the base of the fire and

the center of the web (m)z' = Virtual source origin correction (m)LW = Flame extension along web center from


The incident heat flux levels measured byWakamatsu et al.102 on the different faces of theI-beam are shown in Figure 43. On the downwardface of the lower flange (where the fire was directlyimpinging), heat flux levels along the flame lengthwere measured to be similar to the incident heatfluxes measured along a flame under an unboundedceiling. However, all other surfaces of the I-beamhad heat fluxes somewhat lower than those meas-ured along a flame under an unbounded ceiling.

The study of Myllymaki and Kokkala98 con-sidered the effects of larger fires (up to 3.9 MW) onthe heat flux incident on the different faces of theI-beam. Some of the heat flux measurements madein this study are shown in Figure 44. In this study,Myllymaki and Kokkala98 found that, for fires over2.0 MW, the incident heat fluxes onto all faces ofthe I-beam were equivalent to or slightly higherthan those measured along an unbounded ceiling.

Data from these studies demonstrate that the heat flux to the I-beam can be conservatively esti-mated using the bounding heat flux correlation inEquation 85 using the appropriate expression for wprovided in Equations 82 through 84:

(Eq. 85a)

(Eq. 85b)

65

Data Requirements



bottom flange, center of the web, and the top ofthe flange.

4. Distance out from impingement point on theI-beam where the heat flux is needed, r.

Data Sources


Assumptions

The I-beam being analyzed should have similardimensions to the one considered in these twostudies (3.6 m long, a web 150 mm high and 5 mmthick, and flanges 75 mm wide and 6 mm thick),and the fire is assumed to be impinging directlyonto the bottom flange of the I-beam. The I-beam is

66

(r + HB + z')/(LB + HB + z') (-)

100

10

00.1

. q"

(kW

/m2 )

1.0 10.0

flame tips

100

10

0

100

10

0

100

10

0

(r + HC + z')/(LC + HC + z') (-)0.1 1.0 10.0

(r + HC + z')/(LC + HC + z') (-)0.1 1.0 10.0

(r + HC + z')/(LC + HC + z') (-)0.1 1.0 10.0

Lower Flange Downward

Lower Flange Upward Upper Flange Downward

Web

H = 1.0m Q = 100 kW Q = 150k Q = 200k

H = 0.6m Q = 95 kW Q = 130k Q = 160k

H = 1.2m Q = 540 kW Q = 750k Q = 900k

Flat Ceiling MaximumFlat Ceiling Minimum

. q"

(kW

/m2 )

. q"

(kW

/m2 )

. q"

(kW

/m2 )

FIGURE 43. Heat Flux Measured Onto the Surfaces of an I-Beam Mounted Below an UnboundedCeiling for Fires 95 to 900 kW102

also assumed to be located remote from any wallsor ceiling obstructions.

Validation

These two studies provide a good validation ofthe heat fluxes experienced by the particular I-beamtested. Results produced using propane fuel firesagreed well with the larger liquid heptane pool fire tests.

Limitations

The height of the webbing and the width of theflanges may affect the heat fluxes to the I-beam.Other size I-beams have not been tested to evaluatethe impact of I-beam dimensions on heat flux. Cor-relations for incident heat fluxes were developedusing luminous flames in an open environment with

the fire directly impinging on the I-beam. Usingthese relations inherently assumes negligible heatingfrom a hot gas layer in the surroundings and that the I-beam is not located near any boundaries. Thepresence of a hot gas layer may increase the totalincident flux onto the I-beam, and, if significant, thiscontribution should be added to the total incidentheat flux from the fire plume.90 Moving the fireaway from the I-beam so that it does not impinge onthe lower flange will change the heat flux distribu-tion on the I-beam. These test data were developedwith 0.48 < QH* < 1.27, fire distance below thelower flange of 0.6 < HB < 1.9, fire diameters up to1.6 m, and heat release rates up to 3.9 MW. Thoughresults in this section indicate the heat flux isbounded by the correlation in Equation 85, heatfluxes from large pool fires (D > 1.6 m) impingingon an I-beam may be higher due to the changes ingas emissivity and flame temperature.67,68

67

w (- -)0.1 1 10

0.1

1

10

100

1000

.

Hea

t F

lux,

q"

(kW

/m2 )

FIGURE 44. Heat Flux Measured on the ●●-Bottom Flange, ■■-Web, and ▲▲-Upper Flange ofan I-Beam Mounted Below an Unbounded Ceiling for Fires 565 to 3,870 kW.98 The line inthe plot is the curve given in Equation 85.

SUMMARY AND RECOMMENDATIONS

The motivation for the work in the BoundingHeat Flux section has been the effect of the fire onobjects in flames. Those studies were interested inthe ability of nuclear waste casks or structural ele-ments in offshore drilling facilities to withstand fire.On the other hand, the motivation for the workreported in the section on Heat Fluxes for SpecificGeometries was primarily the effect of fire on igni-tion and fire growth (except for the I-beam studies).As a consequence, smaller exposure fires are con-sidered in the latter section. For example, in the for-mer, fires of up to 9 by 18 m were used (more than300 MW) as compared to fires of up to 1 m at mostor about 500 kW for the latter section. For the I-beam study, data include larger fires of 3.9 MW atmost. The differences in the two sections are pro-found, and the reader should be aware of these dis-tinctions in using the correlations. It is clear thatpool-like fires exhibit higher temperatures andtherefore higher heat fluxes as they become bigger.For example, a flame temperature of 1200°C corre-sponds to a radiant heat flux of 267 kW/m2. Yet inthe Bounding section (Table 4), most measurementsare more generally in the range of 150 kW/m2,

while for the smaller fires in the section on specificgeometries, the upper limit of the heat flux measure-ments is more like 120 kW/m2. Therefore, the userof this information must take into account the sizeand configuration of the fire. The type of fuel is lesslikely to be a factor.

Another issue that should be recognized in apply-ing these results is that they are presented in termsof incident heat flux, or the heat flux as measured toa cold target. In a design application, the heat fluxthat is absorbed into the structural element willdecrease as the surface temperature increases. Theboundary condition that should be used for thestructure should account for the radiation loss forelements impacted by a fire plume:

Where:= Incident heat flux given herein

ε = Surface emissivityTo = Cold target temperature

No factor of safety is addressed, and the usermust be aware that that is not implicit in any ofthese results.

68

Theoretical Examination of Methods

69

Appendix A

As can be seen in Figure A.1, predictions ofburning rate vary markedly among the differentmethods. Some of the methods assume stoichio-metric or ventilation-limited burning, while othersaccount for fuel-controlled burning.

Results by Harmathy for Wood Cribs

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 5 10 15 20 25 30 35 40 45 50 55

Tanaka

Eurocode

Lie

Harmathy

Magnusson

Babrauskas

Law

Ma

A/AoHo1/2 (m–1/2)

. mF(D

/W)1

/2/A

oH

o1/

2 (k

g/s

– m

5/2 )

FIGURE A.1. Comparison of Burning Rate Predictions

From Equation 20 similar results can be derived:

or

Results by Bullen and Thomas for Pool Fires

, s ≈ 4 for wood, s ≈ 7-10 for liquids.

The pool fire results are explained by thestoichiometry and thermal feedback.

The locus of

s and both are larger for liquids compared to wood.

70

FIGURE A.2. Wood Crib and Liquid Pool Fires

71

CIB DataIn the CIB experiments the fuel is placed over

the entire floor; therefore, AF ≈ A.Here the theory gives

From Equation 26, for and

for which agrees with thetrends in Figure A.2.

EurocodeThe Eurocode prescribes

Here t* is essentially Qw*–2

or only.

This specification must assume a ventilation-limited fire and ignores the other variables.

LieOnly a ventilation-limited fire is assumed.

Lie gives .

The theoretical development gives

.

For temperature, T = f(F, t) and C, a constantwhich takes into account the properties of thebounding materials of the enclosure.

This represents kρc since k ~ ρ.

Magnusson, Thelandersson,and Petersson

Magnusson, Thelandersson, and Petersson com-pute a result for temperature based on a similartheory. They augment it with a rate of rise fordeveloping fire and a prescribed cooling phase.They use only the ventilation-controlled fire forcribs from Kawagoe and Sekine:

.

(The theoretical development gives.)

They compute results for and kρc for various fuel loads.

based on a distribution of fuel over

the entire compartment surface area A. (Normally

fuel loading is based on floor area, i.e. .)

BabrauskasA computer solution was correlated to give an

analytical result:

72

Both θ2 and θ3 correspond to Qw*, but not

exactly, since the powers are different in θ3 for each term. Since the dimensionalization of theequations must be consistent, it suggests that thereis an inconsistency in θ3.

This corresponds to .

θ5 pertains to combustion efficiency and is onlyrelevant if the theoretical heat of combustion is used.

It is interesting that the maximum temperaturegiven by the correlation is 1425ºC. The theory sug-gests this is 1500ºC at most.

LawLaw developed a correlation based on the CIB

data.16 A fit giving the maximum or upper values of data is

where A is the heat transfer area of the boundary sur-faces, not including the vents (as used in the theory).

An adjustment is made if the fuel load is low.

The mass loss rate is correlated as

for

Where:D = Compartment depthW = Compartment width

Ma and MäkeläinenThese authors develop a correlation based on

the CIB and other data. Its novel feature is that itincludes a prediction of temperature over timestarting at the onset of the fully developed stage.They use Harmathy’s result for the burning rate inthe fuel-controlled regime, and his demarcation ofthe regime change to ventilation-limited:

They use Law’s correlation for the ventilation-limited burning rate.

The temperature is given as

Where:δ = 0.5 for the ascending phase and

1.0 for the descending phase

The maximum temperature is given as linear fitsto the CIB and other data in terms of .The time at the maximum temperature is selected as

.This model does not include the effect of the wall

thermal properties.

Comparisons of Enclosure FirePredictions with Data

73

Appendix B

Predictions of compartment fire temperature andduration are compared to two sets of data. The firstset of data is from 321 experiments conductedunder the auspices of CIB.48 See the section entitledCIB beginning on page 31 for more information on these experiments. The compartments in theseexperiments were roughly cubic, although some ofthe compartments had aspect ratios (length towidth) of 1/2 or 2.

In these experiments, the stage of fully devel-oped burning was defined as the period from whenthe mass of fuel was between 80% and 30% of theoriginal, unburned fuel mass. Average temperaturesduring the period of fully developed burning fromthese experiments were presented as a function

of .

Average burning rate data during the fullydeveloped stage was presented as

as a function of . Data was

also included where the average burning rate duringthe fully developed burning stage was presented in

tables of as a function of .

Although both the CIB report48 and the Cardingtondata103 show that the aspect ratio of a compartmentcan influence the burning rate for fully developed,ventilation-limited fires, most predictive methods do not explicitly account for this effect. Therefore,predictive methods that do not account for compart-ment aspect ratio were evaluated using the CIBburning rate data, which was normalized by the areaand square root of the height of the ventilationopening, but not by the square root of the ratio ofcompartment depth to width. Methods that dospecifically account for the compartment aspectratio were evaluated using the CIB data that wasnormalized by both the area and square root of theheight of the ventilation opening and the square rootof the ratio of compartment depth to width. When

the CIB data was not normalized by the square rootof the ratio of compartment depth to width, therewas more scatter in the data.

The methods presented in this guide were evalu-ated by plotting predictions of average temperatureduring the fully developed stage along with the CIB data. When comparing predictions to data,averages were taken of what appeared to be thefully developed stage from the temperature data.Similarly, predictions of duration were compared tothe CIB data by dividing the initial mass of fuel,mf , by the predicted duration, τ, and plotting thisquantity along with the CIB data.

Some of the predictive methods required as inputthe surface area of the fuel. The ratio of fuel surfacearea to total room surface area (defined as includingthe area of the ceiling and walls, but not the area ofthe ventilation opening or the floor) was calculatedfor each of the CIB experiments. The average ratioof fuel surface area to total room surface area inthese experiments was 0.75, with a standard devia-tion of 0.90. Figure B.1 shows a histogram of theratio of fuel surface area to the enclosure surfacearea for the CIB experiments. For methods thatrequire as input the fuel surface area, the value of0.75A was used for comparing predictions to theCIB data.

To explicitly analyze the effect of long, narrowcompartments, temperature data as a function oftime from a series of experiments that were con-ducted in a compartment that was approximately23 meters long, 2.7 meters high, and 5.5 meterswide103 were compared to predictions. In theseexperiments, the ventilation opening ranged from1/8 to 1/1 of the small side of the compartment. The fuel loading consisted of wood cribs with atotal density of 20 or 40 kg/m2. Additionally, forone experiment, the compartment size was reducedto approximately 5.6 x 5.6 x 2.75 meters (high). Thefull details of the experiments may be found inreference 103.

CIB DataThe experiments in the CIB study were con-

ducted in a variety of enclosures since multiplelaboratories participated. Statistical means wereused to overcome systematic differences betweenthe laboratories. The majority of the laboratoriesused a test enclosure constructed of 10 mm thickasbestos millboard with a reported thermal conduc-tivity of 0.15 W/m°C, and this is the value that wasused for methods that required specific heat as aninput. The density of the asbestos millboard and the specific heat were not reported, so values of 816 J/kg°C and 1100 kg/m3 were selected.27,104

In the CIB study, separate graphs of temperatureand burning rate data were presented for cribs with20 mm thick wood sticks spaced 20 mm apart, andfor cribs with 20 mm wide sticks spaced 60 mmapart, or with 10 mm wide sticks spaced 30 mmapart. However, for purposes of comparing predic-tions with the CIB data, all temperature and burningrate data was aggregated into single graphs.

Cardington DataA total of nine experiments were conducted

under a collaborative project between British Steeland the British Research Establishment’s FireResearch Station. The experiments were conductedin a purpose-built compartment within the BritishResearch Establishment’s ex-airship hanger.

The floor of the compartment was made of75 mm thick concrete covered with sand. The wallswere made of lightweight concrete blocks thatmeasured 440 x 215 x 215 mm. In most tests thewalls were lined with a 50 mm thick ceramic fiberblanket. However, in one of the tests (test #8) thewalls were lined with two 12.5 mm thick plaster-board sheets affixed onto 47 x 47 mm wood studsspaced 600 mm apart. The ceiling was constructedof 200 mm thick aerated concrete slabs and waslined in the same manner as the walls.

The opening of the compartment was located onone of the smaller walls, and concrete blocks wereused to restrict the opening to 100%, 50%, 25%, or12.5% of the wall size. Additionally, in some of thetests a 400 mm insulated steel column was placedflush with the opening, which further reduced theopening size.

74

Distribution of Area Ratio

0

10

20

30

40

50

60

70

80

90

100

0.04 0.40 0.76 1.11 1.47 1.83 2.18 2.54 2.90 3.25 3.61 3.97 4.32 4.68

Area of Fuel/Total Area

Freq

uenc

y

FIGURE B.1. Histogram of Ratio of Fuel Surface Area to Enclosure SurfaceArea for the CIB Experiments

The dimensions of the enclosure are provided inTable B.1,103 the dimensions of the opening arelisted in Table B.2,103 and the properties of theenclosure materials are listed in Table B.3.103

The fuel for the Cardington tests was wood cribs,constructed of 1 m long sticks of 50 x 50 mm westernhemlock spaced 50 mm apart. The heat of combustionof the wood was reported as 19.0 MJ/kg. The fuelloading for each of the tests can be found in Table B.4.

In all but tests #7 and #9, the fires were ignited atthe rear of the compartment (opposite the end withthe ventilation opening). In tests #7 and #9, all cribswere ignited simultaneously. In all the tests, the firespread to the cribs nearest the ventilation opening,and, once the fire reached the cribs nearest the ven-tilation opening, the cribs further away from theventilation opening ceased burning. The cribs

nearest the ventilationopening continuedburning, and, as the fuelwas depleted, the firesprogressed toward therear of the enclosure.As a result, the temper-atures were not hori-zontally homogeneous,and higher temperaturesat any given time weremeasured above thelocation where the firewas burning.

75

Test # Total Width (mm) Height (mm)

1 5595 2750

2 5595 2750

3 5195 1470

4 5195 1470

5 2139 1730

6 5195 375

7 1370 2750

8 5065 2680

9 5195 2750

TABLE B.2. Opening Dimensions of theCardington Tests

Test # Length (m) Width (m) Height (m)

1 22.855 5.595 2.750

2 22.855 5.595 2.750

3 22.855 5.595 2.750

4 22.855 5.595 2.750

5 22.855 5.595 2.750

6 22.855 5.595 2.750

7 5.595 5.595 2.750

8 22.780 5.465 22.780

9 22.855 5.595 2.750

TABLE B.1. Compartment Dimensions of theCardington Tests

Test # Fuel Load (kg/m2)

1 40

2 20

3 20

4 40

5 20

6 20

7 20

8 20.6

9 20

TABLE B.4. Fuel Loading for the Cardington Tests

Thermal Density Specific Heat Conductivity

Structure Material (kg/m3) (J/kg K) (W/m K)

Walls Lightweight 1375 753 0.42concrete blocks

Roof Aerated 450 1050 0.16concrete slabs

Floor Sand 1750 800 1.0

Fiber lining Ceramic fiber 128 1130 0.02

Plasterboard Fireline 900 1250 0.24lining plasterboard

TABLE B.3. Properties of Enclosure Materials

The temperature data from the Cardington testswas compared to predictions made using themethods identified in this guide by comparing themeasured temperatures to predictions. Temperatureswere measured at locations approximately 3, 11,and 19 m (measured horizontally) from the ventila-tion opening. In the graphs, averages of the thermo-couple measurements are plotted, with error barsindicating the range of the measured temperatures.

Predictions were made using each of the methodsidentified in this guide at 3-minute intervals for tests#1, 2, 3, 7, and 9; at 6-minute intervals for tests #4,5, and 8; and at 25-minute intervals for test #6.

For predictive methods that have distinct correla-tions for fuel-controlled and ventilation-controlledburning, the fire was assumed to be ventilation con-trolled. Given the behavior of the burning, this is areasonable assumption.

Eurocode

CIB DATA

In the CIB experiments, the mass of fuel per unitarea ranged from 20 to 40 kg/m2. (A few tests useda mass of fuel per unit area of 10 kg/m2 but, sincethe CIB report indicated that only a “few” testswere conducted at this density, this value was notmodeled.) For an effective heat of combustion forpine of 12.4 MJ/kg,33 qt,f would range from 248 to496 MJ/m2, and multiplying this by the ratio ofAfloor/A in the CIB compartments results in a rangeof qt,d of approximately 50 to 100 MJ/m2.

Predictions of temperature as a function of timewere made using the Eurocode method for values

of ranging from 5 to 50 m–1/2. Predictions

were made at time increments ranging from 0.005 hours to 5 hours, depending on the values of

qt,d and used. For each value of ,

averages of the temperature predictions during thetime in which t* < td* were compared to the CIBdata. The Eurocode method was evaluated aspresented, and the modifications suggested byBuchannan and Franssen also were evaluated. Agraph of Eurocode predictions and the CIB data ispresented in Figures B.2 and B.3.

The predicted duration of the fully developedburning stage is when t* = td*. Given that

, and t* = tΓ,

the predicted duration in hours would be

,

where τ is in hours and can be rewritten as

.

Substituting ,

Since ,

.

Since

, which can be

rearranged as (kg/h) or

(kg/s). Substituting ∆Hc =

12.4 MJ/kg, the predicted burning rate would be . This is compared to the CIB

burning rate data in Figure B.4.

76

77

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

CIB Data

Eurocode

Buchanan

Franssen

FIGURE B.2. Comparison of CIB Temperature Data to Predictions Made Using Eurocode,Buchanan, and Franssen Methods, qt,d = 100 MJ/m2

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

CIB Data

Eurocode

Buchanan

Franssen

FIGURE B.3. Comparison of CIB Temperature Data to Predictions Made Using Eurocode,Buchanan, and Franssen Methods, qt,d = 50 MJ/m2

78

Franssen’s modification results in a calculatedburning duration of 20 minutes when t*

d /Γ is less than 20 minutes. For the CIB data and qt,d = 50 MJ/m2, t*

d /Γ is less than 20 minutes

for cases where was less than or equal to

10 m–1/2. With qt,d = 100 MJ/m2, t*d /Γ is less than

20 minutes for cases where was less than

or equal to 30 m–1/2.

CARDINGTON DATA

Inputs were created in accordance with the rec-ommendations of the Eurocode. When calculatingqt,d, the area of the ventilation opening was notincluded in the calculation of the total surface area;however, the area of the openings was included incalculations of the total surface area of the enclo-sure. Predictions less than 20°C were assumed toindicate that the decay period had completed andthe temperature in the compartment was ambient.The results of the comparisons of predictions usingthe Eurocode to the Cardington data are presentedin Figures B.5 through B.13.

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 10 20 30 40 50

A/AoHo1/2 (m–1/2)

. mf/A

oH

o1/

2 (k

g/s

– m

5/2 )

121221211441Eurocode

FIGURE B.4. Comparison of CIB Burning Rate Data to Predictions Made Using the Eurocode Method

79

0

200

400

600

800

1000

1200

1400

0 0.5 1.0 1.5 2.0

Time (h)

Tem

per

atu

re (

°C)

Measured

Eurocode

Buchanan

Franssen

FIGURE B.5. Comparison of Predictions Made Using Eurocode, Buchanan, and FranssenMethods to Data from Cardington Test #1

Time (h)

Tem

per

atu

re (

°C)

Measured

Eurocode

Buchanan

Franssen

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6


80

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

Time (h)

Tem

per

atu

re (

°C)

Measured

Eurocode

Buchanan

Franssen


Measured

Eurocode

Buchanan

Franssen

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (h)

Tem

per

atu

re (

°C)


81

Measured

Eurocode

Buchanan

Franssen

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3

Time (h)

Tem

per

atu

re (

°C)


Measured

Eurocode

Buchanan

Franssen

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8

Time (h)

Tem

per

atu

re (

°C)


82

Measured

Eurocode

Buchanan

Franssen

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1.0 1.2

Time (h)

Tem

per

atu

re (

°C)


0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3

Time (h)

Tem

per

atu

re (

°C)

Measured

Eurocode &Franssen

Buchanan


LieSince it was not possible to determine the dura-

tion of burning for each data point in the CIB datain a straightforward manner, to compare predictionsusing Lie’s method to the CIB data average temper-ature predictions were made for a fire of 2 hours’

duration with opening factors F =

ranging from 0.02 to 1. Because the density of the

enclosures used in the CIB tests was assumed to be1100 kg/m3, the C factor used in Lie’s methodwould equal 1. A comparison of Lie’s predictionsand the CIB data can be found in Figure B.14.

Lie gives (kg/s). This is com-pared to the CIB burning rate data in Figure B.15.

Comparisons of predictions using Lie’s methodto the Cardington data can be found in Figures B.16through B.24.

83

Time (h)

Tem

per

atu

re (

°C)

200

400

600

800

1000

1200

1400

0 0.5 1.0 1.5 2

Measured

Eurocode

Buchanan

Franssen

0


84

A/AoHo1/2 (m–1/2)

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

Tem

per

atu

re (

°C)

CIB Data

Lie

FIGURE B.14. Comparison of CIB Temperature Data to Predictions Made Using Lie’s Method

A/AoHo1/2 (m–1/2)

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 10 20 30 40 50

121221211441LieLie * 1.8Lie / 1.8

. mf/A

oH

o1/

2 (k

g/s

– m

5/2 )

FIGURE B.15. Comparison of CIB Burning Rate Data to Predictions Made Using Lie’s Method

85

1400

1200

1000

800

600

400

200

0

Time (h)

Tem

per

atu

re (

°C)

Measured

Lie

0 0.5 1 1.5 2

FIGURE B.16. Comparison of Predictions Made Using Lie’s Method to Data from Cardington Test #1

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time (h)

Tem

per

atu

re (

°C)

Measured

Lie


86

Measured

Lie

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

Time (h)

Tem

per

atu

re (

°C)


Measured

Lie

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (h)

Tem

per

atu

re (

°C)


87

Measured

Lie

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3

Time (h)


Measured

Lie

Tem

per

atu

re (

°C)

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8

Time (h)


88

Measured

Lie

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1

Time (h)


Measured

Lie

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3

Time (h)


TANAKA

For Tanaka’s methods, it was not possible todetermine the duration of burning for each point inthe CIB data in a straightforward manner. To com-pare predictions using Tanaka’s method and hisrefined method to the CIB data, average tempera-ture predictions were made for a fire of 2 hours’

duration with ranging from 1 to 50 m–1/2.

For = 1 m–1/2, Tanaka’s refined method

produced rapidly declining temperatures, and anytemperature below 600°C was neglected. The resultof this comparison can be seen in Figure B.25.

Both Tanaka’s method and Tanaka’s refinedmethod predict the mass loss rate as .This is compared with the CIB data in Figure B.26.

Comparisons of predictions using Tanaka’smethod, both the simple and refined versions, to theCardington data can be found in Figures B.27through B.35.

89

Measured

Lie

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2

Time (h)


90

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

0

1000

2000

3000

4000

5000

0 10 20 30 40 50

CIB DataTanakaRefined Tanaka

FIGURE B.25. Comparison of CIB Temperature Data to Predictions Made Using Tanaka’s Methods

A/AoHo1/2 (m–1/2)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 10 20 30 40 50

121221211441TanakaTanaka * 1.6Tanaka / 1.9

. mf/A

oH

o1/

2 (k

g/s

– m

5/2 )

FIGURE B.26. Comparison of CIB Burning Rate Data to Predictions Made Using Tanaka’s Methods

91

Tem

per

atu

re (

°C)

0

Measured

Tanaka

Refined Tanaka

200

400

600

800

1000

1200

1400

1600

0 0.5 1 1.5 2

Time (h)

FIGURE B.27. Comparison of Predictions Made Using Tanaka’s Methods to Data from Cardington Test #1

Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

500

1000

1500

2000

2500

3000

3500

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6


92

Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

500

1000

1500

2000

2500

3000

0 0.5 1 1.5 2 2.5


Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

500

1000

1500

2000

2500

3000

3500

0 1 2 3 4


93

Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

500

1000

1500

2000

2500

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 1 2 3 4 5 6 7 8 9 10


94

Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

500

1000

1500

2000

2500

3000

0 0.2 0.4 0.6 0.8 1


Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

200

400

600

800

1000

1200

1400

1600

0 0.5 1 1.5 2 2.5 3


MAGNUSSON AND THELANDERSSON

The enclosures that were used in the CIB testswere modeled as Type C (as defined by Magnussonand Thelandersson38) since the Type C enclosuremost closely represents the material properties ofthe CIB enclosures.

Given that it was not possible to estimate theburning rates applicable to the CIB data in astraightforward manner, a duration of 2 hours wasarbitrarily selected. This selection should have onlya minor influence on the comparison with the CIBdata since only the average temperature during thefully developed stage is of interest. A comparison ofpredictions made in this manner with the CIB datais shown in Figure B.36.

Magnusson and Thelandersson’s method predictsburning duration as follows:

where q is the fuel load in Mcal/m2 related to thesurface area of the enclosure. Using a heat of com-bustion of 12.4 MJ/kg and converting units, this can

be reduced to .

Since , the burning rate predicted using

Magnusson and Thelandersson’s method would

be , which is identical to themethod that Babrauskas recommends for ventila-tion-controlled burning. A comparison of predic-tions of burning rate made using Magnusson andThelandersson’s method to the CIB data is shown inFigure B.37.

With the exception of test #8, which was modeledas Type G, the Cardington enclosure was modeledas Type C. The area of the ventilation opening wasnot included in calculations of the surface area of

the enclosure. Where values of or the

burning duration were not sufficiently close to the values presented in the tables, linear interpola-tion was performed. It was not possible to modeltest #6 using Magnusson and Thelandersson’smethod since no table or graph was provided thatresembled the conditions associated with test #6.Comparisons of predictions made using Magnussonand Thelandersson’s method to the Cardington datacan be found in Figures B.38 through B.45.

95

Tem

per

atu

re (

°C)

Measured

Tanaka

Refined Tanaka

Time (h)

0

500

1000

1500

2000

2500

3000

3500

0 0.5 1 1.5 2


96

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

CIB DataMagnusson

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

FIGURE B.36. Comparison of CIB Temperature Data to Predictions Made Using Magnusson andThelandersson’s Method

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 10 20 30 40 50

121221211441MagnussonMagnusson * 1.3Magnusson / 2.3

A/AoHo1/2 (m–1/2)

. mf/A

oH

o1/

2 (k

g/s

– m

5/2 )

FIGURE B.37. Comparison of CIB Burning Rate Data to Predictions Made Using Magnusson andThelandersson’s Method

97

Tem

per

atu

re (

°C)

Measured

Magnusson (Type C)

0 0.5 1 1.5 2

1400

1200

1000

800

600

400

200

0

Time (h)

FIGURE B.38. Comparison of Predictions Made Using Magnusson and Thelandersson’s Method(Type C) to Data from Cardington Test #1

Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Measured

Magnusson (Type C)


98

Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

Measured

Magnusson (Type C)


Tem

per

atu

re (

°C)

Time (h)

Measured

Magnusson (Type C)

0

200

400

600

800

1000

1200

1400

0 1 2 3 4


99

Time (h)

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3

Measured

Magnusson (Type C)


Time (h)

Tem

per

atu

re (

°C)

Measured

Magnusson (Type C)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1


100

Time (h)

Tem

per

atu

re (

°C)

Measured

Magnusson (Type G)

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3


Time (h)

Tem

per

atu

re (

°C)

Measured

Magnusson (Type C)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


HARMATHY

Because of the iterative nature of Harmathy’smethod for predicting compartment fire tempera-tures, it is not possible to compare predictions usingHarmathy’s method to the CIB data in a straight-forward manner.

Harmathy distinguishes fuel-limited burningfrom ventilation-limited burning as the point where

= 0.263. Substituting ρ0 = 1.2 kg/m3

and g = 9.8 m/s2, = 0.07. In the CIB tests,

the average value of AF/A was approximately 0.75.Substituting and inverting, the threshold betweenfuel-limited and ventilation-limited burning would

be = 19.0.

For fuel-limited burning Harmathy gives:

. Substituting Af = 0.75A and

= mf /τ yields = 0.00465A.

For ventilation-limited burning, Harmathy gives:

.

Substituting ρ0 = 1.2 kg/m3 and g = 9.8 m/s2,

. Substituting this into = mf /τ

yields . This is compared to theCIB data in Figure B.46.

Comparisons of predictions using Harmathy’smethod to the Cardington data are presented inFigures B.47 through B.55. Predictions for timesless than the burning duration were created by usingthe iterative method recommend by Harmathy, anda minimum resolution of 1°C was required for theprediction to be accepted.

101

A/AoHo1/2 (m–1/2)

121221211441HarmathyHarmathy * 1.8Harmathy / 1.8Harmathy / 1.5Harmathy * 2.8

0

. mf/A

oH

o1/

2 (k

g/s

– m

5/2 )

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 10 20 30 40 50

FIGURE B.46. Comparison of CIB Burning Rate Data to Predictions Made Using Harmathy’s Method

102

Tem

per

atu

re (

°C)

Measured

Harmathy

0 0.5 1 1.5 2

1400

1200

1000

800

600

400

200

0

Time (h)

FIGURE B.47. Comparison of Predictions Made Using Harmathy’s Method to Data fromCardington Test #1

Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Measured

Harmathy


103

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Harmathy

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Harmathy

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4


104

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Harmathy

200

400

600

800

1000

1200

1400

1600

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

1600

0 1 2 3 4 5 6 7 8

Measured

Harmathy


105

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Harmathy

200

400

600

800

1000

1200

1400

1600

0 0.2 0.4 0.6 0.8 1


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Harmathy

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3


BABRAUSKAS

Babrauskas provides the equivalence ratio as

where and s is the ratio such

that 1 kg fuel + s kg air = (1 + s) kg products.Harmathy39 notes that a typical wood would havethe chemical formula CH1.455O0.645•0.233H2O,which would result in a value of s of 6.0, which isslightly larger than the value of 5.7 proposed by

Babrauskas.46 Using .Substituting this into the correlation for the

equivalence ratio yields .

Babrauskas provides methods for modelingburning rate for ventilation-controlled burning, andfor fuel-controlled burning, for wood cribs, andthermoplastic or liquid pools.45 Babrauskas’ modelfor calculating the burning rate of ventilation-con-trolled fires is used here; however, in most designsituations, the input data needed to use Babrauskas’models for fuel-controlled burning is not available.Therefore, Harmathy’s model for the burning rate ofover-ventilated fires was used for the present analysis.

For fuel-controlled burning, Harmathy estimatesthe burning rate as = 0.0062Af. Substituting thisinto the above yields:

. For stoichiometric burning, φ = 1.

In the CIB tests, the average value of AF /A wasapproximately 0.75. Substituting and solving for

, the threshold between fuel-limited and

ventilation-limited burning would be = 18.0.

Substituting in the relevant values for enclosureproperties from the CIB tests and assuming that Ho ≈ 1 m (in the CIB tests, Ho ranged from 0.5 m to1.5 m, but, given that Babrauskas’ method varieswith Ho

–0.3, predictions are not highly sensitive tothis parameter) and bp = 0.9 results in the predic-tions of the CIB temperatures shown in Figure B.56.

For ventilation-controlled burning, Babrauskasestimates the burning rate as45:

106

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Harmathy

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


107

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

CIB Data

Babrauskas

FIGURE B.56. Comparison of CIB Temperature Data to Predictions Made Using Babrauskas’ Method

Given that Harmathy’s method of estimatingburning rate for fuel-controlled burning was used,the evaluation of that method is applicable to theassumption made here. A comparison of burningrate predictions using Babrauskas’ method to theCIB data for ventilation-controlled fires is presentedin Figure B.57.

The closed form approximation was used tocreate predictions of compartment fire temperaturesfor the Cardington tests. In these tests, it was appar-ent that the fires were ventilation controlled fromthe observed burning behavior. While Babrauskas’method is capable of predicting burning rate andcompartment fire temperatures during the growthand decay stages of a fire, these stages wereneglected. The burning rate was calculated as45:

Once the fuel was depleted, the fire was con-sidered to cease, and the temperature assumed to

immediately return to ambient. Thus, the only time-dependent variable remaining was θ3, which veryquickly equaled one. Therefore, compartment firetemperatures were modeled as a square wave.

The value of s was calculated as 6.0, based onthe chemical formula for typical wood provided byHarmathy39 of CH1.455O0.645•0.233H2O.

Calculations of the wall area did not includeeither the area of the floor or the area of the ventila-tion opening. The lining properties used were thoseof the ceramic fiber lining. For calculation of θ5, avalue of 0.9 was used for bp. The burning durationwas calculated by dividing the mass of unburnedfuel by the burning rate.

Comparisons of predictions using Babrauskas’method to the Cardington data are presented inFigures B.58 through B.66.

108

Tem

per

atu

re (

°C)

Measured

Babrauskas

0 0.5 1 1.5 2

1400

1200

1000

800

600

400

200

0

Time (h)

FIGURE B.58. Comparison of Predictions Made Using Babrauskas’ Method to Data fromCardington Test #1

A/AoHo1/2 (m–1/2)

0

121221211441BabrauskasBabrauskas * 1.3Babrauskas / 2.3

. mf/A

oH

o1/

2 (k

g/s

– m

5/2 )

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 10 20 30 40 50

FIGURE B.57. Comparison of CIB Burning Rate Data to Predictions Made Using Babrauskas’ Method

109

Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Measured

Babrauskas


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5


110

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3


111

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1


112

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Babrauskas

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


Ma and MäkeläinenMa and Mäkeläinen define the critical value of

that separates the fuel-controlled and

ventilation-controlled regimes as

In the CIB tests, the ratio Afloor/A ranged from0.18 to 0.25. Ma and Mäkeläinen noted that Af /mf

typically ranges from 0.1 to 0.4 m2/kg, and that in aseries of Japanese tests Af /mf = 0.131 m2/kg.Substituting Afloor/A = 0.2, Af /mf = 0.131 m2/kg,

and m"f = 40 kg/m2, the critical value of

that separates the fuel-controlled and ventilation-

controlled regimes would be = 13.68.

Ma and Mäkeläinen estimate the maximumtemperature that would be achieved for ventilation-controlled fires would be:

For fuel-controlled fires, Ma and Mäkeläinenstate that the maximum temperature would be

where ηcr is the value

of that differentiates between fuel- and

ventilation-controlled burning (for the CIB data, ηcr was calculated as 13.68 m–1/2) and Tgmcr is thevalue of Tgm for η = ηcr. It should be noted that the above temperature correlations provide an esti-mation of the maximum temperature that would beattained during a fire; for the majority of the fireduration the temperature would be lower, and, hence,the average temperature during the fire would belower. Figure B.67 provides a comparison of pre-dicted maximum temperatures with the CIB data.

Ma and Mäkeläinen use Harmathy’s correlationto predict the burning rate for fuel-controlled burn-ing and Law’s correlation to predict the burning ratefor ventilation-controlled burning. See the discus-sion of those methods for an evaluation of theirburning rate predictions.

Comparisons of predictions to the Cardingtondata are presented in Figures B.68 through B.75.For test #6, Ma and Mäkeläinen’s method predictedtemperatures below ambient.

113

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

CIB Data

Ma (Max)

FIGURE B.67. Comparison of CIB Burning Rate Data to Predictions Made Using Ma andMäkeläinen’s Method

114

Tem

per

atu

re (

°C)

Measured

Ma

1400

1200

1000

800

600

400

200

0

Time (h)0 0.5 1 1.5 2

FIGURE B.68. Comparison of Predictions Made Using Ma and Mäkeläinen’s Method to Data fromCardington Test #1

Tem

per

atu

re (

°C)

Time (h)

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Measured

Ma


115

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Ma

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Ma

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4


116

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Ma

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Ma

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1


117

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Ma

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Ma

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


CIBThe temperature data from the Cardington tests

was compared to the temperature data from the CIBtests by averaging the temperatures measured at dif-ferent horizontal locations in the Cardington tests.These average temperatures were averaged over theduration of maximum burning and plotted alongwith the CIB data. Error bars on the Cardingtondata are included to show the range of temperaturesmeasured during the period of maximum burning.The results are shown in Figure B.76, with theabscissa plotted in logarithmic scale.

Predictions using the CIB method are comparedto data from the Cardington tests in Figures B.77through B.83. The compartment temperature andburning duration were predicted using the graphspresented earlier in this guide for cribs with 20 mmthick wood sticks spaced 20 mm apart. No decayrate was imposed, and for times greater than theduration the compartment temperature was assumedto be ambient.

118

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1 10 100 1000

CIB DataCardingtonCIB Curve

FIGURE B.76. Comparison of Cardington and CIB Temperature Data

119

Tem

per

atu

re (

°C)

Measured

CIB

1400

1200

1000

800

600

400

200

0

Time (h)0 0.5 1 1.5 2

FIGURE B.77. Comparison of Predictions Made Using the CIB Data to Cardington Test #1

Tem

per

atu

re (

°C)

Time (h)

0

Measured

CIB

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


120

Tem

per

atu

re (

°C)

Time (h)

0

Measured

CIB

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5


Tem

per

atu

re (

°C)

Time (h)

0

Measured

CIB

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4


121

Tem

per

atu

re (

°C)

Time (h)

0

Measured

CIB

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1


Tem

per

atu

re (

°C)

Time (h)

0

Measured

CIB

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3


LawFigure B.84 shows predictions of maximum tem-

perature using Law’s method compared to the CIBdata. Law’s method includes a means of reducingthe predicted temperature based on the fuel loading.However, for the range of conditions in the testsfrom which the CIB data were collected, utilizingthis factor would result in unrealistically low tem-peratures for some combinations of scale, openingfactor, and ventilation area. Therefore, this methodof reducing the temperature was not utilized.

Figure B.85 shows a comparison of burning rate predictions made using Law’s method to the

CIB data. Note that, because Law’s method con-siders the effect of compartment depth and width, the CIB burning rate data that was normalized by was utilized.

Comparisons of predictions made using Law’smethod to the Cardington data are shown inFigures B.86 through B.94. For times less than thecalculated burning duration, the temperature wascalculated using Law’s adjustment for fuel load. No decay rate was imposed, and for times greaterthan the duration the compartment temperature wasassumed to be ambient.

122

Tem

per

atu

re (

°C)

Time (h)

0

Measured

CIB

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


123

A/AoHo1/2 (m–1/2)

Tem

per

atu

re (

°C)

0

200

400

600

800

1000

1200

1400

0 10 20 30 40 50

CIB DataLaw (max)

FIGURE B.84. Comparison of CIB Temperature Data to Predictions Made Using Law’s Method

A/AoHo (m–1/2)

0

121221211441LawLaw X 1.4Law / 1.4

. mf/A

oH

o1/

2 (D

/W)1

/2 (

kg/s

– m

5/2 )

0.05

0.10

0.15

0.20

0.25

0 10 20 30 40 50 60

FIGURE B.85. Comparison of CIB Burning Rate Data to Predictions Made Using Law’s Method

124

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2

FIGURE B.86. Comparison of Predictions Made Using Law’s Method to Data from Cardington Test #1

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


125

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4


126

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8


127

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1


Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3


128

Tem

per

atu

re (

°C)

Time (h)

0

Measured

Law

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2


Time-Equivalent Methods

129

Appendix C

As stated in ASTM E119,105 standard furnacetests such as ASTME119; BS 476, Part 20106; andISO 834107 provide a relative measure of the firetest response of comparable assemblies understandardized fire exposure conditions. The exposureis not representative of all fire conditions becauseconditions vary with changes in the amount, nature,and distribution of fire loading; ventilation; com-partment size and configuration; and thermal char-acteristics of the compartment. Real fires can bemore or less severe in terms of duration, rate ofheating, and peak temperature than the standardtemperature–time relationship in a furnace test.

Real fires are a function of fuel load, compartmentdimensions, thermal properties of the compartmentboundaries, and the quantity of unprotected open-ings that allow ventilation in a post-flashover fire.

Also of importance is that the standard furnacetest does not assess real structural response in fireconditions because single elements of structure aretested in the furnace even though they form compo-nent parts of complex three-dimensional frames inreal buildings.

Various methods exist for designers to derivemore realistic temperature–time relationships forcompartments. For example, as a result of concernswith the standard furnace test temperature–timerelationship, work was carried out by Ingberg, Law, and Pettersson, among others, to determinewhat is known as an equivalent fire resistance. Forthese methods, the heating effect in a compartmentis based on real compartment fire behavior andtherefore takes into account fuel load density, venti-lation openings, compartment dimensions, andenclosure thermal properties. This allows someimprovement in the grading method based on thestandard furnace test that is currently assumed inbuilding codes worldwide.

This section describes various calculation pro-cedures for these time-equivalent methods. Limi-tations and assumptions for each method are

described. Pettersson’s method is put forward as the preferred time-equivalent method, and its rangeof use is outlined.

Real Structural ResponseIt is important to note that time-equivalent meth-

ods do not assess local or global structural response.They relate only to heating effects and their rela-tionship to the standard furnace test.

The t-equivalent methods do not address transienttemperature gradients or associated load-bearingcapacities. The ratings derived do not relate toactual frame performance in fire. These methods aresimply refined versions of performance of a singleelement in fire, but only relative to the standard fur-nace test. They normally assume insulated struc-tures only (protected steel or reinforced concrete).Pettersson’s work, however, does address uninsu-lated steel also. In the work carried out for theNatural Fire Safety Concept,108 good correlationwas achieved when the t-equivalent results werecompared to real fire test data for insulated steelstructures. The results for uninsulated steel struc-tures gave very poor correlation, as would beexpected. Bare steel tends to follow the furnace testcurve, so use of bare steel elements, in terms ofstandard fire resistance, would not be expectedbeyond 20 to 30 minutes depending on section size.

A time-equivalent calculation does not apply ifthe pre-flashover calculations show that flashoverwill not occur, i.e., the calculation is no longerrelevant if flashover has not occurred. Then, localheating effects are relevant, not temperatures in auniformly heated compartment, as is assumed intime-equivalent analysis methods.

Time-equivalent methods are empirical formulaedeveloped by regression analysis using a selectednumber of tests or calculations. Therefore, theyhave been developed for a certain range of struc-tural steel sizes and thicknesses of insulation and somay not be appropriate outside this range.

They are used for other materials, but beyondprotected steel and reinforced concrete very little isknown of the accuracy in applying this method toother materials.

Note that all t-equivalent methods described hereinvolve combustible solids only.

Discussion of MethodsTime-equivalent methods can be described as

methods that define the thermal exposure of a par-ticular compartment fire in terms of the duration ofthe equivalent standard fire.

Equivalence of thermal exposure has beendefined in two ways:

1. Equal areas under the temperature–time curves2. Equal temperatures at the critical part of a

structural element

The two methods give similar results where theelement selected has a fire resistance of the order ofhalf an hour or more.

FIRE LOAD CONCEPT

By 1918 there was a concern in the fire protec-tion and code enforcement communities that therewas no accepted method for establishing appropri-ate levels of fire endurance for buildings of differentsizes and occupancies.109 The original work hadbeen based on “fireproof” large commercial build-ings. It was recognized that these differed signifi-cantly from residential fires, but it was not under-stood how their severity related to the conditions inthe now-formulated standard fire resistance test.

To develop a solution to this problem, in 1922the National Bureau of Standards investigated thenature of building fires under the direction of SimonIngberg.110 The main aim was to determine theintensity and duration of uncontrolled fires in par-ticular occupancies resulting from different levels of fuel load. Ingberg was also to investigate thevalidity of the standard temperature–time curve.

Ingberg investigated office and record storage-type occupancies. The effects of the building size andfuel load, combustible and noncombustible flooring,plus wood and steel furniture were investigated.

130

Tem

per

atu

re (

°F)

Time (h)

2400

2000

1600

1200

800

400

00 0.5 1.0 1.5 2.0 2.5 3.0

Area 1

Area 2

The "fire severity" is considered to be the same when Area 1 = Area 2

1200

1000

800

600

400

200

Tem

per

atu

re (

°C)

FIGURE C.1. Fire Severity Concept109

As a result of these tests, Ingberg established asimple relationship between the average weight ofcombustible material within a room and the fireendurance necessary to withstand a completeburnout of the contents. This is known as the “fuelload concept.” It assumes that the area under anytemperature–time curve from ignition through decayprovides a comparative measure of fire severity, andthat fire severity is a function of the fuel load only.

Ingberg compared the area under the tempera-ture–time curves generated in the burnout tests to anequivalent area under the standard temperature–timecurve. The areas below a threshold temperature ofabout 300°C were not taken into account. The graphin Figure C.1 shows the basis for Ingberg’s work.

Ingberg developed the following relationship fortime-equivalence:

te = k1m"f (Eq. C.1)

Where:te = t-equivalent (min)m"f = Fuel load (wood) per unit floor area k1 = Unity when m"f is in units of kg/m2;

k1 = 5 when m"f is in units of lb/ft2

Ingberg’s work became widely accepted as thegeneral basis for establishing fire endurancerequirements.

KAWAGOE AND SEKINE

In 1963, Kawagoe and Sekine111 went on toshow the importance of the ventilation parameter:

Where:Ho = Window height (m) Ao = Total area of openings (m2) A = Total area of inside surfaces including

opening area (m2)

Kawagoe and Sekine also developed a formulafor fire duration and defined it as the period fromthe beginning of temperature rise until the time thetemperature drops after most of the combustiblematerial is burnt.

This time, τ is approximated as:

(min) (Eq. C.2)

Where:τ = Time (min)m"f = Fuel load (kg/m2)Afloor = Floor area (m2)H = Height of the window (m)Ao = Area of the windows (m2)

LAW

Law developed a t-equivalent formula112,113

from the results of the CIB test program.114

The maximum temperature that would beattained by a protected steel element in a real firecompartment was chosen as a basis for comparisonwith the heating effect in a standard fire.

For a temperature–time curve, the maximumtemperature obtained by a protected steel element ina compartment fire is calculated as:

(Eq. C.3)

Where:Ts = Steel temperature (K)t = Time (s)T = Fire temperature (K)R = δ i /(kiPH)δ i = Thickness of insulating material (m)ki = Thermal conductivity of insulating material

(kW/m-K)P = Heated perimeter of steel member (m)H = Height (or length) of steel member (m)C = AHρscs

A = Cross-sectional area of steel member (m2)ρs = Density of steel (kg/m3)cs = Specific heat of steel (kJ/kg-K)

The temperature of the heated surface of the pro-tective material is assumed to be the same as thefire temperature. The heat transfer through the steelsection can then be calculated.

For a given temperature–time curve, the value RCwas determined so that the maximum temperatureof the protected member was 550°C.

131

The time for the protected member to attain550°C when exposed to the standard temperature–time curve gives the value of t-equivalent.

The best correlation was obtained from theproduct (mf /Ao) and a term taking into account Ao

and the solid surface to which heat is lost:

(Eq. C.4)

Where:Afloor = Floor area of the compartment (m2)mf = Fuel load (wood equivalent) (kg) Ao = Area of ventilation opening (m2)k3 = 1.3 to 1.5, depending on the stick

spacing in the cribs used as fuel (min m2/kg)

A = Surface area of interior of enclosure(walls, floor, ceiling, and openings) (m2)

In this correlation, Afloor was not included in theevaluation of solid surfaces because the floors werevery well insulated.

In all experiments the openings were the fullcompartment height.

The values of t-equivalent were found to be inde-pendent of scale and height of ventilation openings.

Law then analyzed temperature–time data from anumber of fires in larger brick and concrete com-partments (approximately 3 m high)115,116 withfuels consisting of wood cribs, furniture, and liquidfuels, and developed

(Eq. C.5)

where k4 is 1.0. This was due to the little effect fuel arrangement appeared to have in these largerscale tests.

In this correlation, the floor area was included inthe evaluation of solid surfaces to which heat is lost.The larger scale data also showed no significanteffect of ventilation opening height on te.

Law concluded this equation (C.5) was most suit-able for engineering purposes for protected steelcolumns and went on to demonstrate that it gavegood results for reinforced concrete also. She dis-covered that it overestimates the time prediction fortightly baled paper and cloth.

PETTERSSON

In 1976, Pettersson117 adopted Law’s approach tot-equivalent, but, instead of the experimental curveson which her work was based, used the family ofcalculated temperature–time curves for particularcompartments as derived by Magnusson andThelandersson.118

When the fuel load is expressed in mass (kg) ofwood instead of “effective calorific value” (MJ),Pettersson’s expression for t-equivalent is as follows:

(Eq. C.6)

Where:Ho = Height of vertical opening (m)A = Total area of internal envelope (walls,

floor, ceiling, and openings). (Note that inhis original heat balance work he excludedAo but for an unstated reason does not in his final equations presented in hisdesign guide.)

This equation includes because of the inputparameters in the method for calculating thetemperature–time curves on which this equation is based.

Equation C.6 can be modified to take intoaccount the thermal properties of the compartmentenclosure by applying the factor kf to each inputparameter.

This yields:

(Eq. C7)

where kf = factor applied to input parameters to takeaccount of the thermal properties kρc of the com-partment enclosure expressed as a proportion of the kρc for Pettersson’s “standard” compartment.This is the compartment, defined in the SwedishBuilding Regulations in 1967, as where the sur-rounding structure has the thermal properties of anaverage of concrete, brick, and lightweight concretewith a thickness of 20 cm. (Note also that the fire isventilation controlled and with a cooling phase of10°C/min.)

132

NORMALIZED HEAT LOAD CONCEPT

In 1983, Harmathy and Mehaffey119 developedthe “normalized heat load” concept. The total heatpenetrating the compartment boundaries is calcu-lated taking into account and the proportionof heat evolution in the compartment, χ. When nounburnt gases emerge from the compartment, χ = 1.

Based on the results of many experiments andtests using the Division of Building Research/National Research Council of Canada floor testfurnace,119 they derived the following relationshipfor t-equivalent:

(s) (Eq. 8)

for0 < HN < 9 × 104

Where:

(Eq. 9)

χ = or 1, whichever is less

H = Compartment height (m)k = Thermal conductivity

(kW/m K)ρ = Density (kg/m3)c = Specific heat (kJ/kg K)

te from Equation C.9 is then given approximately by:

te = 0.0016HN (Eq. C.10)

forHN ≤ 9 × 104

EUROCODE TIME-EQUIVALENT METHOD

The Eurocode120 defines t-equivalent asdescribed in the German standard DIN 18230, ver-sion 94, method.121 The derivation of this formulahas never been published, but it is understood tohave come from an empirical analysis of calculatedsteel temperatures in a large number of simulatedfires computed by the German program “MultiRoom Fire Code.”121 Though this reference refersto an earlier published version of the Eurocode, thebasic formulations remain, and therefore this originis believed to still apply.

This method is dependent on ceiling height of thecompartment but not the opening height. The fueltype assumed in the original work is unknown,though it is widely believed to be cellulosic.

The t-equivalent is defined in the Eurocode as:

te,d = qf,dkbwf kc (Eq. C.11)

Where:qf,d = Fuel load density related to the floor area

(MJ/m2), which can be calculatedaccording to

qf,d = qf,k mδq1δq2δn (Eq. C.12)

Where:qf,k = Fuel load density determined from a fuel

load classification of occupancies (seeTable C.1)

m = Combustion factor, which for cellulosicmaterials is defined as 0.8

δq1 = Safety factor taking account of the risk ofa fire starting due to the size ofcompartment (see Table C.2)

δq2 = Safety factor taking account of the risk ofa fire starting due to the type ofoccupancy (see Table C.3)

δn = Factor taking account of the differentactive fire-fighting measures such assprinklers, detection, fire fighters, etc.)(see Table C.4)

133

134

Compartment Danger of Floor Area Fire Activation

Af (m2) (δq1)

25 1.1

250 1.5

2500 1.9

5000 2.0

10000 2.13

TABLE C.3. Safety Factor Taking Account of the Risk of a Fire Starting Due to the Type of Occupancy120

Danger of Fire Starting (δq2) Examples of Occupancies

0.78 Art gallery, swimming pool

1.00 Offices, hotel, residential

1.22 Manufacturing for machinery and engines

1.44 Chemical lab, panting workshop

1.66 Manufacturing of fireworks or paints

TABLE C.2. Safety Factor Taking Account of the Risk of aFire Starting Due to the Size of Compartment120

80% Occupancy Average Fractile

Dwelling 780 948

Hospital (room) 230 280

Hotel (room) 310 377

Library 1500 1824

Office 420 511

Classroom of a school 285 347

Shopping center 600 730

Theater (cinema) 300 365

Transport (public space) 100 122

Gumbel distribution is assumed for the 80% fractile

TABLE C.1. Fuel Load Density Determinedfrom a Fuel Load Classification ofOccupancies120

kb is a conversion factor = 0.07 (min m2 /MJ)when no detailed assessment of the thermal proper-ties of the boundary is pursued, and when qd isgiven in MJ/m2.

Otherwise kb may be related to the thermal

property in accordance with Table C.5:

wf is calculated as:

(Eq. C.13)

Where:αv = Ao /Afloor = Area of vertical openings A0

in the façade related to the floor area of the compartment where the limit 0.025 ≤ αv ≤ 0.25 should be observed

αh = Ah /Afloor = Area of horizontal opening inthe roof related to the floor area of thecompartment

bv = 12.5(1+10α v – α v2) ≥ 10

H = Height of the compartment (m)

For small fire compartments (defined in theEurocode as Afloor < 100 m2) without openings inthe roof, the factor wf may also be calculated as:

(Eq. C.14)

Where:

0.02 ≤ ≤ 0.20 with the default value

kb = 0.07 and assuming 18 MJ/kg forwood, Equation C.14 becomes the same as Equation C.7.

Kc = A correction factor that is a function of the material composing structural cross sections and is defined as

13.7 for unprotected steel.

Reinforced concrete and protected concrete remain as 1.

135

δni Function of Active Fire-Fighting Measures

Automatic Fire Suppression Automatic Fire Detection Manual Fire Suppression

(δn1) 0 1 2 By By (δn5) (δn6) (δn7) (δn8) (δn9) (δn10)heat smoke(δn3) (δn4)

0.61 1 0.87 0.7 0.87 or 0.73 0.87 0.61 or 0.78 0.9/1/1.5 1/1.5 1/1.5

Note: According to the Eurocode, for “normal fire-fighting measures” such as safe access routes,firefighting devices, and smoke exhaust systems in staircases, the factors should be taken as 1.0, andif these measures have not been foreseen but provided, then the values can be taken as 1.5.

TABLE C.4. A Factor Taking Account of the Different Active Fire-Fighting Measures (Sprinklers,Detection, Fire Fighters, Etc.)120

Auto waterextinguishing

system

Independentwater

supplies(δq2)

Auto firedetection and alarm

Autotransmission

to fire brigade

Work fire

brigade

Off-sitefire

brigade

Safeaccessroutes

Fire-fightingdevices

Smokeexhaustsystem

Kb

J/m2 s1/2 K min m2 /MJ

b > 2500 0.04

720 ≤ b ≤ 2500 0.055

b < 720 0.07

TABLE C.5. Relationship Between kb and theThermal Property b

The basis of this method is that it should be veri-fied that te,d < tfi,d where tfi,d is the design value ofthe standard fire resistance of the members, assessedaccording to the relevant parts of the Eurocode.

This method could therefore be used for otherdefined periods of fire resistance such as in U.S. codes.

NEW ZEALAND CODE

The New Zealand Fire Engineering DesignGuide122 gives the same empirical expression forequivalent fire severity te (min) as the Eurocode.

The upper and lower kb values have beenincreased by a factor of 1.3 compared to the Euro-code due to what it declares inherent uncertaintiesin the Eurocode formula, the use of fuels other thanwood, structures other than steel, and deep compart-ment effects. The values for kb recommended by theNew Zealand Fire Engineering Design Guide areshown in Table C.6. If the properties of the liningsare not known, a value of kb = 0.09 is suggested.This formula is based on cellulosic-type fuels.

The ventilation factor limits of use are retained,though the small compartment formula in theEurocode does not form part of the New Zealandguidance.

ComparisonsTime-equivalent methods are an improvement on

the grading method in building codes worldwide,which is based on the standard fire temperature–time relationship (such as ASTM E119, BS 476 Part 20, or ISO834). This is because they attempt to

account for compartment geometry, ventilationopenings, fuel load density, and compartmentboundary materials in addition to fuel load density,the key factors that affect full-scale fire develop-ment. However, the temperatures calculated onthese principles are then related back to the standardtemperature–time relationship. It is also importantto note that they are based on specific compartmenttest data rather than generalized heat balance solu-tions. Time-equivalent methods are therefore unlikenatural temperature time relationships, which repre-sent a real temperature–time relationship and areused as such, independent of the standard fire resist-ance test formulation.

Drysdale14 describes a comparison Harmathycarried out where the Ingberg, Law, Pettersson, andHarmathy equations for te were compared. Drysdalerejects Ingberg’s method since radiative heat fluxvaries with T4, which makes simple scaling impos-sible when heat transfer is dominated by radiation.He concluded that Law and Harmathy providedmore conservative solutions than the others. Notethat Ingberg’s method ignores ventilation, unlike theother methods presented here.

Law compared results using the time-equivalentrelationships by Ingberg, Kawagoe, Law,Pettersson, Harmathy, and Mehaffey, plus the 1993Eurocode formula with experimental data frompost-flashover fires in full-scale compartments.115

These consisted of small insulated compartments,30 m2 area, 2.5 to 3 m high, with brick or concreteenclosures,113 and larger, deeper rooms 128 m2 inarea (depth to width ratio 4:1).123 Law concludedLaw, Pettersson, Harmathy, and Mehaffey were themost promising methods.

136

(J/m2Ks1/2) Construction Materials kb Value

400 Very light insulating materials 0.10

700 Plasterboard ceiling and walls, timber floor 0.09

1100 Lightweight concrete ceiling and floor, plasterboard walls 0.09

1700 Normal concrete ceiling and floor, plasterboard walls 0.065

2500 Thin sheet steel roof 0.045

TABLE C.6. Values for kb Recommended by the New Zealand Fire EngineeringDesign Guide

Limitations and Assumptions

THE DEEP COMPARTMENT EFFECT

Law examined deep compartments further sinceall her derived time-equivalent formulae gave oddresults when deep compartments were studied. Indeep compartments, temperatures and local burningrates are not uniform, but rather progress from theopening toward the back of the enclosure as fuel isdepleted.103 Law also discovered that the 1993Eurocode t-equivalent method gives poor correla-tion for both small and deep compartments.115

She concluded that the depth of the compartmenthas an effect on time-equivalent over and abovewhat can be allowed for by increase in insulationand in internal surface area A. Thomas andHeselden114 had already shown that the ventilation-controlled rate of burning is affected by the com-partment depth to width ratio.

Recent research on this phenomenon has alsoresulted in the New Zealand code’s recommendingfactors of safety that have been increased by 30% toaccount for this effect in its time-equivalent formula.

THE EUROCODE

The Eurocode formulae do not reference thesource of the equation derivations, particularly theventilation factor needed in the time-equivalent cal-culation, as well as the correction factor to takeaccount of cross section material types, plus theother factors of safety recommended for applicationto the calculated time-equivalent value.

kc is defined for unprotected steel as 13.7 times

the opening factor. Since 0.02 < < 0.2, this

gives 0.27 < kc < 2.7 for unprotected steel.For small compartments (Afloor < 100 m2)

wf = , which can be written for

unprotected steel in small compartments:

Ted = 13.7 qfd × kb ×

(fuel load given as per unit floor area)Pettersson calculated Te (h) for unprotected steel

for values of ranging from 0.2 to 0.12 m1/2.

137

200

180

160

140

120

100

80

60

40

20

0

L" Af/[Av(At – Av )]1/2

0 10 20 30 40 50 60 70 80 90 100

Small StandardCompartment

Deep InsulatedCompartment

Small InsulatedCompartment

t e

FIGURE C.2. Law’s Correlation Between Fire Resistance Requirements (tf) and L/(AW AT)1/2 115

L = Fuel load (kg)AW = Area of the ventilationAT = Total internal surface area (m2)

of the compartment

Pettersson’s time-equivalent formula is comparedwith the Eurocode formula for resultant emmisivity,εr, of 0.5 for unprotected steel as follows.

Pettersson

Section factor = 50 m-1

Qt,d Te (h)

= 0.02 0.04 0.08 0.12

42 0.29 0.25 0.21 0.16

84 0.42 0.43 0.36 0.29

126 0.51 0.58 0.525 0.41

Section factor = 150 m-1

Qt,d Te (h)

= 0.02 0.04 0.08 0.12

42 0.21 0.23 0.19 0.16

84 0.27 0.37 0.38 0.30

126 0.30

It can be seen that, while the values of Te tend toincrease with increasing fuel load, they tend todecrease with ventilation factor. They are not inde-pendent of section factor.

Yet when the Eurocode formula with kb = 0.055for Pettersson’s compartment type A, for all sec-tions, is used to do a similar check, the followingdata is produced.

Eurocode

Qt,d Te (h)

= 0.02 0.04 0.08 0.12

42 0.07 0.11 0.15 0.18

84 0.15 0.21 0.30 0.37

126 0.22 0.32 0.45 0.55

The trends in the Eurocode are different, and noexplanation as to why has been provided.

For unprotected steel, if the steel is assumed tobe at a uniform temperature, the post-flashover firetemperatures are also perfectly stirred, and thetemperature of the exposed surface is the same asthe fire temperatures, this implies the steel tempera-ture is always the same as the fire temperature.Therefore, section factors of 100 m–1 or morewould not be expected to survive a post-flashoverfire, but a localized fire.

Until suitable justification of such difference is made, it seems the original work of Pettersson or Law is best suited to this type of time-equivalent calculation.

138

Examples

139

Appendix D

Example 1A room 5 m wide, 4 m deep, and 2.5 m high has

one vent that is 2 m high and 3 m wide. The fuelload is 10 lb/ft2. The enclosure is made of gypsumplaster on metal with the following properties:

k = 0.47 W/m°Cρ = 1440 kg/m3c = 0.84 kJ/kg°C

Find the maximum temperature of the fire and itsburning duration.

Law’s method is recommended for all roughlycubic compartments and in long, narrow compart-

ments where is approximately less than or

equal to 18 m–1/2. Since this room is roughly cubic,Law’s method is applicable.

A = 2(5 * 4) + 2(5 * 2.5) + 2(4 * 2.5) = 85 m2

Ao = 2 * 3 = 6 mHo = 2 m

MAXIMUM TEMPERATURE

Thus,

BURNING DURATION

Where:mf = m"f × Afloor

m"f = 10 lb/ft2 or 49 kg/m2

Afloor = 20 m2

mf = 49 × 20 = 980 kg

and

where W is width of the room and D is depth of the room.

This equation is valid for

and in this case

To ensure that predictions are sufficiently conser-vative using Law’s method, the predicted burningrate should be reduced by a factor of 1.4.

The adjusted burning rate is then

The burning duration can be found by

Example 2A room 7 m wide, 28 m deep, and 4 m high has

one vent that is 4 m high and 3.5 m wide in one of the small end walls. The fuel load is 35 kg/m2.The enclosure is made of brick with the followingproperties:

k = 0.69 W/m°Cρ = 1600 kg/m3c = 0.84 kJ/kg°C

Find the burning duration, and plot the tempera-ture–time curve.

For long, narrow spaces in which the value

of is in the range of 45 to 85 m–1/2,

Magnusson and Thelandersson’s method is recommended.

A = 2(7 * 4) + 2(7 * 28) + 2(4 * 28) = 672 m2

Ao = 1.9 * 3.7 = 7 mHo = 3.7 m

In this case

and Magnusson and Thelandersson’s method is used.

The first step is to decide which of the sevenmodels in Magnusson and Thelandersson’s methodis applicable to the problem.

Type A enclosed spaces consist of a material, 20 cm in thickness, whose thermal properties arecharacterized by the following average values.

These characteristics normally belong to concrete,brick, and lightweight concrete.

Thermal conductivity: k = 0.7 kcal/m-h-°CProduct of the specific heat and the density,c * ρ = 400 kcal/ m3°C

In this case k = 0.5937 kcal/mh°C and c * ρ =321.22 kcal/ m3°C. Thus, Type A enclosed space is used.

The next step is to calculate the burning duration, τ.

q, the fire load per bounding surface area, iscalculated using the fuel load and the heat of com-bustion, ∆Hc. A heat of combustion of cellulosicmaterials, 15 MJ/kg, is used for this example.

q = 35 kg/m2 *(7 * 28)/672 * 15 MJ/kg = 153 MJ/m2 = 37 Mcal/m2

Therefore,

The type of enclosure, opening factor, and theburning duration can be used to referenceMagnusson and Thelandersson’s tables. The tablesgive the temperature at 0.05-, 0.10-, and 0.20-hourintervals up to 6.00 hours for various burningdurations. The temperatures for a burning durationof 1.5 hours and an opening factor of 0.02 m1/2

were used to create the temperature–time curve inFigure D.1.

140

141

0

100

200

300

400

500

600

700

800

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Time (h)

Tem

per

atu

re (

°C)

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

143

Glossary

Nomenclature Used in the Enclosure Fires Section

A Surface area of interior of enclosure (m2)Af Surface area of fuel (m2)Afloor Surface area of floor (m2)Ao Area of ventilation opening (m2)b Stick width (m)bp Factor (-)C Wood constant (g/m1.5-s)c Specific heat of enclosure lining (J/kg-K)cp Specific heat of air (J/kg-K)D Depth of compartment (m)F View factor (-) or opening factor (m1/2)G Gravitational constant (9.81 m/s2)h Equivalent conductance (W/m2-K)hc Convection coefficient (W/m2-K)hk Conduction coefficient (W/m2-K)hr Radiation coefficient (W/m2-K)H Height of compartment (m)Ho Height of ventilation opening (m)k Thermal conductivity of enclosure lining (W/m-K)ko Coefficient (-)L Latent heat of vaporization (kJ/g)mf Mass of fuel (kg)

Mass burning rate of fuel (kg/s)m"f Mass of fuel per unit area (kg/m2)

Mass burning rate of fuel per unit area (kg/m2-s)Free burning mass loss rate of fuel per unit area (kg/m2-s)Asymptotic mass loss rate of fuel per unit area (kg/m2-s)Mass flow rate of air (kg/s)

p Pressure (Pa)q Fuel load density (Mcal/m2)

Heat loss rate (kW)Heat loss through walls (kW)Heat flux from fire (kW/m2)Effective heat flux (W/m2)Heat release rate (kW)Rate of the heat energy stored in the gas volumeRate of heat energy withdrawn from the enclosed space due to air flowRate of heat energy withdrawn from the enclosed space by radiationRate of heat energy withdrawn from enclosed space through the wall, floor, or ceiling

Q* Dimensionless heat release rate (-)

144

Qf* Dimensionless radiation rate to fuel (-)

Qr* Dimensionless radiation loss rate (-)

Qw* Dimensionless heat loss rate to walls (-)

R Universal gas constant (8.31 J/kMol-K)s Atoichiometric air to fuel ratio (-)t Time (units as stated)tm Time corresponding to maximum temperature (units as stated)T Temperature in compartment (units as stated)Tb Fuel boiling point (units as stated)Tf Flame temperature (units as stated)Tgm Maximum temperature (units as stated)Tgmcr Maximum temperature in the critical region (units as stated)To Ambient temperature (units as stated)Tw Wall temperature (units as stated)V Volume (m3)W Width of wall containing ventilation opening (m)YO2 Mass fraction of O2 (-)

GREEK

β Factor (-)Γ Scaling factorφ Equivalence ratio (-)δ Thickness (m) or shape factor (-)∆Hp Heat of vaporization of liquid (kJ/kg)∆Hc Heat of combustion (MJ/kg)∆Hair Heat of combustion per unit mass of air (MJ/kg)ε Gas emissivity (-)εw Wall emissivity (-)κ Absorbsion coefficient (m–1) or factor (-)θ1-θ5 Variable (-)η Factor (-)ηcr Factor (-)ρ Density of enclosure lining (kg/m3)ρ0 Density of air (kg/m3)σ Stefan-Boltzmann Constant [5.67 × 10–11 kW/(m2 K4)]τ Duration of fully developed fire (units as stated)ν Kinematic viscosity (m2/s)ς Factor (-)Ψ Factor (kg/m2)

Nomenclature Used in the Plumes Section

A Surface area of noncircular fuel package (m2)bu Plume width (m)bt Thermal plume width (m)Cp Specific heat capacity of air at 300 K [1.0 kJ/(kg K)]D Length of single side of square burner, diameter (m)g Acceleration of gravity (9.81 m/s2)h Convective heat transfer coefficient [kW/(m2 K)]H Distance between base of fire and ceiling (m)HB Distance between base of fire and lower flange of I-beam (m)HC Distance between base of fire and upper flange of I-beam (m)HW Distance between base of fire and center of web on I-beam (m)h Convective heat transfer coefficient [kW/(m K)]LB Distance from fire centerline to flame tip along lower flange of an I-beam (m)LC Distance from fire centerline to flame tip along upper flange of an I-beam (m)LH Distance from fire centerline to flame tip length along ceiling or upper flange of an I-beam (m)LW Distance from fire centerline to flame tip length along the web center of an I-beam (m)Lf Average flame length or unconfined flame tip length (m)Lf,tip Flame tip length (m)Lf,tipB Flame tip length along lower flange of I-beam (m)Lf,tipC Flame tip length along upper flange of I-beam (m)Lf,tipW Flame tip length along web center of an I-beam (m)Q Fire heat release rate (kW)Q* Dimensionless parameter, , with D being length scale

r Distance from corner or stagnation point to measurement location or radial distance for plume centerline (m)Heat flux (kW/m2)

T Temperature (K)Tm,c Centerline plume temperature (K)Tg Room gas temperature (K)Ts Material surface temperature (K)T∞ Ambient temperature (300 K)U Plume velocity (m/s)Um,c Centerline plume velocity (m/s)w Dimensionless distance along ceiling or I-beam, w = (r + H + z' )/(LH + H + z' ) x Horizontal distance from corner or fire centerline or width distance into the material thickness (m)y Horizontal distance from corner (m)z Vertical distance above base of fire (m)z' Virtual source origin correction in tests with fires impinging on ceilings and I-beams (m)zo Virtual source origin correction for plumes (m)

145

GREEK

α Absorbtivity (- -)χ r Radiative fraction (- -)ε Emissivity (- -)ρ∞ Ambient density of air (1.2 kg/m3)π Constant (3.14159)σ Stefan-Boltzmann constant [5.67 × 10–11 kW/(m2 K4)]

SUBSCRIPTS

cl Centerlineconv ConvectiveD Defined using D as length scalef Flamehfg Heat flux gaugeH Defined using H as length scaleHB Defined using HB as length scaleHC Defined using HC as length scaleinc Incident m Measuredmax Maximum levelnet Net peak Peakrad Radiative rr Reradiateds Material surfacew Centerline of web

146

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114 Thomas, P., and A. Heselden, “Fully Developed Fires in Single Compartments – a Co-operative ResearchProgramme of the Conseil International du Batiment,” CIB Report No.20, Research Note No. 923,Borehamwood, England, Fire Research Station, 1972.

115 Law, M., “A Review of Formulae for T-Equivalent,” Fire Safety Science – Proceedings of the FifthInternational Symposium, International Association of Fire Safety Science, 1997.

116 Law, M., “Prediction of Fire Resistance,” Fire-Resistance Requirements for Buildings – A New Approach,Proceedings, Symposium No. 5, Borehamwood, England, Fire Research Station, 1971.

117 Pettersson, O., S. Magnusson, and J. Thor, “Fire Engineering Design of Steel Structures,” Swedish Institute ofSteel Construction, 1976.

118 Magnusson, S., and S. Thelandersson, “Temperature–Time Curves of Complete Process of Fire Development,”Civil Engineering and Building Construction Series No.65, Stockholm, ACTA Polytechnica Scandinavica, 1970.

119 Harmathy, T., and J. Mehaffey, “Post-Flashover Compartment Fires,” Fire and Materials 7:2 (1983) 49–61.120 British Standards Institute, “Actions on Structures Exposed to Fire,” Part 1.2: General Actions, Eurocode 1:

Actions on Structures, BS EN 1991-1-2:2002. 121 Thomas, G., A. Buchanan, and C. Fleischmann, “Structural Fire Design: The Role of Time Equivalence,”

Fire Safety Science – Proceedings of the Fifth International Symposium, International Association of FireSafety Science, 1997.

122 Fire Engineering Design Guide, Report of a Study Group of the New Zealand Structural Engineering Societyand the New Zealand National Fire Protection Association, A. Buchanan, ed., 1994.

123 Kirby, B., et al., “Natural Fires in Large Scale Compartments,” A British Steel Technical, Fire Research StationCollaborative Project, British Steel Technical Swinden Laboratories, 1994.

151

Organized in 1950, the Society of Fire Protection Engineers is the professional society for engineersinvolved in the multifaceted field of fire protection engineering. The purposes of the society are to advance the science and practice of fire protection engineering, to maintain a high ethical standing among itsmembers, and to foster fire protection engineering education. Its worldwide members include engineers inprivate practice; in industry; and in local, regional, and national government, as well as technical membersof the insurance industry. Chapters of the society are located in Canada, France, Italy, Sweden, Japan,Hong Kong, New Zealand, and the United States.

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