NIST Special Publication 1026

April 2009 Revision

CFAST – Consolidated Model of Fire Growth

and Smoke Transport (Version 6)

Technical Reference Guide

Walter W. Jones

Richard D. Peacock

Glenn P. Forney

Paul A. Reneke

NIST Special Publication 1026

April 2009 Revision

CFAST – Consolidated Model of Fire Growth

and Smoke Transport (Version 6)

Technical Reference Guide

Walter W. Jones

Richard D. Peacock

Glenn P. Forney

Paul A. Reneke

Fire Research Division

Building and Fire Research Laboratory

April 29, 2009

SVN Repository Revision: 125

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

National Institute of Standards and Technology

James M. Turner, Deputy Director

Disclaimer

The U. S. Department of Commerce makes no warranty, expressed or implied, to users of CFAST

and associated computer programs, and accepts no responsibility for its use. Users of CFAST

assume sole responsibility under Federal law for determining the appropriateness of its use in any

particular application; for any conclusions drawn from the results of its use; and for any actions

taken or not taken as a result of analyses performed using these tools. CFAST is intended for use

only by those competent in the field of fire safety and is intended only to supplement the informed

judgment of a qualified user. The software package is a computer model which may or may not

have predictive value when applied to a specific set of factual circumstances. Lack of accurate

predictions by the model could lead to erroneous conclusions with regard to fire safety. All results

should be evaluated by an informed user.

i
ii
Intent and Use

The algorithms, procedures, and computer programs described in this report constitute a methodology for predicting some of the consequences resulting from a prescribed fire. They have been

compiled from the best knowledge and understanding currently available, but have important limitations that must be understood and considered by the user. The program is intended for use by

persons competent in the field of fire safety and with some familiarity with personal computers. It

is intended as an aid in the fire safety decision-making process.

iii
iv
Preface

CFAST is a two-zone fire model used to calculate the evolving distribution of smoke, fire gases

and temperature throughout compartments of a constructed facility during a fire. In CFAST, each

compartment is divided into two gas layers.

The modeling equations used in CFAST take the mathematical form of an initial value problem

for a system of ordinary differential equations (ODEs). These equations are derived using the

conservation of mass, the conservation of energy (equivalently the first law of thermodynamics),

the ideal gas law and relations for density and internal energy. These equations predict as functions

of time quantities such as pressure, layer height and temperatures given the accumulation of mass

and enthalpy in the two layers. The CFAST model then consists of a set of ODEs to compute the

environment in each compartment and a collection of algorithms to compute the mass and enthalpy

source terms required by the ODEs.

In general, this document provides the technical documentation for CFAST along with significant information on validation of the model. It follows the ASTM E1355 guide for model

assessment. The guide provides several areas of evaluation:

• Model and scenarios definition: CFAST is designed primarily to predict the environment

within compartmented structures which results from unwanted fires. These can range from

very small containment vessels, on the order of 1 m3to large spaces on the order of 1000

m3. The appropriate size fire for a given application depends on the size of the compartment

being modeled. A range of such validation exercises is discussed in chapter 6.

• Theoretical basis for the model: Details of the underlying theory, governing equations, correlations, and organization used in the model are presented. The process of development

of the model is discussed with reference to a range of NIST memorandums, published reports, and peer-reviewed journal articles on the model. In addition to overall limitations of

zone-fire modeling, limitations of the individual sub-models are discussed.

• Mathematical and numerical robustness: CFAST has been subjected to extensive use and

review both internal to NIST and by users worldwide in a broad range of applications. In

addition to review within NIST independent of the model developers, the model has been

published in international peer-reviewed journals worldwide, and in industry-standard handbooks referenced in specific consensus standards. Besides formal internal and peer review,

CFAST is subjected to continuous scrutiny because it is available to the general public and

is used internationally by those involved in fire safety design and post-fire reconstruction.

• Model sensitivity: Many of the outputs from the CFAST model are relatively insensitive to

uncertainty in the inputs for a broad range of scenarios. Not surprisingly, the heat release

v
rate is the most important variable because it provides the driving force for firedriven flows.

For CFAST, the heat release rate is prescribed by the user. Thus, careful selection of the fire

size is necessary for accurate predictions. Other variables related to compartment geometry

such as compartment height or vent sizes, while deemed important for the model outputs,

are typically more easily defined for specific design scenarios than fire related inputs.

• Model evaluation: The CFAST model has been subjected to extensive validation studies by

NIST and others. Although some differences between the model and the experiments were

evident in these studies, they are typically explained by limitations of the model and uncertainty of the experiments. Most prominent in the studies reviewed was the overprediction

of gas temperature often attributed to uncertainty in soot production and radiative fraction.

Still, studies typically show predictions accurate within about 30 % of measurements for a

range of scenarios. Like all predictive models, the best predictions come with a clear understanding of the limitations of the model and of the inputs provided to the calculations.

vi
Nomenclature

A area, m2

Aslab cross-sectional area of vent slab in horizontal vent flow, m2

Av cross-sectional area of a vent, m2

C vent constriction (or flow) coefficient, dimensionless

CLOL lower oxygen limit coefficient, fraction of the available fuel which can be burned

with the available oxygen, dimensionless

CT constant from plume centerline temperature calculation, 9.115

cp heat capacity of air at constant pressure, J/kg K

cv heat capacity of air at constant volume, J/kg K

D fire diameter, m

D

∗

characteristic fire diameter parameter, Qf /



ρ∞cpT∞

√g

2/5

vent diameter, m

d0 inlet ceiling jet depth in corridor flow, m

EO energy release per unit mass of oxygen consumed, J/kg

Eiinternal energy in layer i, W

Fk−j configuration factor, fraction of radiation given off by surface k intercepted by surface j, dimensionless

g gravitational constant, 9.8 m/s2

h convective heat transfer coefficient (W/m2 K)

h ̇

i rate of addition of enthalpy into layer i, J/s

h ̇

L rate of addition of enthalpy into lower layer in a compartment, J/s

h ̇U rate of addition of enthalpy into upper layer in a compartment, J/s

H height of a compartment, m

flame height, m

H1 distance from fire source to target location in plume centerline temperature calculation, m

H2 distance from virtual fire source to target location in plume centerline temperature

calculation, m

Hc heat of combustion of the fuel, J/kg

k equivalent thermal conductivity of air, W/m K, with subscripts c,e and s

k equilibrium coefficient for HCl transport and deposition

L length of a compartment, m

characteristic length for radiation calculation, m

m mass, kg

m ̇ e entrainment rate, kg/s

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m ̇ ex bi-directional vent flow in vertical flow vent, kg/s

m ̇ f pyrolysis rate of the fire, kg/s

mitotal mass in gas layer i, kg

m ̇io mass flow rate through a vent, kg/s

mL total mass in lower gas layer in a compartment, kg

m ̇ O oxygen required for full combustion of available fuel, kg/s

m ̇ p plume flow rate, kg/s

mU total mass in upper gas layer in a compartment, kg

P pressure at floor level of a compartment, Pa

Pb cross-vent differential pressure at the bottom of a vent flow slab in horizontal vent

flow, Pa

Pt cross-vent differential pressure at the top of a vent flow slab in horizontal vent flow,

Pa

Qceil average convective heat transfer from the ceiling jet to the ceiling surface, W

Qftotal heat release rate of the fire, W

Qf,C heat release rate of the fire released as convective energy, W

Qf,eq effective heat release rate of a vent fire, kW

Qf,R heat release rate of the fire released as radiation, W

Qspray spread density of a sprinkler

Q

∗

I,1

original fire strength for plume temperature calculation, dimensionless

Q

∗

I,2 modified fire strength for plume temperature calculation when target location is in

the upper layer, dimensionless

∆qˆ

00

k

net radiative flux at wall segment k

q

00 heat flux, W/m2

R universal gas conference, J/kg K

RT I thermal characteristic response time index of a sprinkler or heat detector (m1/2 s

1/2)

r radial distance from the fire, m

S vent coefficient for vertical flow vents, dimensionless

T∞ ambient gas temperature in compartment well removed from a target, K

Tamb ambient temperature, K

Ti gas temperature of layer i, K

TL gas temperature of lower layer in a compartment, K

Tp gas temperature in the plume, K

TU gas temperature of upper layer in a compartment, K

T0 inlet gas temperature of the ceiling jet in corridor flow, K

T

∗

1

calculated plume temperature at the transition between continuous flaming and intermittent flaming, K

T

∗

2

calculated plume temperature at the transition intermittent flaming and the fire

plume, K

t time, s

V total volume of a compartment, m3

VL total volume of lower layer in a compartment, m3

VU total volume of upper layer in a compartment, m3

Vi volume of gas layer i, m3

viii
v velocity, m/s

v0 inlet ceiling jet velocity in corridor flow, m/s

W width of a compartment, m

Wf molar mass of fuel, kg/mol

Ws molar mass of soot, kg/mol

YLOL mass fraction of oxygen below which combustion will no longer occur, dimensionless

YO2 mass fraction of oxygen in a gas layer, dimensionless

ys soot yield, mass of soot produced by the fire per unit mass of fuel, kg/kg

ZI,1 distance from fire source to the interface between upper and lower layers, m

ZI,2 distance from virtual fire source to the interface between upper and lower layers, m

z height above the base of the fire, m

zν height of the virtual origin of a vent fire, m

zp reduced height of the plume of a vent fire, m

z0 height of the virtual origin of fire, m

z

∗

1

height above the base of the fire at the transition between continuous flaming and

intermittent flaming, m

z

∗

2

height above the base of the fire at the transition intermittent flaming and the fire

plume, m

α gas absorptance, dimensionless

thermal diffusivity in conduction (m2/s)

β experimentally-determined constant in plume centerline temperature calculation,

β

2 = 0.913

γ ratio of cp/cv, dimensionless

∆T temperature rise, K

ε emissivity

ν stoichiometric coefficients for combustion reaction, dimensionless

kinematic viscosity, m2/s

ρ density, kg/m3

ρ∞ density of gas well removed from a target, kg/m3

ρc j density of the ceiling jet gas, kg/m3

ρi density of gas layer i, kg/m3

σ Stefan-Boltzman constant (5.67 x 10-8 W/m2K

4

)

τ transmittance, dimensionless

χC fraction of the fire’s heat release rate released as convective energy, dimensionless

χR fraction of the fire’s heat release rate released as radiation, dimensionless

ξ ratio of upper layer gas temperature to lower layer gas temperature, dimensionless

ix
x
Acknowledgments

Continuing support for CFAST is via internal funding at NIST. In addition, support is provided by

other agencies of the U.S. Federal Government, most notably the Nuclear Regulatory Commission

Office of Research and the U.S. Department of Energy. The U.S. NRC Office of Research has

funded key validation experiments, the preparation of the CFAST manuals, and the continuing

development of sub-models that are of importance in the area of nuclear power plant safety. Special

thanks to Mark Salley and Jason Dreisbach for their efforts and support. Support to refine the

software development and quality assurance process for CFAST has been provided by the U.S.

Department of Energy (DOE). The assistance of Subir Sen and Debra Sparkman in understanding

DOE software quality assurance programs and the application of the process to CFAST is gratefully

acknowledged. Thanks are also due to Allan Coutts, Washington Safety Management Solutions

for his insight into the application of fire models to nuclear safety applications and detailed review

of the CFAST document updates for DOE.

xi
xii
Contents

Disclaimer i

Intent and Use iii

Preface v

Nomenclature vii

Acknowledgments xi

1 Overview 1

1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Model and Scenario Definition 5

2.1 Model Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Name and Version of the Model . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Type of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.3 Model Developers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.4 Relevant Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.5 Governing Equations and Assumptions . . . . . . . . . . . . . . . . . . . 6

2.1.6 Input Data Required to Run the Model . . . . . . . . . . . . . . . . . . . . 7

2.1.7 Property Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.8 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.9 Uses and Limitations of the Model . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Scenarios for which the Model is Evaluated in this Document . . . . . . . . . . . . 10

2.2.1 Description of Scenarios of Interest . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 List of Quantities Predicted by the Model . . . . . . . . . . . . . . . . . . 11

2.2.3 Degree of Accuracy Required for Each Output Quantity . . . . . . . . . . 11

3 Theoretical Basis for CFAST 13

3.1 Derivation of Equations for a Two-Layer Model . . . . . . . . . . . . . . . . . . . 14

3.2 Equation Set Used in CFAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Limitations of the Zone Model Assumptions . . . . . . . . . . . . . . . . . . . . . 17

3.4 Source Terms for the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 The Fire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

xiii
3.4.2 Plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.3 Vent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.4 Corridor Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.5 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.6 Ceiling Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Heat Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6 Sprinkler Activation and Fire Attenuation . . . . . . . . . . . . . . . . . . . . . . 53

3.7 Species Concentration and Deposition . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7.1 Species Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7.2 HCl Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.8 Single Zone Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.9 Review of the Theoretical Development of the Model . . . . . . . . . . . . . . . . 57

3.9.1 Assessment of the Completeness of Documentation . . . . . . . . . . . . . 58

3.9.2 Assessment of Justification of Approaches and Assumptions . . . . . . . . 58

3.9.3 Assessment of Constants and Default Values . . . . . . . . . . . . . . . . 59

4 Mathematical and Numerical Robustness 61

4.1 Structure of the Numerical Routines . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Code Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Comparison with Analytic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Sensitivity of the Model 65

5.1 Factorial Design Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1 Model Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1.2 Sensitivity to Larger Changes in Model Inputs . . . . . . . . . . . . . . . 69

5.2 Response Surface Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Latin Hypercube Sampling Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Survey of Past Validation Work 77

6.1 Comparisons with Full-Scale Tests Conducted Specifically for the Chosen Evaluation 77

6.2 Comparisons with Previously Published Test Data . . . . . . . . . . . . . . . . . . 79

6.2.1 NIST/NRC Verification and Validation . . . . . . . . . . . . . . . . . . . 80

6.2.2 Fire Plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2.3 Multiple Compartments . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2.4 Large Compartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.5 Prediction of Flashover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 Comparison with Documented Fire Experience . . . . . . . . . . . . . . . . . . . 83

6.4 Comparison with Experiments Which Cover Special Situations . . . . . . . . . . . 84

6.4.1 Nuclear Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4.2 Small Scale Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4.3 Unusual Geometry and Specific Algorithms . . . . . . . . . . . . . . . . . 85

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xiv
7 Conclusion 89

References 101

xv
xvi
List of Figures

1.1 Zone Model Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Schematic of control volumes in a two-layer zone model. . . . . . . . . . . . . . . 14

3.2 Entrainment and burning in a two-layer, multi-compartment model. . . . . . . . . 19

3.3 Excess plume centerline temperature from Baum and McCaffrey correlation. . . . 26

3.4 Geometry for plume centerline temperature calculation. . . . . . . . . . . . . . . . 27

3.5 Vent opening size fraction as a function of time. . . . . . . . . . . . . . . . . . . . 29

3.6 Geometry and notation for horizontal flow vents in a two-zone fire model. . . . . . 30

3.7 Flow patterns and layer number conventions for horizontal flow through a vertical

vent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.8 Some simple fan-duct systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.9 Relative excess downstream temperature in a corridor using an adiabatic temperature boundary condition for several inlet depths and inlet temperature boundary

conditions. The inlet velocity is given by eq (3.55). . . . . . . . . . . . . . . . . . 38

3.10 Radiation Exchange in a two-zone fire model. . . . . . . . . . . . . . . . . . . . . 41

3.11 An example of the calculated two-wall (RAD2) and four-wall (RAD4) contributions to radiation exchange on a ceiling and wall surface. . . . . . . . . . . . . . . 42

3.12 Setup for a configuration factor calculation between two arbitrarily oriented finite

areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.13 Radiative heat transfer from a point source fire to a target. . . . . . . . . . . . . . . 45

3.14 Radiative heat transfer from the upper and lower layer gas to a target in the lower

layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.15 Convective heat transfer to ceiling and wall surfaces via the ceiling jet. . . . . . . . 51

3.16 Schematic of hydrogen chloride deposition region. . . . . . . . . . . . . . . . . . 55

4.1 Subroutine structure for the CFAST model. . . . . . . . . . . . . . . . . . . . . . 62

5.1 Building Geometry for base case scenario. . . . . . . . . . . . . . . . . . . . . . . 68

5.2 An example of time dependent sensitivity of fire model outputs to a 10 % change

in room volume for a single room fire scenario. . . . . . . . . . . . . . . . . . . . 69

5.3 Layer temperatures and volumes in several rooms resulting from variation in heat

release rate for a four-room growing fire scenario. . . . . . . . . . . . . . . . . . . 70

5.4 Comparison of the time dependent heat release rate and layer temperatures in several rooms for a four-room growing fire scenario. . . . . . . . . . . . . . . . . . . 71

5.5 Sensitivity of temperature to heat release rate for a four-room growing fire scenario. 72

5.6 Sensitivity of temperature to heat release rate for a four-room growing fire scenario. 73

xvii
5.7 Sensitivity of temperature to heat release rate for a four-room growing fire scenario. 74

6.1 Comparison of correlations, CFAST predictions, and experimental data for the

prediction of flashover in a compartment fire. . . . . . . . . . . . . . . . . . . . . 83

xviii
List of Tables

3.1 Conservative zone model equations . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Recommended compartment dimension limits . . . . . . . . . . . . . . . . . . . . 17

3.3 Transfer coefficients for HCl deposition . . . . . . . . . . . . . . . . . . . . . . . 56

5.1 Typical Inputs for a Two-Zone Fire Model . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Typical Outputs for a Two-Zone Fire Model . . . . . . . . . . . . . . . . . . . . . 67

6.1 Summary of Model Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xix
xx
Chapter 1

Overview

1.1 History

Analytical models for predicting fire behavior have been evolving since the 1960s. Over the past

decade, the completeness of the models has grown considerably. In the beginning, the focus of

these efforts was to describe in mathematical language the various phenomena which were observed in fire growth and spread. These separate representations have typically described only a

small part of a fire. However, when combined they can create a complex computational model that

can predict the expected course of a fire.

Once a mathematical representation of the underlying physics has been developed, the conservation equations can be re-cast into predictive equations for temperature, smoke and gas concentration and other parameters of interest, and solved numerically. The equations are usually in the

form of differential equations. A complete set of equations can describe the conditions produced

by the fire at a given time in a specified volume of air. Referred to as a control volume, the model

assumes that the predicted conditions within this volume are uniform at any time. Thus, the control

volume has one temperature, smoke density, gas concentration, etc. Different models divide the

building into different numbers of control volumes depending on the desired level of detail. The

most common fire model, known as a zone model, generally uses two control volumes to describe

a compartment an upper layer and a lower layer. In the compartment with the fire, additional

control volumes for the fire plume or the ceiling jet may be included to improve the accuracy of

the prediction (see figure 1.1).

This two-layer approach has evolved from observation of such layering in real-scale fire experiments. Hot gases collect at the ceiling and fill the compartment from the top. While these

experiments show some variation in conditions within the layer, these are small compared to the

differences between the layers. Thus, the zone model can produce a fairly realistic simulation

under many common and important conditions.

Other types of models include network models and field models. Network models use one

control volume per compartment and are used to predict conditions in spaces far removed from

the fire compartment where temperatures are near ambient and layering does not occur. The field

model goes to the other extreme, dividing the compartment into thousands or millions of control

volumes. Such models can predict the variation in conditions within the layers, but typically

require far longer run times than zone models. They are used when a highly detailed prediction of

1
Lower Layer

Upper Layer

Plume

Ceiling jet

Figure 1.1: Zone Model Terms.

the flow itself is of interest.

1.2 Model Evaluation

The process of model evaluation is critical to establishing both the acceptable uses and limitations

of fire models. It is not possible to evaluate a model in total; instead, available guides such as

ASTM E1355 [1] are intended to provide a methodology for evaluating the predictive capabilities

for a specific use . Validation for one application or scenario does not imply validation for different scenarios. Several alternatives are provided for performing the evaluation process including

comparison of predictions against standard fire tests, full-scale fire experiments, field experience,

published literature, or previously evaluated models.

The use of fire models currently extends beyond the fire research laboratory and into the engineering, fire service and legal communities. Sufficient evaluation of fire models is necessary to

ensure that those using the models can judge the adequacy of the scientific and technical basis for

the models, select models appropriate for a desired use, and understand the level of confidence

which can be placed on the results predicted by the models. Adequate evaluation will help prevent

the unintentional misuse of fire models. Verification is a process to check the correctness of the

solution of the governing equations. Verification does not imply that the governing equations are

appropriate; only that the equations are being implemented and solved correctly. Validation is a

process to determine the appropriateness of the governing equations as a mathematical model of

the physical phenomena of interest. Typically, validation involves comparing model results with

experimental measurement. Differences that cannot be explained by numerical errors in the model

or uncertainty in the experiments are attributed to the assumptions and simplifications of the physical model. These terms are used together to perform a model assessment. The more general term,

model assessment, encompasses both verification and validation of a computer model.

In general, this document follows the ASTM E1355 [1] guide for model assessment and provides a model assessment for the zone fire model CFAST. The guide provides four areas of evalu2
ation for predictive fire models:

• Defining the model and scenarios for which the evaluation is to be conducted (chapter 2),

• Assessing the appropriateness of the theoretical basis and assumptions used in the model

(chapter 3),

• Assessing the mathematical and numerical robustness of the model (chapter 4), and

• Quantifying the uncertainty and accuracy of the model results in predicting the course of

events in similar fire scenarios (chapters 5 and 6).

3
4
Chapter 2

Model and Scenario Definition

Sufficient documentation of calculation models is necessary to assess the adequacy of the scientific

and technical basis of the model and the accuracy of the computational procedures for scenarios

of interest. In addition, adequate documentation will help prevent the unintentional misuse of the

model. The documentation in this document follows the guidelines in ASTM E1355-04 [1].

2.1 Model Documentation

2.1.1 Name and Version of the Model

The name of the model is the Consolidated Fire Growth and Smoke Transport Model or CFAST.

The first public release was version 1.0 in June of 1990. This version was restructured from FAST

[2] to incorporate the ”lessons learned” from the zone model CCFM developed by Cooper and Forney [[3], namely that modification is easier and more robust if the components such as the physical

routines are separated from the solver. chapter 4 (Mathematical and Numerical Robustness) discusses this in more detail. Version 2 was released as a component of Hazard 1.2 in 1994 [4, 5]. The

first of the 3.x series was released in 1995 and included a vertical flame spread algorithm, ceiling

jets and nonuniform heat loss to the ceiling, spot targets, and heating and burning of multiple objects (ignition by flux, temperature or time) in addition to multiple prescribed fires. As it evolved

over the next five years, version 3 included smoke and heat detectors, suppression through heat

release reduction, better characterization of flow through doors and windows, vertical heat conduction through ceiling/floor boundaries, and non-rectangular compartments. In 2000, version 4

was released and included horizontal heat conduction through walls, and horizontal smoke flow in

corridors. Version 5 improved the combustion chemistry. Version 6, released in July, 2005, incorporates a more consistent implementation of vents, fire objects and event processing and includes

a graphical user interface which substantially improves its usability.

The code is written in FORTRAN 90.

2.1.2 Type of Model

CFAST is a model that predicts the environment within compartmented structures resulting from

a fire prescribed by the user. It is an example of the class of models called finite element. This

5
particular implementation is called a zone model, and essentially the space to be modeled is broken down to a few elements. The physics of the compartment fire phenomena is driven by fluid

flow, primarily buoyancy. The usual set of elements or zones are the upper and lower gas layers,

partitioning of the wall/ceiling/floor to an element each, one or more plumes and objects such as

fires, targets, and detectors. One feature of this implementation of a finite element model is that

the interface between the elements (in this case, the upper and lower gas layers) can move, with its

position defined by the governing equations.

2.1.3 Model Developers

CFAST was developed and is maintained primarily by the Fire Research Division of the National

Institute of Standards and Technology. The developers are Walter Jones, Richard Peacock, Glenn

Forney, Rebecca Portier, Paul Reneke, and John Hoover 1.

There have been contributions through research and published papers at Worcester Polytechnic Institute, University of California at Berkeley, VTT of Finland and CITCM of France. An

important guide to development of the model has been from many people around the world who

have provided ideas, suggestions, comments, detailed questions, opinions on what should happen

in particular scenarios, what physics and chemistry are needed and what types of problems must

be addressed by such a model in order to be useful for real world applications.

2.1.4 Relevant Publications

To accompany the model and simplify its use, NIST has developed this Technical Reference Guide

[6], a User’s Guide [7] and a Software and Validation Guide [8]. The Technical Reference Guide

describes the underlying physical principles and summarizes sensitivity analysis, model validation,

and model limitations consistent with ASTM E 1355 [1]. The Users Guide describes how to use

the model.

The U.S. Nuclear Regulatory Commission has published a verification and validation study of

five selected fire models commonly used in support of risk-informed and performance-based fire

protection at nuclear power plants [9]. In addition to an extensive study of the CFAST model, the

report compares the output of several other models ranging from simple hand calculations to more

complex CFD codes such as the Fire Dynamics Simulator (FDS) developed by NIST.

There are documents available (http://cfast.nist.gov) that are applicable to versions 2, 3, 5 as

well as 6 of both the model and user interface.

2.1.5 Governing Equations and Assumptions

For CFAST, as for most zone fire models, the equations solved are for conservation of mass and

energy. The momentum equation is not solved explicitly, except for use of the Bernoulli equation for the flow velocity at vents. Based on an integration over the volume of an element, these

equations are solved as ordinary differential equations.

There are two assumptions which reduce the computation time dramatically. The first is that

relatively few zones or elements per compartment is sufficient to model the physical situation. The

1Naval Research Laboratory, Washington, DC 20375.

6
second assumption is to close the set of equations without using the momentum equation in the

compartment interiors. This simplification eliminates acoustic waves. Though this prevents one

from calculating gravity waves in compartments (or between compartments), coupled with only a

few elements per compartment allows for a prediction in a large and complex space very quickly.

The equations themselves and the algorithms and sub-models used are discussed in detail in

chapter 3.

2.1.6 Input Data Required to Run the Model

All of the data required to run the CFAST model reside in a primary data file, which the user

creates. Some instances may require databases of information on objects, thermophysical properties of boundaries, and sample prescribed fire descriptions. In general, the data files contain the

following information:

• compartment dimensions (height, width, length)

• construction materials of the compartment (e.g., concrete, gypsum)

• material properties (e.g., thermal conductivity, specific heat, density, thickness, heat of combustion)

• dimensions and positions of horizontal and vertical flow openings such as doors, windows,

and vents

• mechanical ventilation specifications

• fire properties (e.g., heat release rate, lower oxygen limit, and species production rates as a

function of time)

• sprinkler and detector specifications

• positions, sizes, and characteristics of targets

Sample data files are provided which encompass many of the validation exercises described in

chapter 6 and in the various articles and reports referenced in chapter 6. These examples range

from simple one-compartment simulations to a large multi-story hotel scenario that includes an

elevator shaft and stairwell pressurization. A complete description of the input parameters required

by CFAST can be found in the CFAST Users Guide [7]. Some of these parameters have default

values included in the model, which are intended to be representative for a range of fire scenarios.

2.1.7 Property Data

Any simulation of a real fire scenario involves prescribing material properties for the walls, floor,

ceiling, and furnishings. CFAST treats all of these materials as homogeneous solids, thus the

physical parameters for many real objects can only be viewed as approximations to the actual

properties. Describing these materials in the input data file is a challenging task for the model

user. Thermal properties for the most common barrier materials used in construction, e.g. gypsum

wall board, are included in a database, thermal.df, included with the model. These properties come

directly from handbook values for typical materials [10].

7
2.1.8 Model Results

The output of CFAST are the sensible variables that are needed for assessing the environment

in a building subjected to a fire. Once the simulation is complete, CFAST produces an output

file containing all of the solution variables. Typical outputs include (but are not limited to) the

following:

• environmental conditions in the room (such as hot gas layer temperature; plume centerline

temperature; oxygen and smoke concentration; and ceiling, wall, and floor temperatures)

• heat transfer-related outputs to walls and targets (such as incident convective, radiated, and

total heat fluxes)

• fire intensity and flame height

• flow velocities through vents and openings

• detector and sprinkler activation times

There is more extensive discussion of the output in chapter 6 of this technical reference manual

and the users guide. The output is always in the metric system of units.

2.1.9 Uses and Limitations of the Model

CFAST has been developed for use in solving practical fire problems in fire protection engineering.

It is intended for use in system modeling of building and building components. A priori prediction

of flame spread or fire growth on objects is not modeled. Rather, the consequences of a specified

fire is estimated. It is not intended for detailed study of flow within a compartment, such as is

needed for smoke detector siting. It includes the activation of sprinklers and fire suppression by

water droplets.

The most extensive use of the model is in fire and smoke spread in complex buildings. The

efficiency and computational speed are inherent in the few computation cells needed for a zone

model implementation. The use is for design and reconstruction of time-lines for fire and smoke

spread in residential, commercial, and industrial fire applications. Some applications of the model

have been for design of smoke control systems.

• Compartments: CFAST is generally limited to situations where the compartment volumes

are strongly stratified. However, in order to facilitate the use of the model for preliminary estimates when a more sophisticated calculation is ultimately needed, there are algorithms for

corridor flow, smoke detector activation, and detailed heat conduction through solid boundaries. This model does provide for non-rectangular compartments, although the application

is intended to be limited to relatively simple spaces. There is no intent to include complex

geometries where a complex flow field is a driving force. For these applications, computational fluid dynamics (CFD) models are appropriate.

• Gas Layers: There are also limitations inherent in the assumption of stratification of the gas

layers. The zone model concept, by definition, implies a sharp boundary between the upper

8
and lower layers, whereas in reality, the transition is typically over about 10 % of the height

of the compartment and can be larger in weakly stratified flow. For example, a burning

cigarette in a normal room is not within the purview of a zone model. While it is possible

to make predictions within 5 % of the actual temperatures of the gas layers, this is not the

optimum use of the model. It is more properly used to make estimates of fire spread (not

flame spread), smoke detection and contamination, and life safety calculations.

• Heat Release Rate: CFAST does not predict fire growth on burning objects. Heat release

rate is specified by the user for one or more fire objects. The model does include the ability

to limit the specified burning based on available oxygen. There are also limitations inherent

in the assumptions used in application of the empirical models. As a general guideline,

the heat release should not exceed about 1 MW/m3. This is a limitation on the numerical

routines attributable to the coupling between gas flow and heat transfer through boundaries

(conduction, convection, and radiation). The inherent two-layer assumption is likely to break

down well before this limit is reached.

• Radiation: Because the model includes a sophisticated radiation model and ventilation algorithms, it has further use for studying building contamination through the ventilation system,

as well as the stack effect and the effect of wind on air circulation in buildings. Radiation

from fires is modeled with a simple point source approximation. This limits the accuracy of

the model near fire sources. Calculation of radiative exchange between compartments is not

modeled.

• Ventilation and Leakage: In a single compartment, the ratio of the area of vents connecting

one compartment to another to the volume of the compartment should not exceed roughly

2 m. This is a limitation on the plug flow assumption for vents. An important limitation

arises from the uncertainty in the scenario specification. For example, leakage in buildings

is significant, and this affects flow calculations especially when wind is present and for tall

buildings. These effects can overwhelm limitations on accuracy of the implementation of

the model. The overall accuracy of the model is closely tied to the specificity, care, and

completeness with which the data are provided.

• Thermal Properties: The accuracy of the model predictions is limited by how well the user

can specify the thermophysical properties. For example, the fraction of fuel which ends up

as soot has an important effect on the radiation absorption of the gas layer and, therefore, the

relative convective versus radiative heating of the layers and walls, which in turn affects the

buoyancy and flow. There is a higher level of uncertainty of the predictions if the properties

of real materials and real fuels are unknown or difficult to obtain, or the physical processes

of combustion, radiation, and heat transfer are more complicated than their mathematical

representations in CFAST.

User feedback indicates that using CFAST to predict the transport of heat and combustion

products from a prescribed fire is straightforward, easily and quickly accomplished, and the results

are within expectations. Any user of a computer based (numerical) model must be aware of the

assumptions and approximations being employed. Except for those few materials supplied in the

property databases, the user must supply the thermal properties of the materials, and then assess

9
the performance of the model compared with experiments to ensure that the model is valid for a

specific application. Only then can the model be expected to predict the outcome of fire scenarios

that are similar to those that have actually been tested.

In addition, there are specific limitations and assumptions made in the development of the

algorithms. These are detailed in the discussion of each of these sub-models:

In addition, there are specific limitations and assumptions made in the development of the

algorithms. These are detailed in the discussion of each of these sub-models:

• section 3.3 on zone model assumptions,

• section 3.4.1 on prescribed fires,

• section 3.4.1 on the relationship between fires and mass balance,

• section 3.4.2 on the plume entrainment model,

• section 3.4.4 on the assumptions made for corridor flow correlations,

• section 3.4.5 on the assumptions made for radiation heat transfer,

• section 3.6 on the suppression model, and

• section 3.7.2 on HCl deposition.

2.2 Scenarios for which the Model is Evaluated in this Document

CFAST is used for a wide range of buildings of interest, from glove-box size compartments, to

complex hotels to the vehicle assembly building at Cape Canaveral. The intended use of ASTM

E1355 [1] is to validate a specific scenario of interest so that the model can be used for scenarios

similar to the chosen scenario. The intent of this document, however, is to cover a much wider

range of scenarios which encompass the range of acceptable use of the model. Thus, this section

provides a description of this broader range of scenarios as discussed in this technical reference

guide rather than a single, specific scenario of interest for a validation exercise.

2.2.1 Description of Scenarios of Interest

CFAST is designed primarily to predict the environment within compartmented structures which

results from unwanted fires. These can range from very small containment vessels, on the order of

1 m3to large spaces on the order of 1000 m3. As discussed in the section on limitations and use

(see section 2.1.9), the appropriate size fire depends on the size of the compartment being modeled.

A range of such validation exercises is discussed in chapter 6.

10
2.2.2 List of Quantities Predicted by the Model

CFAST provides a prediction of the plume centerline, gas layer, and boundary temperatures, target

temperatures, species concentration (including soot volume fraction), layer height, fire size and

flame length, floor pressure, flow and fire size at vents, and heat flux (both radiative and convective). There is a more extensive discussion of the output in the CFAST users guide.

2.2.3 Degree of Accuracy Required for Each Output Quantity

The degree of accuracy for each output variable required by the user is highly dependent on the

technical issues associated with the analysis. The user must ask: How accurate does the analysis

have to be to answer the technical question posed? Thus, a generalized definition of the accuracy

required for each quantity with no regard as to the specifics of a particular analysis is not practical

and would be limited in its usefulness.

Returning to the earlier definitions of “design” and “reconstruction,” fire scenarios, design

applications typically are more accurate because the heat release rate is prescribed rather than

predicted, and the initial and boundary conditions are far better characterized. Mathematically, a

design calculation is an example of a “well-posed” problem in which the solution of the governing

equations is advanced in time starting from a known set of initial conditions and constrained by a

known set of boundary conditions. The accuracy of the results is a function of the fidelity of the

numerical solution, which is largely dependent on the quality of the model inputs.

A reconstruction is an example of an “ill-posed” problem because the outcome is known

whereas the initial and boundary conditions are not. There is no single, unique solution to the

problem. Rather, it is possible to simulate numerous fires that produce the given outcome. There

is no right or wrong answer, but rather a small set of plausible fire scenarios that are consistent

with the collected evidence and physical laws incorporated into the model. These simulations are

then used to demonstrate why the fire behaved as it did based on the current understanding of fire

physics incorporated in the model. Most often, the result of the analysis is only qualitative. If there

is any quantification at all, it could be in the time to reach critical events, like a roof collapse or

room flashover.

The CFAST validation guide [8] includes efforts to date involving well-characterized geometries and prescribed fires. These studies show that CFAST predictions vary from being within

experimental uncertainty to being about 30 % different than measurements of temperature, heat

flux, gas concentration, etc (see, for example, reference [9]). In general, this is adequate for its intended uses which are life-safety calculations and estimation of the environment to which building

elements are subjected in a fire environment. Applied design margins are typically larger than this

level of accuracy and may be appropriate to insure an adequate factor of safety.

11
12
Chapter 3

Theoretical Basis for CFAST

Adequately detailed documentation of the theoretical basis of the model allows the model user

to understand the underlying theory behind the model implementation and thus be able to assess

the appropriateness of the model to specific problems. This chapter presents a derivation of the

predictive equations for zone fire models and explains in detail the ones used in CFAST.

The modeling equations used in CFAST take the mathematical form of an initial value problem

for a system of ordinary differential equations. These equations are derived using the conservation

of mass, the conservation of energy (equivalently the first law of thermodynamics), the ideal gas

law. These equations predict as functions of time quantities such as pressure, layer height and

temperatures given the accumulation of mass and enthalpy in the two layers. The assumption of

a zone model is that properties such as temperature can be approximated throughout a control

volume by an average value.

Many formulations based upon these assumptions can be derived. One formulation can be converted into another using the definitions of density, internal energy and the ideal gas law. Though

equivalent analytically, these formulations differ in their numerical properties. Each formulation

can be expressed in terms of mass and enthalpy flow. These rates represent the exchange of mass

and enthalpy between zones due to physical phenomena such as plumes, natural and forced ventilation, convective and radiative heat transfer, and so on. For example, a vent exchanges mass and

enthalpy between zones in connected rooms, a fire plume typically adds heat to the upper layer and

transfers entrained mass and enthalpy from the lower to the upper layer, and convection transfers

enthalpy from the gas layers to the surrounding walls.

As discussed in references [11] and [12], the zone fire modeling ordinary differential equations

(ODEs) are stiff. The term stiff means that large variations in time scales are present in the ODE

solution. In our problem, pressures adjust to changing conditions more quickly than other to solve

zone fire modeling ODEs because of this stiffness. Runge-Kutta methods or predictor-corrector

methods such as Adams-Bashforth require prohibitively small time steps in order to track the

short-time scale phenomena (pressure in our case). Methods that calculate the Jacobian (or at least

approximate it) have a much larger stability region for stiff problems and are thus more successful

at their solution.

13
3.1 Derivation of Equations for a Two-Layer Model

A compartment is divided into two control volumes, a relatively hot upper layer and a relatively

cool lower layer, as illustrated in figure 3.1. The gas in each layer has attributes of mass, internal

energy, density, temperature, and volume denoted respectively by mi, Ei, ρi, Ti, and Vi where i=L

for the lower layer and i=U for the upper layer. The compartment as a whole has the attribute of

pressure P. These 11 variables are related by means of the following seven constraints (counting

density, internal energy and the ideal gas law twice, once for each layer).

ρi =

mi

Vi

Density (3.1)

Ei = cvmiTi Internal Energy (3.2)

P = RρiTi Ideal Gas Law (3.3)

V = VL +VU Total Volume (3.4)

Lower Layer

Upper Layer

Plume

Natural Vent

Components of mass and

enthalpy entering or leaving

a control volume

Layer Interface

Figure 3.1: Schematic of control volumes in a two-layer zone model.

The specific heat at constant volume and at constant pressure cv and cp, the universal gas

constant, R, and the ratio of specific heats, γ, are related by γ = cp/cv and R = cp −cv. For ambient

air, cp ≈ 1 kJ/kg K and γ = 1.4. Four additional equations obtained from conservation of mass and

energy for each layer are required to complete the equation set. The differential equations for mass

in each layer are

dmL

dt

= m ̇ L (3.5)

14
dmU

dt

= m ̇U (3.6)

The first law of thermodynamics states that the rate of increase of internal energy plus the rate

at which the layer does work by expansion is equal to the rate at which enthalpy is added to the

gas. In differential form this is

internal energy work enthalpy

z}|{ z}|{ z}|{

dEi

dt

+ P

dVi

dt

= h ̇

i

(3.7)

where cp is taken as constant in the enthalpy term,

h ̇ = cpm ̇U TU +E ̇U +cpm ̇ LTL +E ̇

L (3.8)

A differential equation for pressure can be derived by adding the upper and lower layer versions

of eq (3.7), noting that dVU

dt = −

dVL

dt , and that

dEi

dt

=

d(cvm ̇iTi)

dt

=

cv

R

d(PVi)

dt

(3.9)

to obtain

dP

dt

=

γ−1

V



h ̇

L +h ̇U



(3.10)

Differential equations for the layer volumes can be obtained by substituting equation 3.9 into

3.7 to obtain

dVi

dt

=

1

Pγ



(γ−1)h ̇

i −Vi

dP

dt 

(3.11)

Equation 3.2 can be rewritten using eq 3.11 to eliminate dV/dt to yield

dEi

dt

=

1

γ



h ̇

i +V

dP

dt 

(3.12)

A differential equation for density can be derived by applying the quotient rule to dρi

dt =

d

dt 

mi

Vi



and using eq 3.11 to eliminate dVi

dt to obtain

dρi

dt

=

−1

cpTiVi





h ̇

i −cpm ̇iTi



−

Vi

γ−1

dP

dt 

(3.13)

Temperature differential equations can be obtained from the equation of state by applying the

quotient rule to dTi

dt =

d

dt 

P

Rρi



and using eq 3.13 to eliminate dρi

dt to obtain

dTi

dt

=

1

cpρiVi





h ̇

i −cpm ̇iTi



+Vi

dP

dt 

(3.14)

These equations for each of the 11 variables are summarized in table 3.1. The time evolution

of these solution variables can be computed by solving the corresponding differential equations

15
Table 3.1: Conservative zone model equations

Equation Type Differential Equation

i’th layer mass dmi

dt

= m ̇i

pressure

dP

dt

=

γ−1

V



h ̇

L +h ̇U



i’th layer energy dEi

dt

=

1

γ



h ̇

i +V

dP

dt 

i’th layer volume dVi

dt

=

1

Pγ



(γ−1)h ̇

i −Vi

dP

dt 

i’th layer density dρi

dt

=

−1

cpTiVi





h ̇

i −cpm ̇iTi



−

Vi

γ−1

dP

dt 

i’th layer temperature dTi

dt

=

1

cprhoiVi





h ̇

i −cpm ̇iTi



+Vi

dP

dt 

together with appropriate initial conditions. The remaining seven variables can be determined

from the four solution variables using eqs (3.1) to (3.4).

There are, however, many possible differential equation formulations. Indeed, there are 330

different ways to select four variables from eleven. Many of these systems are incomplete due

to the relationships that exist between the variables given in eqs (3.1) to (3.4). For example the

variables, ρU , VU , mU , and P form a dependent set since ρU = mU /VU .

The number of differential equation formulations can be considerably reduced by not mixing

variable types between layers; that is, if upper layer mass is chosen as a solution variable, then

lower layer mass must also be chosen. For example, for two of the solution variables choose mL

and mU , or ρL and ρU , or TL and TU . For the other two solution variables pick EL and EU or P

and VL or P and VU . This reduces the number of distinct formulations to nine. Since the numerical

properties of the upper layer volume equation are the same as a lower layer one, the number of

distinct formulations can be reduced to six.

3.2 Equation Set Used in CFAST

The current version of CFAST is set up to use the equation set for layer temperature, layer volume,

and pressure as shown below.

dP

dt

=

γ−1

V



h ̇

L +h ̇U



(3.15)

dVU

dt

=

1

Pγ



(γ−1)h ̇

i −VU

dP

dt 

(3.16)

dTU

dt

=

1

cprhoiVU





h ̇U −cpm ̇U TU



+VU

dP

dt 

(3.17)

16
dTL

dt

=

1

cprhoiVL





h ̇

L −cpm ̇ LTL



+VL

dP

dt 

(3.18)

In these equations, the pressure is actually modeled with the pressure difference relative to an

ambient reference pressure to minimize numerical instability.

3.3 Limitations of the Zone Model Assumptions

The basic assumption of all zone fire models is that each compartment can be divided into a small

number of control volumes, each of which is uniform in temperature and composition. In CFAST

all compartments have two zones except for the fire room which has an additional zone for the

plume. Since a real upper/lower interface is not as sharp as this, one has a spatial error of about

10 % in determining the height of the layer [13, 14].

The zone model concept best applies for an enclosure in which the width and length are not

too different. If the horizontal dimensions of the room differ too much (i.e., the room looks like a

corridor), the flow pattern in the room may become asymmetrical. If the enclosure is too shallow,

the temperature may have significant radial differences. The width of the plume may at some height

become equal to the width of the room and the model assumptions may fail in a tall and narrow

enclosure. Therefore, the user should recognize approximate limits on the ratio of the length (L),

width (W), and height (H) of the compartment.

If the aspect ratio (length/width) is over about 10, the corridor flow algorithm should be used.

This provides the appropriate filling time. Similarly, for tall shafts (elevators and stairways), a

single zone approximation is more appropriate. It was found experimentally [15] that the mixing

between a plume and lower layer due to the interaction with the walls of the shaft, caused complete mixing. The is the flip side of the corridor problem and occurs at a ratio of the height to

characteristic floor length of about 10. The following quantitative limits are recommended:

Table 3.2: Recommended compartment dimension limits

Group Acceptable Special consideration Corridor flow

required algorithm

(L/W)max L/W < 3 3 < L/W < 5 L/W > 5

(L/H)max L/H < 3 3 < L/H < 6 L/H > 6

(W/H)max W/H > 0.4 0.2 < L/W < 0.4 L/W < 0.2

3.4 Source Terms for the Model

This section discusses each of the sub-models in CFAST. In general, the sections are similar to the

way the model itself is structured. The sub-sections which follow discuss the way the actual phenomena are implemented numerically. For each of the phenomena discussed below, the physical

basis for the model is discussed first, followed by a brief presentation of the implementation within

17
CFAST. For all of the phenomena, there are basically two parts to the implementation: the physical interface routine (which is the interface between the CFAST model and the algorithm) and the

actual physical routine(s) which implement the physics. This implementation allows the physics

to remain independent of the structure of CFAST and allows easier insertion of new phenomena.

3.4.1 The Fire

A fire in CFAST is implemented as a source of fuel mass which is released at a prescribed rate (the

pyrolysis rate). Energy is released by the fuel and combustion products are created as it burns.

The model can simulate multiple fires in one or more compartments of the building. These fires

are treated as totally separate entities, with no interaction of the plumes. These fires are generally

referred to as “objects” and can be ignited at a prescribed time, temperature or heat flux.

CFAST does not include a pyrolysis model to predict fire growth. Rather, pyrolysis rates for

each fire are prescribed by the user. While this approach does not directly account for increased

pyrolysis due to radiative feedback from the flame or compartment, in theory these effects could

be prescribed by the user. In an actual fire, this is an important consideration, and the specification

used should consider the experimental conditions as closely as possible.

Constrained Fire

A fire releases energy based on the pyrolysis of fuel, but may be constrained by the oxygen available for combustion depending on the compartment conditions. Complete burning will take place

only where there is sufficient oxygen. When insufficient oxygen is entrained into the fire plume,

unburned fuel will be transported from zone to zone until there is sufficient oxygen and a high

enough temperature to support combustion. In general, CFAST uses a simple definition of a combustion reaction that includes major products of combustion for hydrocarbon fuels:

CxHyOzNaClb +νO2 O2 → νCO2 CO2 +νH2O H2O+νCO CO+νS Soot+νN2 N2 +νHClHCl+νHCNHCN

(3.19)

where the stoichiometric coefficients νO2, νCO2, etc. represent appropriate molar ratios for a stoichiometric balance of the equation. For example, for soot, it is related to the soot yield, yS

, via the

relation:

νS =

WF

WS

yS (3.20)

For complete combustion of the simplest hydrocarbon fuel, methane reacts with oxygen to form

carbon dioxide and water. The only input required is the pyrolysis rate and the heat of combustion.

For fuels that contain oxygen, nitrogen, or chlorine, the reaction becomes more complex. In this

case, production yields for the species are prescribed by the user. Stoichiometry is used to insure

conservation of mass and elements in the reaction. The species which are calculated are oxygen,

carbon dioxide, carbon monoxide, water, and soot. Gaseous nitrogen is included, but only acts as

a diluent. Production of hydrogen cyanide and hydrogen chloride are tracked solely based on user

prescribed yields. There is a separate calculation of the concentration-time product Ct. Finally, a

user-specified trace species can be specified to follow the transport that results from fire-induced

flow for an arbitrary species. This may be of particular interest for radiological releases [16], but

may be useful for any trace amounts released by a fire.

18
The heat release rate for a constrained fire may be reduced below its prescribed value based

upon the oxygen available for combustion. When there is not enough oxygen to support complete

combustion, some of the fuel will be transported to the gas layers and through vents as unburned

hydrocarbons.

As fuel and oxygen are consumed, heat is released and various products of combustion are

formed. The heat is released as radiation and convected enthalpy:

Qf,R = χRQf (3.21)

Qf,C = (1−χR)Qf (3.22)

where, χR is the fraction of the fires heat release rate given off as radiation. The convective enthalpy, Qf,C then becomes the driving term in the plume flow. For a constrained fire there is

radiation to both the upper and lower layers, whereas the convective part contributes only to the

upper layer.

Limiting Combustion by Available Oxygen

For any individual fire, the heat release rate is limited by available oxygen in the layer where the

fire is located. This limit is applied in three places, which are shown schematically in figure 3. The

first is burning in the portion of the plume which (at least initially) is typically in the lower layer

of the room of fire origin (region # 1). The second is the portion of the plume in the upper layer,

also in the room of origin (region # 2). The third is in the vent flow which entrains air from a lower

layer into an upper layer in an adjacent compartment (region # 3). The unburned hydrocarbons are

tracked in this model. Further combustion of CO to CO2 is not included in the model.

p2 e

2 

e1 

e3 

p

3 

Door jet 

Equivalent 

virtual plume 

vp p1 

Region #1

Lower Layer

Region #2

Upper Layer

Region #3

Vent Flow

p

3 

Figure 3.2: Entrainment and burning in a two-layer, multi-compartment model.

Initially,  ̇mfis just the pyrolysis rate of the source fire in kg/s (region # 1). For subsequent

regions, the burning rate  ̇mfis the unburned fuel from a previous region,  ̇ mtuhc = m ̇f −m ̇b where

the subscript tuhc is total unburned hydrocarbons, f is the fire source, and b is the amount burned.

The first step is to limit the actual burning which takes place in the combustion zone. In each

combustion zone, there is a quantity of fuel available. At the source this results from the pyrolysis

19
of the material,  ̇mf

. In other situations such as a plume or door jet, it is the net unburned fuel

available,  ̇mtuhc. In each case, the fuel which is available but not burned is then deposited into the

“  ̇mtuhc ” category. This provides a consistent notation. In the discussion below,  ̇mfis the amount

of fuel burned. This value is initially specified as to the available fuel, and then reduced if there

is insufficient oxygen to support complete combustion. Subsequently, the available fuel,  ̇mtuhc,

is reduced by the final value of  ̇mf burned or  ̇mb. Thus we have a consistent description in each

burning region, with an algorithm that is invoked independent of the region being analyzed.

Qf = m ̇ f Hc (3.23)

with the mass of oxygen required to achieve this energy release rate of

m ̇ O =

Qf

EO

= m ̇ f

Hc

E

(3.24)

where EO is the heat release per mass unit of oxygen consumed, taken to be 1.31 x 107J/kg1

(based on oxygen consumption calorimetry for typical fuels [17, 18, 19]). If the fuel contains

oxygen (available for combustion), the oxygen needed to achieve full combustion is less:

m ̇ O,needed = m ̇ O −m ̇ O,inthe f uel (3.25)

If sufficient oxygen is available, then it is fully burned. However, if the oxygen concentration is

low enough, it will constrain the burning and impose a limit on the amount of fuel actually burned,

as opposed to the amount pyrolyzed. The actual limitation is discussed below and is:

m ̇ O.actual = minm ̇ O,available,m ̇ O,needed(3.26)

m ̇ f,actual = m ̇ O,actual

EO

Hc

(3.27)

The relationship between oxygen and fuel concentration defines a range in which burning will take

place. In the CFAST model, a limit is incorporated by limiting the burning rate as the oxygen level

decreases until a “lower oxygen limit” (LOL) is reached. The lower oxygen limit is incorporated

through a smooth decrease in the burning rate near the limit:

m ̇ O,available = m ̇ eYO2CLOL (3.28)

where  ̇me is the mass entrainment flow rate, YO2is the mass fraction of oxygen, and the lower

oxygen limit coefficient, CLOL , is the fraction of the available fuel which can be burned with the

available oxygen and varies from 0 at the limit to 1 above the limit. The functional form that utilizes

the hyperbolic tangent was determined empirically to provide a smooth cutoff of the burning over

a narrow range above the limit.

CLOL =

tanh(800(YO2 −YLOL)−400) +1

2

(3.29)

1The units for oxygen consumption calorimetry are J/kg. The value 1.31 x 107

J/kg is representative of typical

fuels such as furniture (see reference [17]) and implies these units. The variation or uncertainty (2σ) associated with

this value is on the order of ± 5 %

20
A temperature criterion is also imposed so that no burning will take place when the temperature is

below a user prescribed temperature.

In summary, it is possible to follow the formation of the major products of combustion (carbon

dioxide, carbon monoxide, soot, water, hydrogen cyanide, and hydrogen chloride) using appropriate measured product yields (e.g., [17]) to define product yields for eq (33). Actual combustion

chemistry is not considered in CFAST due to the complexities associated with detailed kinetics

and transport.

Flame Height

CFAST includes a calculation of average flame height based on the work of Heskestad [20]. Valid

for a wide range of hydrocarbon and gaseous fuels, the correlation is given by

H = −1.02D+0.235

Qf

1000

(3.30)

where H is the average flame height, D the diameter of the fire and Qfis the fire size. The mean

flame height is defined as the distance from the fuel source to the top of the visible flame where the

intermittency is 0.5. A flame intermittency of 0.5 means that the visible flame is above the mean

50 % of the time and below the mean 50 % of the time. This average flame height is included in

the printed output from CFAST. Note that Qfin Eq. (8) in Ref. [20] is in kW.

Limitation of the Algorithm for Fires and Mass Balance

CFAST depends on pyrolysis data for the source term for a fire. The usual way to obtain this data

is a large-scale calorimeter, e.g., reference [[21]. Generally, a product (e.g., chair, table, bookcase)

is placed under a large collection hood and ignited by a burner (≈ 50 kW simulating a wastebasket)

placed adjacent to the item. The combustion process then proceeds under assumed free-burning

conditions, and the heat release rate is measured. Potential sources of uncertainty include measurement errors related to the instrumentation and the degree to which free-burn- ing conditions

are not achieved (e.g., radiation from the gases under the hood or from the hood itself, and restrictions in the air entrained by the object causing locally reduced oxygen concentrations affecting the

combustion chemistry). There are limited experimental data for upholstered furniture which suggest that prior to the onset of flashover in a compartment, the influence of the compartment on the

burning behavior of the item is small. The differences obtained from the use of different types or

locations of ignition sources have not been explored. These factors are discussed in reference [22].

Where small-scale calorimeter data are used, procedures are available to extrapolate to the

behavior of a full-size item. These procedures are based on empirical correlations of data which

exhibit significant scatter, thus limiting their accuracy. For example, for upholstered furniture, the

peak heat release rates estimated by the triangular approximation method averaged 91 % (range

46 % to 103 %) of values measured for a group of 26 chairs with noncombustible frames, but only

63 % (range 46 % to 83 %) of values measured for a group of 11 chairs with combustible frames

[23]. Also, the triangle neglects the tails of the curve; these are the initial time from ignition to

significant burning of the item, and the region of burning of the combusti- ble frame, after the

fabric and filler are consumed.

21
The provided data and procedures only relate directly to burning of items initiated by relatively

large flaming sources. Little data are currently available for release rates under smoldering combustion, or for the high external flux and low oxygen conditions characteristic of post- flashover

burning. While the model allows multiple items burning simultaneously, it does not account for

the synergy of such multiple fires. Thus, for other ignition scenarios, multiple items burning simultaneously (which exchange energy by radiation and convection), combustible interior finish,

and post-flashover conditions, the model can give estimates which are often non- conservative (the

actual release rates would be greater than estimated). At present, the only sure way to account for

all of these complex phenomena is to conduct a full-scale compartment burn and use the pyrolysis

rates directly.

Burning can be constrained by the available oxygen. However, this constrained fire is not

subject to the influences of radiation to enhance its burning rate, but is influenced by the oxygen

available in the compartment. If a large mass loss rate is entered, the model will follow this input

until there is insufficient oxygen available for that quantity of fuel to burn in the compartment. The

unburned fuel (sometimes called excess pyrolysate) is tracked as it flows out in the door jet, where

it can entrain more oxygen. If this mixture is within the user-constrained flammable range, it burns

in the door plume. If not, it will be tracked throughout the building until it eventually collects as

unburned fuel or burns in a vent. The enthalpy released in the fire compartment and in each vent, as

well as the total enthalpy released, is detailed in the output of the model. Since mass and enthalpy

are conserved, the total will be correct. However, since combustion did not take place adjacent

to the burning object, the actual mass burned could be lower than that specified by the user. The

difference will be the unburned fuel.

An oxygen combustion chemistry scheme is employed only in constrained fires. Here userconstrained hydrocarbon ratios and species yields are used by the model to predict concentrations. A balance among hydrogen, carbon, and oxygen molecules is maintained. Under some

conditions, low oxygen can change the combustion chemistry, with a resulting increase in the

yields of products of incomplete combustion such as carbon monoxide. However, not enough

is known about these chemical processes to build this relationship into the model at the present

time. Some data exist in reports of full-scale experiments (e.g., reference [24]) which can assist in

making such determinations.

3.4.2 Plumes

A plume is formed above any burning object. It acts as a pump transferring mass and enthalpy

from the lower layer into the upper layer. A correlation is used to predict the amount of mass and

enthalpy that is transferred. A more complete plume model would predict plume entrainment by

creating a separate zone and solving the appropriate equations.

Two sources exist for moving enthalpy and mass between the layers within and between compartments. Within the compartment, the fire plume provides one source. The other source of

mixing between the layers occurs at vents such as doors or windows. Here, there is mixing at the

boundary of the opposing flows moving into and out of the compartment. The degree of mixing is

based on an empirically-derived mixing relation. Both the outflow and inflow entrain air from the

surrounding layers. The flow at vents is also modeled as a plume (called the door plume or jet),

and uses the same equations as the fire plume, with two differences. First, an offset is calculated

to account for entrainment within the doorway and second, the equations are modified to account

22
for the rectangular geometry of vents compared to the round geometry of fire plumes. All plumes

within the simulation entrain air from their surroundings according to an empirically-derived entrainment relation. Entrainment of relatively cool, non-smoke laden air adds oxygen to the plume

and allows burning of the fuel. It also causes it to expand as the plume moves upward in the shape

of an inverted cone. The entrainment in a vent is caused by bi- directional flow and results from

vortices formed near a shear layer. This phenomenon is called the Kelvin-Helmholtz instability

[25]. It is not exactly the same as a normal plume, so some error (not measured) arises when this

entrainment is approximated by a normal plume entrainment algorithm.

While experiments show that there is very little mixing between the layers at their interface,

sources of convection such as radiators or diffusers of heating and air conditioning systems, and the

downward flows of gases caused by cooling at walls, will cause such mixing. These are examples

of phenomena which are inconsistent with the two-zone approximation. Also, the plumes are

assumed not to be affected by other flows which may occur. For example, if the burning object

is near the door the strong inflow of air will cause the plume axis to lean away from the door and

affect entrainment of gases into the plume. Such effects are not included in the model.

As discussed above, each compartment is divided into an upper and lower layer. At the start

of the simulation, the layers in each compartment are initialized at ambient conditions and by

default, the upper layer volume set to 0.001 of the compartment volume (an arbitrary, small value

set to avoid the potential mathematical problems associated with dividing by zero). Other values

can be set. As enthalpy and mass are pumped into the upper layer by the fire plume, the upper

layer expands in volume causing the lower layer to decrease in volume and the interface to move

downward. If the door to the next compartment has a soffit, there can be no flow through the vent

from the upper layer until the interface reaches the bottom of that soffit. Thus in the early stages the

expanding upper layer will push down on the lower layer air and force it into the next compartment

through the vent by expansion.

Once the interface reaches the soffit level, a door plume forms and flow from the fire connecting doorway compartment to the next compartment is initiated. As smoke flow from the fire

compartment fills the second compartment, the lower layer of air in the second compartment is

pushed down. As a result, some of this air flows into the fire compartment through the lower part

of the (or vent). Thus, a vent between the fire compartment and connecting compartments can

have simultaneous, opposing flows of air. All flows are driven by pressure and density differences

that result from temperature differences and layer depths. The key to getting the correct flow is to

distribute correctly the fire and plumes mass and enthalpy between the layers.

Buoyancy generated by the combustion processes in a fire causes the formation of a plume.

Such a plume can transport mass and enthalpy from the fire into the lower or upper layer of a

compartment. In the present implementation, we assume that both mass and enthalpy from the fire

are deposited only into the upper layer. In addition the plume entrains mass from the lower layer

and transports it into the upper layer. This yields a net enthalpy transfer between the two layers.

A fire generates energy at a rate Qf. Some fraction, χR, will exit the fire as radiation. The

remainder, χC, will then be deposited in the layers as convective energy or heat additional fuel

which may then pyrolyze. A buoyant plume carries this energy into the upper layer. Within

CFAST, the radiative fraction, χR, defaults to 0.30 [26]; i.e., 30 % of the fires energy is released via

radiation. For other fuels, the work of Tewarson [27], McCaffrey [28], or Koseki [29] is available

for reference. The typical range for the radiative fraction is from about 0.05 to 0.4.

23
Plume Entrainment

All plume models used to date are based on the seminal work of Morton et al. [30]. For an extended source such as a fire, the prescription needs modification. Several studies have devised such

modifications. Two of these are included in CFAST, the work of McCaffrey [31] and Heskestad

[32].

McCaffrey [31] estimated the mass entrained by the fire/plume from the lower into the upper layer. This correlation divides the flame/plume into three regions as given in eq 3.48. This

prescription agrees with the work of Cetegen et al. [33, 34] in the intermittent regions but yields

greater entrainment in the other two regions. This difference is particularly important for the initial

fire since the upper layer is far removed from the fire.

flaming: m ̇ e

Qf

= 0.011

Z

Q

2/5

f

0.566

0.00 ≤





Z

Q

2/5

f



 < 0.08

intermittent: m ̇ e

Qf

= 0.026

Z

Q

2/5

f

0.909

0.08 ≤





Z

Q

2/5

f



 < 0.20 (3.31)

plume m ̇ e

Qf

= 0.124

Z

Q

2/5

f

1.895

0.20 ≤





Z

Q

2/5

f





McCaffrey’s correlation is an extension of the common point source plume model, with a different

set of coefficients for each region. These coefficients are based on experimental correlations.

Heskestad analyzed both his own data [32] and that of Zukoski [35] to develop the correlation

m ̇ e = 0.071Q

1/3

f,C

(z−z0)

5/3



1+0.026Q

2/3

f,C

(z−z0)

−5/3



(3.32)

where Qf,C is the convective heat release rate of the fire and z0 is a virtual origin for the fire plume

defined as z0/D = −1.02+0.083Q

2/5

f

/D, here based on the total heat release rate of the fire. Both

correlations provide similar results in CFAST calculations.

Entrainment Limits

In CFAST, there is a constraint on the quantity of gas which can be entrained by a plume arising

from a fire. The constraint arises from the physical fact that a plume can rise only so high for a

given size of a heat source. Early in a fire, when the energy flux is very small, the plume may not

have sufficient energy to reach the compartment ceiling. The correct sequence of events is for a

small fire to generate a plume which does not reach the ceiling or upper layer initially. The plume

entrains enough cool gas to decrease the buoyancy to the point where it no longer rises. When

there is sufficient energy present in the plume, it will penetrate the upper layer. To this end the

following prescription has been incorporated: for a given size fire, a limit is placed on the amount

of mass which can be entrained, such that no more is entrained than would allow the plume to

reach the layer interface. The result is that the interface falls at about the correct rate, although it

24
starts a little too soon, and the upper layer temperature is over predicted, but follows experimental

data after the initial phase.

For the plume to be able to penetrate the inversion formed by a hot gas layer over a cooler gas

layer, the density of the gas in the plume at the point of intersection must be less than the density of

the gas in the upper layer. In practice, this places a maximum on the air entrained into the plume.

From conservation of mass and enthalpy

m ̇ p = m ̇ f +m ̇ e (3.33)

m ̇ pcpTp = m ̇ f cpTf +m ̇ ecpTl (3.34)

where the subscripts p, f , e, and l refer to the plume, fire, entrained air, and lower layer, respectively.

The criterion that the density in the plume region be lower than the upper layer implies that

Tu < Tp. Solving eq 3.34 for Tp and substituting for  ̇mp from 3.33 yields

Tp =

Tf m ̇ f +Tlm ̇ e

m ̇ f +m ̇ e

> Tu (3.35)

or

m ̇ e <



Tf −Tu

Tu −Tl



m ̇ f <

Tf

Tu −Tl

m ̇ f (3.36)

Substituting the convective energy released by the fire, Qf,C = m ̇ f cpTf, into eq 3.36 yields the form

of the entrainment limit use in the CFAST model:

m ̇ e <

Qf,C

cp(Tu −Tl)

(3.37)

which is incorporated into the model. It should be noted that both the plume and layers are assumed

to be well mixed with negligible mixing and transport time for the plume and layers.

Plume Centerline Temperature

Used to calculate convective heat transfer to targets located directly above a fire source, CFAST

includes an empirical calculation of plume centerline gas temperature taken from the work of

Davis [36] using the work of Baum and McCaffrey [37] and Evans [38]. Baum and McCaffrey

[37] provide an experimental correlation for plume centerline temperature consistent with plume

theory. The correlation gives the excess temperature as a function of height above a fire as

∆Tp = B

 z

D∗

2n−1

(3.38)

where D

∗ =



Qf

ρ∞cpT∞

√g

2/5

, Qfis the total heat release rate, z is the height above the base of the

fire, and T∞, cp, and ρ∞ are the temperature, heat capacity, and density of the ambient gas at height

z. Depending on the location z, this may be either the lower gas layer temperature or upper gas

layer temperature. The constants B and n depend on the height above the base of the fire so that

25
Figure 3.3: Excess plume centerline temperature from Baum and McCaffrey correlation.

flaming: ∆Tp = 2.91T∞ 0.00 ≤ z/D

∗ < 1.32

intermittent: ∆Tp = 3.81T∞(z/D

∗

)

−1

1.32D

∗ ≤ z/D∗ < 3.30

plume ∆Tp = 8.41T∞(z/D

∗

)

−5/3

3.30 ≤ z/D

∗

Figure 3.3 shows the correlation. This is used directly whenever both the fire and target location

are within a single layer. When a hot layer forms so that the fire source and target location are not in

the same gas layer, the correlation must be modified since the plume now includes added enthalpy

by entraining hot layer gas as it moves through the upper layer to the ceiling. Evans [38] defines a

virtual source and heat release rate to extend the plume into the upper layer. Evans’ method defines

the strength and location of the substitute source with respect to the interface between the upper

and lower layers by

Q

∗

I,2 =

 

1+CTQ

∗

I,1

2/3

ξCT

−

1

CT

!3/2

(3.39)

ZI,2 =





ξQ

∗

I,1CT

Q∗

I,2

1/3



(ξ−1)(β

2 +1) +x1CTQ∗

I,2

2/3







2/5

ZI,1 (3.40)

Q

∗

I,1 =

Qf,C

ρ∞cpT∞

√gZ5/2

I,1

(3.41)

where ZI,1 is the distance from the fire to the interface between the upper and lower gas layers,

ZI,2 is the distance from the virtual source to the layer interface, ξ is the ratio of the upper to lower

26
Virtual Source

Fire Source

Layer Interface

Target Location

ZI,2

ZI,1

Z Z eff

Figure 3.4: Geometry for plume centerline temperature calculation.

layer temperature, β is an experimentally determined constant [35] (β

2 = 0.913), and CT = 9.115.

The effective source strength and distance between the virtual source and target position is given

by

Qf,C,e f f = Q

∗

I,2ρ∞cp∞T∞

√

gZ5/2

I,2

(3.42)

ze f f = z−ZI,1 +ZI,2 (3.43)

(see figure 3.4). The new values of the fire source and target location are then used in the standard

plume correlation where the ambient conditions are now those of the upper layer.

Limitations of the Plume Algorithms

The entrainment coefficients are empirically determined values from the work of McCaffrey [31]

and Heskestad [32]. Small errors in these values will have a small effect on the fire plume or the

flow in the plume of gases exiting the door of that compartment. In a multi-compartment model

such as CFAST, however, small errors in each door plume are multiplicative as the flow proceeds

through many compartments, possibly resulting in a significant error in the furthest compartments.

The data available from validation experiments [39] discussed in the CFAST Validation Guide [8]

indicate that the values for entrainment coefficients currently used in most zone models produce

good agreement for a three-compartment configuration. More data are needed for larger numbers

of compartments to study this further.

In real fires, smoke and gases are introduced into the lower layer of each compartment primarily due to mixing at connections between compartments and from the downward flows along walls

(where contact with the wall cools the gas and reduces its buoyancy). Doorway mixing has been

27
included in CFAST, using the same empirically derived mixing coefficients as used for calculating

fire plume entrainment. Downward wall flow has not been included. This could result in underestimates of lower layer temperatures and species concentration. Entrainment at a vent (doors,

windows, ...) yields mixing into the lower and upper layers. The latter has been studied more extensively than the former. The door jets are not symmetric for these mixing phenomena, however.

We have constrained the phenomenon for CFAST to be in the range as predicted by Zukoski et al.

[40].

The plume centerline temperature is based on an experimental correlation that has been subjected to considerable validation [36]. When the desired location of the plume centerline temperature is within the flaming region of the fire, the temperature is likely to be underestimated.

3.4.3 Vent Flow

Flow through vents is a dominant component of any fire model because it is sensitive to small

changes in pressure and transfers the greatest amount of enthalpy on an instantaneous basis of all

the source terms (except of course for the fire and plume). Its sensitivity to environmental changes

arises through its dependence on the pressure difference between compartments which can change

rapidly.

CFAST models two types of vent flow, vertical flow through horizontal vents (such as ceiling

holes or hatches) and horizontal flow through vertical vents (such as doors or windows). Horizontal

flow is the flow which is normally thought of when discussing fires. Vertical flow is particularly

important in two disparate situations: a ship, and the role of fire fighters doing roof venting.

Horizontal vent flow is determined using the pressure difference across a vent. Flow at a given

elevation may be computed using Bernoullis law by first computing the pressure difference at that

elevation. The pressure on each side of the vent is computed using the pressure at the floor, the

height of the floor and the density.

Atmospheric pressure is about 100 000 Pa. Fires produce pressure changes from 1 Pa to

1 000 Pa and mechanical ventilation systems typically involve pressure differentials of about 1 Pa

to 100 Pa. The pressure variables are solved to a higher accuracy than other solution variables

because of the subtraction (with resulting loss of precision) needed to calculate vent flows from

pressure differences.

Mass flow (in the remainder of this section, the term “flow” will be used to mean mass flow)

is the dominant source term for the predictive equations because it fluctuates most rapidly and

transfers the greatest amount of enthalpy on an instantaneous basis of all the source terms (except

of course the fire). Also, it is most sensitive to changes in the environment. Horizontal flow

encompasses flow through doors, windows and so on. Horizontal flow is discussed in section

3.4.3.1. Vertical flow occurs in ceiling vents. It is important in two separate situations: on a ship

with open hatches and in house fires with roof venting. Vertical flow is discussed in section 3.4.3.2.

Flow through vents can be forced (mechanical) or natural (convective). Force flow can occur

through either vertical or horizontal vents. The differences are primarily the selection rules for

the source of the gases or whether the resultant plume enters the lower or upper layer of each

compartment.

There is a special case of horizontal flow for long corridors. A corridor flow algorithm is

incorporated to calculate the time delay from when a plume enters a compartment to when the

effluent is available for flow into adjacent compartments.

28
Figure 3.5: Vent opening size fraction as a function of time.

Flow through vents can be modified, that is turned on or off. This applies to the three types

of vents discussed below, horizontal flow through vertical vents (HVENT), vertical flow through

horizontal vents (VVENT) and forced flow (MVENT). For each key word, there is a an initial

opening fraction which is reflected in the first region in figure 3.5. This initial opening fraction

can be modified by by the EVENT key word to change the fraction. This change occurs over a

transition time which defaults to one second. The final fraction is the third region depicted in figure

3.5. There can be only a single transition per vent.

Horizontal Flow Through Vertical Vents

Flow through normal vents such as windows and doors is governed by the pressure difference

across a vent. A momentum equation for the zone boundaries is not solved directly. Instead

momentum transfer at the zone boundaries is included by using an integrated form of Euler’s

equation, namely Bernoulli’s solution for the velocity equation. This solution is augmented for

restricted openings by using flow coefficients [14] to allow for constriction from finite size doors.

The flow (or orifice) coefficient is an empirical term which addresses the problem of constriction

of velocity streamlines at an orifice.

Bernoulli’s equation is the integral of the Euler equation and applies to general initial and final

velocities and pressures. The implication of using this equation for a zone model is that the initial

velocity in the doorway is the quantity sought, and the final velocity in the target compartment

vanishes. That is, the flow velocity vanishes where the final pressure is measured. Thus, the

29
Lower Layer

Upper Layer Upper Layer

Lower Layer

Hf 

Zi 

Bf 

Zo 4 

2 

1 

3 

Opening Between

Compartments

Z = Layer interface position

H = Vent soft height

B = Vent sill height

(layer numbers refer to nomenclature used in text)

Figure 3.6: Geometry and notation for horizontal flow vents in a two-zone fire model.

pressure at a stagnation point is used. This is consistent with the concept of uniform zones which

are completely mixed and have no internal flow. The general form for the velocity of the mass flow

is given by

v = C

s

2∆P

ρ

(3.44)

where C is the constriction (or flow) coefficient (0.7), ρ is the gas density on the source side,

and ∆P is the pressure across the interface. (Note: at present we use a constant value for C for all

gas temperatures).

The simplest means to define the limits of integration is with neutral planes, that is the height

at which flow reversal occurs, and physical boundaries such as sills and soffits. By breaking the

integral into intervals defined by flow reversal, a soffit, a sill, or a zone interface, the flow equation

can be integrated piecewise analytically and then summed.

The approach to calculating the flow field is of some interest. The flow calculations are performed as follows. The vent opening is partitioned into at most six slabs where each slab is

bounded by a layer height, neutral plane, or vent boundary such as a soffit or sill. The most

general case is illustrated in figure 3.6.

The mass flow for each slab can be determined from

m ̇io =

1

3

C(8ρ)Aslab

x

2 +xy+y2

x+y



(3.45)

where x =

p

|Pt| and y =

p

|Pb|. Pt and Pb are the cross-vent pressure differential at the top and

bottom of the slab respectively and Aslab is the cross-sectional area of the slab. The value of the

density, ρ, is taken from the source compartment.

30
Compartment 1 Compartment 2

Lower Layer

Upper Layer

Lower Layer

Upper Layer

N Zi T2 

T4 Zo 

T1 T3 

m13 

m42 

m12 

m43 

mij 

Mass ow from zone i

to zone j

Figure 3.7: Flow patterns and layer number conventions for horizontal flow through a vertical vent.

A mixing phenomenon occurs at vents which is similar to entrainment in plumes. As hot

gases from one compartment leave that compartment and flow into an adjacent compartment a

door jet can exist which is analogous to a normal plume. Mixing of this type occurs for  ̇m13 > 0

as shown in figure 3.7. To calculate the entrainment (  ̇m43 in this example), once again we use a

plume description consistent with the work of McCaffrey, but with an extended point source. The

estimate for the point source extension is given by Cetegen et al. [34]. This virtual point source

is chosen so that the flow at the door opening would correspond to a plume with the heating for a

equivalent doorway fire source (with respect to the lower layer) given by

Qf,eq = cp(Tl −T4)m ̇ 13 (3.46)

where Qf,eq is the heat release rate of the doorway fire. The concept of the virtual source is that

the enthalpy flux from the virtual point source should equal the actual enthalpy flux in the door jet

at the point of exit from the vent using the same prescription. Thus the entrainment is calculated

the same way as was done for a normal plume. The reduced height of the plume, zp, is

zp =

z13

Q

2/5

f,eq

+vp (3.47)

where vp the virtual point source, is defined by inverting the entrainment process to yield

31
vP =



90.9  ̇m

Qf,eq 1.76

0.00 < vp ≤ 0.08

vP =



38.5  ̇m

Qf,eq 1.001

0.08 < vp ≤ 0.20 (3.48)

vP =



8.10  ̇m

Qf,eq 0.528

0.20 < vp

Although outside of the normal range of validity of the plume model, a level of agreement

with experiment is apparent (section 6 includes discussion of validation experiments for the plume

model). Since a door jet forms a flat plume whereas a normal fire plume will be approximately

circular, strong agreement is not expected.

The other type of mixing is much like an inverse plume and causes contamination of the lower

layer. It occurs when there is flow of the type  ̇m42 > 0. The shear flow causes vortex shedding

into the lower layer and thus some of the particulates end up in the lower layer. The actual amount

of mass or energy transferred is usually not large, but its effect can be large. For example, even

minute amounts of carbon can change the radiative properties of the gas layer, from negligible to

something finite. It changes the rate of radiation absorption significantly and invalidates the simplification of an ambient temperature lower layer. This term is predicated on the Kelvin-Helmholz

flow instability and requires shear flow between two separate fluids. The mixing is enhanced for

greater density differences between the two layers. However, the amount of mixing has never

been well characterized. Quintiere et al. [14] discuss this phenomena for the case of crib fires in

a single room, but their correlation does not yield good agreement with experimental data in the

general case [41]. In the CFAST model, it is assumed that the incoming cold plume behaves like

the inverse of the usual door jet between adjacent hot layers; thus we have a descending plume. It

is possible that the entrainment is overestimated in this case, since buoyancy, which is the driving

force, is not nearly as strong as for the usually upright plume.

Vertical Flow Through Horizontal Vents

Flow through a ceiling or floor vent can be somewhat more complicated than through door or

window vents. The simplest form is uni-directional flow, driven solely by a pressure difference.

This is analogous to flow in the horizontal direction driven by a piston effect of expanding gases.

Once again, it can be calculated based on the Bernoulli equation, and presents little difficulty.

However, in general we must deal with more complex situations that must be modeled in order to

have a proper understanding of smoke movement. The first is an occurrence of puffing. When a

fire exists in a compartment in which there is only one hole in the ceiling, the fire will burn until

the oxygen has been depleted, pushing gas out the hole. Eventually the fire will die down. At this

point ambient air will rush back in, enable combustion to increase, and the process will be repeated.

Combustion is thus tightly coupled to the flow. The other case is exchange flow which occurs when

the fluid configuration across the vent is unstable (such as a hotter gas layer underneath a cooler

gas layer). Both of these pressure regimes require a calculation of the onset of the flow reversal

mechanism.

32
Normally a non-zero cross vent pressure difference tends to drive unidirectional flow from the

higher to the lower pressure side. An unstable fluid density configuration occurs when the pressure

alone would dictate stable stratification, but the fluid densities are reversed. That is, the hotter

gas is underneath the cooler gas. Flow induced by such an unstable fluid density configuration

tends to lead to bi-directional flow, with the fluid in the lower compartment rising into the upper

compartment. This situation might arise in a real fire if the room of origin suddenly had a hole

punched in the ceiling. No pretense is made of being able to do this instability calculation analytically. Cooper’s algorithm [42] is used for computing mass flow through ceiling and floor vents. It

is based on correlations to model the unsteady component of the flow. What is surprising is that

we can find a correlation at all for such a complex phenomenon. There are two components to the

flow. The first is a net flow dictated by a pressure difference. The second is an exchange flow based

on the relative densities of the gases. The overall flow is given by [42, 43, 44]

m ̇ = C f(γ, ε)

s

∆P

ρ

Av (3.49)

where γ = cp/cv is the ratio of specific heats, C = 0.68+0.17ε, ε =

∆P

P

, and f us a weak function

of both γ and ε [42]. In the situation where we have an instability, we use Cooper’s correlations for

the function f . The resulting exchange flow is given by

m ̇ ex = 0.1

 

g∆ρA

5/2

v

ρav !

1.0−

2A

2

v∆P

S

2g∆ρD5



(3.50)

where D = 2

p

Av/π and S is 0.754 for round or 0.942 for square openings, respectively [42]. Vertical flow through horizontal vents is governed by the VFLOW routines. VENTCF is the module

which calculates the mass flow from one compartment to another. The values returned are  ̇mincoming

and  ̇moutgoing through each vent. These terms are symmetric: the outgoing flow from compartment

1 to 2 is the same as incoming flow from compartment 2 to 1, though source and destination layers

may be different.

The energy flux into a compartment is then determined by the relative size and temperature of

the layers of the compartment from which the mass is flowing (incoming, u and l):

q ̇incoming = cpm ̇ uTu +cpm ̇lTl (3.51)

m ̇ u = m ̇incoming

Vu

V

(3.52)

m ̇l = m ̇incoming

Vl

V

(3.53)

The mass and energy are then deposited into the upper or lower layer of the receiving compartment based on the effective temperature of the incoming flow relative to the upper and lower

layers of the receiving compartment. If the temperature of the incoming flow is higher than the

temperature of the lower layer, then the flow is deposited into the upper layer. This is similar to the

usual plume from a fire or a doorway jet. These rules are implemented in VFLOW.

33
(b) Kitchen Exhaust (c) Space With Cross Ventilation

Fire

Plume

(a) Atrium Smoke Management

Figure 3.8: Some simple fan-duct systems.

Forced Flow

Forced flow in this version of CFAST is a supply (or exhaust) system based on constant flow

through a opening/fan/opening triplet . These systems are commonly used in buildings for heating,

ventilation, air conditioning, pressurization, and exhaust. Figure 3.8(a) shows smoke management

by an exhaust fan at the top of an atrium, and figure 3.8(b) illustrates a kitchen exhaust. Cross

ventilation, shown in figure 3.8(c), is occasionally used without heating or cooling. Generally

systems that maintain comfort conditions have either one or two fans.

Further information about these systems is presented in Klote and Milke [45] and the American

Society of Heating, Refrigerating and Air Conditioning Engineers (ASHRAE) [46].

This version of the model does not include duct work or variable fans. These equations are

high-order, non-linear and in some cases ill-posed, which caused a great deal of difficulty in reaching a numerical solution.

34
The flow through mechanical vents can be filtered. Filtering affects particulates such as smoke

and the trace species. Filtering can be turned on at any time. Effectiveness is from 0 % (no effect)

to 100 % which completely blocks flow of these two species.

3.4.4 Corridor Flow

A standard assumption in zone fire modeling is that once hot smoke enters a compartment, a well

defined upper layer forms instantly throughout the compartment. This assumption breaks down

in large compartments and long corridors due to the time required to fill these spaces. A simple

procedure is described for accounting for the formation delay of an upper layer in a long corridor

by using correlations developed from numerical experiments generated with the NIST fire model

Large Eddy Simulation Model (LES) , which is now the Fire Dynamic Simulation Model (FDS)

[47]. FDS is a computational fluid dynamics model capable of simulating fire flow velocities and

temperatures with high (≈0.1 m) resolution. Two parameters related to corridor flow are then

estimated: the time required for a ceiling jet to travel in a corridor and the temperature distribution

down the corridor. These estimates are then used in CFAST by delaying flow into compartments

connected to corridors until the ceiling jet has passed these compartments.

FDS was used to estimate ceiling jet characteristics by running a number of cases for various

inlet layer depths and temperatures. The vent flow algorithm in CFAST then uses this information

to compute mass and enthalpy flow between the corridor and adjacent compartments. This is

accomplished by presenting the vent algorithm with a one layer environment (the lower layer)

before the ceiling jet reaches the vent and a two layer environment afterwards. Estimated ceiling

jet temperatures and depths are used to define upper layer properties.

The problem is to estimate the ceiling jet temperature and depth as a function of time until it

reaches the end of the corridor. The approach used here is to run a field model as a pre-processing

step and to summarize the results as correlations describing the ceiling jet’s temperatures and

velocities. The steps used in this process are as follows:

1. Model corridor flow for a range of inlet ceiling jet temperatures and depths. Inlet velocities

are derived from the inlet temperatures and depths.

2. For each model run calculate average ceiling jet temperature and velocity as a function of

distance down the corridor.

3. Correlate the temperature and velocity distribution down the hall.

The zone fire model then uses these correlations to estimate conditions in the corridor as follows:

1. Estimate the inlet temperature, depth and velocity of the ceiling jet. If the corridor is the fire

room then use a standard correlation. If the source of the ceiling jet is another room then

calculate the inlet ceiling jet flow using Bernoulli’s law for the vent connecting the source

room and the corridor.

2. Use correlations in 3. above to estimate the ceiling jet arrival time at each vent.

3. For each vent in the corridor use lower layer properties to compute vent flow before the

ceiling jet arrives at the vent and lower/upper layer properties afterwards.

35
Assumptions

The assumptions made in order to develop the correlations are:

• The time scale of interest is the time required for a ceiling jet to traverse the length of the

corridor. For example, for a 100 m corridor with 1 m/s flow, the characteristic time would be

100 s.

• Cooling of the ceiling jet due to mixing with adjacent cool air is large compared to cooling

due to heat loss to walls. Equivalently, we assume that walls are adiabatic. This assumption

is conservative. An adiabatic corridor model predicts more severe conditions downstream

in a corridor than a model that accounts for heat transfer to walls, since cooler ceiling jets

travel slower and not as far.

• We do not account for the fact that ceiling jets that are sufficiently cooled will stagnate.

Similar to the previous assumption, this assumption is conservative and results in over predictions of conditions in compartments connected to corridors (since the model predicts that

a ceiling jet may arrive at a compartment when in fact it may have stagnated before reaching

it).

• Ceiling jet flow is buoyancy driven and behaves like a gravity current. The inlet velocity of

the ceiling jet is related to its temperature and depth.

• Ceiling jet flow lost to compartments adjacent to the corridor is not considered when estimating ceiling jet temperatures and depths. Similarly, a ceiling jet in a corridor is assumed

to have only one source.

• The temperature and velocity at the corridor inlet is constant in time.

• The corridor height and width do not effect a ceiling jet’s characteristics. Two ceiling jets

with the same inlet temperature, depth and velocity behave the same when flowing in corridors with different widths or heights as long as the ratio of inlet widths to corridor width are

equal.

• Flow entering the corridor enters at or near the ceiling. The inlet ceiling jet velocity is

reduced from the vent inlet velocity by a factor of wvent/wroom where wvent and wroom

are the width of the vent and room, respectively.

Corridor Jet Flow Characteristics

Ceiling jet flow in a corridor can be characterized as a one dimensional gravity current. To a first

approximation, the velocity of the current depends on the difference between the density of the gas

located at the leading edge of the current and the gas in the adjacent ambient air. The velocity also

depends on the depth of the current below the ceiling. A simple formula for the gravity current

velocity may be derived by equating the potential energy of the current, mgd0/2, measured at the

half-height d0/2 with its kinetic energy, mV2/2 to obtain

v =

p

gd0 (3.54)

36
where m is mass, g is the acceleration of gravity, d0 is the height of the gravity current and V

is the velocity. When the density difference, between the current and the ambient fluid is small,

the velocity, V, is proportional to p

gd0∆ρ/ρc j =

p

gd0∆T/Tamb where ρamb and Tamb are the

ambient density and temperature and ρc j and Tc j are the density and temperature of the ceiling

jet and ∆T = Tc j − Tamb is the temperature difference. Here use has been made of the ideal gas

law, ρambTamb ≈ ρc jTc j. This can be shown using an integrated form of Bernoulli’s law noting

that the pressure drop at the bottom of the ceiling jet is Pb = 0, the pressure drop at the top is

Pt = gd0(ρc j −ρamb) and using a vent coefficient cvent of 0.74, to obtain

v0 = C

√

8

3

1

√ρc j

Pt +

√

PtPb +Pb

√

Pt +

√

Pb

= C

√

8

3

s

Pt

ρc j

= C

√

8

3

s

gd0

ρamb −ρc j

ρc j

≈ 0.7

s

gd0

∆T

Tamb

(3.55)

Formulas of the form of the above equation lead one to conclude that a ceiling jet’s characteristics in a corridor depend on its depth, d0, and relative temperature difference, ∆T/Tamb. Therefore,

as the jet cools, it slows down. If no heat transfer occurs between the ceiling jet and the surrounding

walls, then the only mechanism for cooling is mixing with surrounding cool air.

Twenty numerical experiments were performed using FDS in order to better understand the

effects of the inlet ceiling jet temperature and depth on ceiling jet characteristics downstream in a

corridor. These cases were run with five different inlet depths and four different inlet temperatures.

The inlet ceiling jet temperature rise, ∆T0, and depth, d0, were used to define an inlet velocity, v0

using eq (3.55). The inlet ceiling jet depths, d0, used in the parameter study are 0.15 m, 0.30 m,

0.45 m, 0.60 m and 0.75 m. The inlet ceiling jet temperature rises, T0, used in the parameter study

are 100 ◦C , 200 ◦C , 300 ◦C and 400 ◦C .

Correlations

Ceiling jet functions were plotted as a function of distance down a corridor for each of the 20 test

cases. These results are shown in figure 3.9. Note that all but the 0.15 m ceiling jet data lie on

essentially the same line.

The best fit line is given in the form of log ∆T

∆T0

= a+bx. This is equivalent to

∆T

∆T0

= C110bx = C1



1

2

x/h1/2

(3.56)

where C1 = 10aand h1/2 = −log 2/b. The parameter h1/2 has a physical interpretation. It is the

distance down the corridor where the temperature rise ∆T , falls off to 50 % of its original value or

equivalently, T(x+h1/2) = T(x)/2.

37
Figure 3.9: Relative excess downstream temperature in a corridor using an adiabatic temperature

boundary condition for several inlet depths and inlet temperature boundary conditions. The inlet

velocity is given by eq (3.55).

38
The half-distance, h1/2

, can be approximated by h1/2 = log 2/0.018 = 16.7 m where b =

−0.018 is given in figure 3.9. Similarly, the coefficient C1 is approximated by C1 = 10a =

10−0.003 ≈ 1 where a is determined from figure 3.9. Therefore, the temperature rise, ∆T, may

be approximated by

∆T = ∆T0



1

2

x/16.7

(3.57)

The numerical experiments with FDS demonstrated that for the cases simulated, ceiling jet

characteristics depend on the relative inlet temperature rise and not the inlet depth. Flow in long

corridors (greater than 10 m) need to be better characterized due to the flow stagnation which may

occur because of the ceiling jet’s temperature decay.

3.4.5 Heat Transfer

This section discusses radiation, convection and conduction, the three mechanisms by which heat

is transferred between the gas layers and the enclosing compartment walls. This section also

discusses heat transfer algorithms for calculating target temperatures.

Gas layers exchange energy with their surroundings via convective and radiative heat transfer.

Different material properties can be used for the ceiling, floor, and walls of each compartment

(although all the walls of a compartment must be the same). Additionally, CFAST allows each

surface to be composed of up to three distinct layers. This allows the user to deal naturally with

the actual building construction. Material thermophysical properties are assumed to be constant,

although we know that they actually vary with temperature. The user should also recognize that

the mechanical properties of some materials may change with temperature, but these effects are

not modeled.

Radiative transfer occurs among the fire(s), gas layers and compartment surfaces (ceiling, walls

and floor). This transfer is a function of the temperature differences and the emissivity of the gas

layers as well as the compartment surfaces. Typical surface emissivity values only vary over a

small range. For the gas layers, however, the emissivity is a function of the concentration of

species which are strong radiators, predominately smoke particulates, carbon dioxide, and water.

Thus errors in the species concentrations can give rise to errors in the distribution of enthalpy

among the layers, which results in errors in temperatures, resulting in errors in the flows. This

illustrates just how tightly coupled the predictions made by CFAST can be.

Radiation

Radiation heat transfer forms a significant portion of the energy balance in a zone fire model,

especially in the fire room. Radiative heat transfer is computed from wall and gas temperatures,

emisivities and fire heat release rates. To calculate the radiation absorbed in a zone, a heat balance

must be done accounting for all surfaces that radiate to and absorb radiation from a zone.

A radiation heat transfer calculation can easily dominate the computational requirements of

any fire model. Approximations are then required to perform these calculations in a time consistent

with other zone fire model sources terms. For example, it is assumed that all zones and surfaces

radiate and absorb like a gray body, that the fires radiate as point sources and that the plume does

not radiate at all. Radiative heat transfer is approximated using a limited number of radiating wall

39
surfaces, four in the fire room and two everywhere else. The use of these and other approximations

allows CFAST to perform the radiation computation in a reasonably efficient manner [48].

Modeling Assumptions: The following assumptions are made in order to simplify the radiation heat exchange model used in CFAST and to make its calculation tractable.

• Iso-thermal - Each gas layer and each wall segment is assumed to be at a uniform temperature.

• Equilibrium - The wall segments and gas layers are assumed to be in a quasi-steady state.

In other words, the wall and gas layer temperatures are assumed to change slowly over the

duration of the time step of the associated differential equation.

• Point Soure Fires - The fire is assumed to radiate uniformly in all directions giving off a

fraction, χR, of the total energy release rate. This radiation is assumed to originate from a

single point. Radiation feedback to the fire and radiation from the plume is not modeled in

the radiation exchange algorithm.

• Diffuse and gray surfaces - The radiation emitted is assumed to be diffuse and gray. In

other words, the radiant fluxes emitted are independent of direction and wavelength. These

assumptions allow us to infer that the emittance, ε, absorptance, α and reflectance, ρ, are

related via ε = α = 1−ρ.

• Geometry - Rooms or compartments are assumed to be rectangular boxes. Each wall is either

perpendicular or parallel to every other wall. Radiation transfer through vent openings is lost

from the room.

4-Wall and 2-Wall Radiation Exchange: When computing wall temperatures, CFAST partitions a compartment into four parts; the ceiling, the floor, the wall segments above the layer

interface and the wall segments below the layer interface. The radiation algorithm then computes

a heat flux striking each wall segment using the surface temperature and emissivity.

The four wall algorithm used in CFAST for computing radiative heat exchange is based upon

the equations developed in Siegel and Howell [49] which in turn is based on the work of Hottel

[50]. Siegel and Howell model an enclosure with N wall segments and a homogeneous gas. A

radiation algorithm for a two layer zone fire model requires treatment of an enclosure with two

uniform gases. Hottel and Cohen [51] developed a method where the enclosure is divided into a

number of wall and gas volume elements. An energy balance is written for each element. Each

balance includes interactions with all other elements. Treatment of the fire and the interaction of

the fire and gas layers with the walls is based upon the work of Yamada and Cooper [52]. They

model fires as point heat sources radiating uniformly in all directions and use the Lambert-Beer

law to model the interaction between heat emitting elements (fires, walls, gas layers) and the gas

layers. By implementing a four wall rather than an N wall model, significant algorithmic speed

increases are achieved. This is done by exploiting the simple structure and symmetry of the four

wall problem.

The nomenclature used in this section follows that of Siegel and Howell [49]. The radiation

exchange at the k’th surface is shown schematically in figure 3.10. For each wall segment k from

1 to N, a net heat flux, ∆qˆ

00

k

, must be found such that the energy balance,

40
σAkε kTk

4 1− ε ( k )qk

in qk

in AkΔqk

′′

Outgoing Radiation Incoming Radiation

Gray body

surface radiation

Reflected radiant

energy

Incoming radiation from other

surfaces, fires, and emitting,

absorbing gas layers

Energy added to the kʼth

surface to maintain a

constant temperature

Figure 3.10: Radiation Exchange in a two-zone fire model.

σAkεkT

4

k + (1−εk)q

in

k = q

in

k +Ak∆q

00

k

(3.58)

at each wall segment k is satisfied, where σ is the Stefan-Boltzman constant, Ak is the area of

the kth wall segment , εk is the emissivity of the kth wall segment, Tk is the temperature of the kth

wall segment and q

in

k

is the energy arriving at the kth wall segment from all other wall segments

and heat sources.

Radiation exchange at each wall segment considers the emitted, reflected, incoming and net

radiation terms. The unknown net radiative fluxes, ∆q

00

k

, are found by solving the modified net

radiation equation

∆qˆ

00

k −

N

∑

j=1



1−ε j



∆qˆ

00

jFk−jτ j−k = σT

4

k −

N

∑

j=1



σT

4

k Fk−jτk−j



−

ck

Ak

(3.59)

where ∆qˆ

00

k = ∆qk/ε, Fk−j

is the configuration factor, τ is the transmittance and other terms are

previously defined.

The walls can be modeled using two surfaces or four. The four wall model is necessary for fire

rooms because the temperatures of the ceiling and upper walls differ significantly. The two wall

model is used for compartments that contain no fires.

To simplify the comparison between the two and four wall segment models, assume that the

emissivities of all wall segments are one and that the gas absorptivities are zero. Let the room

dimensions be 4 m by 4 m by 4 m, the temperature of the floor and the lower and upper walls be

300 K, and the ceiling temperature vary from 300 K to 600 K. Figure 3.11 shows a plot of the

heat flux to the ceiling and upper wall as a function of the ceiling temperature [48, 53]. The two

wall model predicts that the extended ceiling (a surface formed by combining the ceiling and upper

wall into one wall segment) cools, while the four wall model predicts that the ceiling cools and the

upper wall warms. The four-wall model moderates temperature differences that may exist between

the ceiling and upper wall (or floor and lower wall) by allowing heat transfer to occur between the

ceiling and upper wall. This problem does not arise when a fire is not present.

Reference [48] documents how to minimize the work required to compute the 16 configuration

factors, Fk−j, required in a 4 wall model.

41
-5000

-4000

-3000

-2000

-1000

0

1000

2000

250 300 350 400 450 500 550 600

Extended ceiling flux using RAD2

Ceiling flux using RAD4

Upper wall flux using RAD4

Flux (W/m2

)

Ceiling Temperature (K)

Figure 3.11: An example of the calculated two-wall (RAD2) and four-wall (RAD4) contributions

to radiation exchange on a ceiling and wall surface.

42
1

2

!1

!2

n1

S

Figure 3.12: Setup for a configuration factor calculation between two arbitrarily oriented finite

areas.

Configuration Factors: A configuration factor between two finite areas denoted F1−2 is the

fraction of radiant energy given off by surface 1 that is intercepted by surface 2 and is given by

F1−2 =

1

A1

Z

A1

Z

A2

cosθ1θ2

πL

2

dA2dA1 (3.60)

where L is the distance along the line of integration, θ1 and θ2 are the angles for surface 1 and 2

between the respective normal vectors and the line of integration, and A1 and A2 are the areas of

the two surfaces. These terms are illustrated in figure 3.12. When the surfaces A1 and A2 are far

apart relative to their surface area, eq (3.60) can be approximated by assuming that θ1, theta2 and

L are constant over the region of integration to obtain

F1−2 =

cosθ1θ2

πL

2

A2 (3.61)

Transmittance and Absorptance: The transmittance of a gas volume is the fraction of radiant

energy that will pass through it unimpeded and is given by

τ(L) = e

−αL

(3.62)

where α is the absorptance of the gas volume and L is a characteristic path length.

The absorptance, α, of a gas volume is the fraction of radiant energy absorbed by that volume.

For a gray gas, α+τ = 1.

Calculating absorption for broad band gas layer radiation: In general, the transmittance

and absorptance are a function of wavelength. This is an important factor to consider for the major gaseous products (CO2 and H2O); however soot has a continuous absorption spectrum which

43
allows the transmittance and absorptance to be approximated as “gray” [49] across the entire spectrum.

The gas absorptance, αG, is due to the combination of the CO2 and H2O and is given by

αG = αH2O +αCO −C (3.63)

where C is a correction for band overlap. For typical fire conditions, the overlap amounts to about

half of the CO2 absorptance [54] so the gas transmittance is approximated by

τG = 1−αH2O −0.5αCO2(3.64)

The total transmittance of a gas-soot mixture is the product of the gas and soot transmittances,

τT = τSτG so that

τT = e

−al(1−αH2O −0.5αCO2

) (3.65)

In the optically thin limit the absorption coefficient, a, may be replaced by the Planck mean

absorption coefficient and in the optically thick limit, it may be replaced by the Rosseland mean

absorption coefficient. For the entire range of optical thicknesses, Tien et al. [54] report that a

reasonable approximation is α = k fvT where k is a constant that depends on the optical properties

of the soot particles, fv is the soot volume fraction and T is the soot temperature in Kelvin. Values

of a, have been found to be about constant for a wide range of fuels [55]. The soot volume fraction,

fv, is calculated from the soot mass, soot density and layer volume. The soot is assumed to be in

thermal equilibrium with the gas layer.

Edwards’ absorptance data for H2O and CO2 are reported [56] as log(emissivity) versus log(pressurepathlength), with log(gas concentration) as a parameter. For each gas, these data were incorporated

into a look-up table, implemented as a two-dimensional array of log(emissivity) values, with indices based on temperature and gas concentration. It is assumed that absorptance and emittance

are equivalent for the gaseous species as well as for soot.

An effective path length ( mean beam length, L) treats an emitting volume as if it were a

hemisphere of a radius such that the flux impinging on the center of the circular base is equal to

the average boundary flux produced by the real volume. The value of this radius is approximated

as [55, 57] L = c4V/A where L is the mean beam length in meters, c is a constant (approximately

0.9, for typical geometries), V is the emitting gas volume m3and A is the surface area (m2) of the

gas volume. The volume and surface area are calculated from the dimensions of the layer.

For each gas, the log(absorptance) is estimated from the look-up table for that gas by interpolating both the log(temperature) and log(concentration) domains. In the event that the required

absorptance lies outside the temperature or concentration range of the look-up table, the nearest

acceptable value is returned. Error flags are also returned, indicating whether each parameter was

in or out of range and, in the latter case, whether it was high or low. This entire process is carried

out for both CO2 and H2O.

Computing Target Heat Flux and Temperature

The calculation of the radiative heat flux to a target is similar to the radiative heat transfer calculation discussed previously. The main difference is that CFAST does not compute feedback from

44
Fire

Target

L

qf,r

nt

Figure 3.13: Radiative heat transfer from a point source fire to a target.

the target to the wall surfaces or gas layers. The target is simply a probe or sensor not interacting

with the modeled environment.

The net flux striking a target can be used as a boundary condition in order to compute the

temperature of the target. If the target is thin, then its temperature quickly rises to a level where

the heat flux to and from the target are in equilibrium.

There are four components of heat flux to a target: fires, walls (including the ceiling and floor),

gas layer radiation and gas layer convection.

Heat Flux from a Fire to a Target: Figure 3.13 illustrates terms used to compute heat flux

from a fire to a target. Let nt be a unit vector perpendicular to the target and θt be the angle between

the vectors L and nt

.

Using the definition that qf,ris the radiative portion of the energy release rate of the fire, then

the heat flux on a sphere of radius L due to this fire is qf,r/



4πL

2



. Correcting for the orientation

of the target and accounting for heat transfer through the gas layers, the heat flux to the target is

q

00

f,r =

qf,r

4πL

2

cos(θt)τU (LU )τL(LL) = −qf,r

ntL

4πL

3

τU (LU )τL(LL) (3.66)

Radiative Heat Flux from a Wall Segment to a Target: Figure 3.14 illustrates terms used to

compute heat flux from a wall segment to a target. The flux, q

00

w,t

, from a wall segment to a target

can then be computed using

q

00

w,t =

Awq

00

w,outFw−t

At

τU (LU )τL(LL) (3.67)

where q

00

w,t

is the flux leaving the wall segment, Aw, At are the areas of the wall segment and target

respectively, Fw−tis the fraction of radiant energy given off by the wall segment that is intercepted

45
LU

LL

Target

Arriving at interface: !U qout

Arriving at interface: !U !L qout

Figure 3.14: Radiative heat transfer from the upper and lower layer gas to a target in the lower

layer.

by the target (i.e., a configuration factor) and τU (LU ) and τL(LL) are defined as before. Equation

(3.67) can be simplified using the symmetry relation AwFw−t = AtFt−w to obtain

q

00

w,t = q

00

w,outFt−wτU (LU )τL(LL) (3.68)

where

q

00

w,out = σT

4

w −(1−εw)

∆q

00

w

εw

, (3.69)

Tw is the temperature of the wall segment, εw is the emissivity of the wall segment and ∆q

00

w is the

net flux striking the wall segment.

Radiation from the Gas Layer to the Target: Figure 3.14 illustrates the setup for calculating

the heat flux from the gas layers to the target. The upper and lower gas layers in a room contribute

to the heat flux striking the target if the layer absorptances are non-zero.

Let q

00

w,t,gas denote the flux striking the target due to the gas g in the direction of wall segment

w. Then

q

00

w,t,gas =



σFt−w



T

4

L αLτU +T

4

U αU



w is in the lower layer

σFt−w



T

4

U αU τL +T

4

L αL



w is in the upper layer (3.70)

The total target flux due to the gas (upper or lower layer) is obtained by summing eq (3.70 over

each wall segment or

q

00

g,t = ∑w

= q

00

w,t,gas (3.71)

Computing the Steady State Target Temperature: The steady state target temperature, Tt

can be found by solving an energy balance on the target; namely

46
εtσT

4

t = εtqr,in +h(Tg −Tt) (3.72)

Note that the local gas temperature, Tg, in the convection calculation, h(Tg −Tt), is taken to be

either the upper layer temperature if the target is located in the upper layer, the lower layer temperature if the target is located in the lower layer, or the plume centerline temperature if the target

is located directly above a fire source.

Let f(Tt) be the difference between the left and right hand side of equation (3.72). Then this

equation may be solved using the Newton iteration

Tnew = Told −

f(Told)

f

0(Told)

(3.73)

Equation (3.73) is iterated until the difference Tnew −Told is sufficiently small.

Computing the Transient Target Temperature: A transient target temperature may be computed using two different methods depending on whether the target is assumed to be thin or thick.

A thin target is presumed to have a constant interior temperature profile. A differential equation

model may then be used to estimate the temperature rise (or fall) based upon the thermal properties

of the target and the heat flux striking the front and back sides; namely

cρV

dT

dt

= A(q

00

f +q

00

b

) (3.74)

where c, ρ and V are the the specific heat, density and volume of the target, A is the crosssectional area of the target and the two q

00 terms are the heat flux (due to all sources) striking the

front and back sides of the target.

Equation (3.74) may be solved implicitly or explicitly. When solved implicitly, the target

temperature is added to the set of solution variables and equation (3.74) is added to the equation

set solved by DASSL. When solved explicitly, equation (3.74) is solved as a stand-alone equation

advancing the target temperature in time.

If the target is thick then it is presumed that the temperature profile within the target varies as

a function of depth and therefore a partial differential equation model must be used to estimate the

changing profile; namely the heat equation

∂T

∂t

=

k

ρC

∂

2T

∂x

2

(3.75)

where k, ρ and C are the thermal conductivity, density and heat capasity of the target. As with

the standard heat conduction model discussed later, the target heat conduction model in CFAST

couples the solid to the gas phase using the relation

q

00 = −k

dT

dx

(3.76)

where q

00 is the heat flux striking the target (again due to all sources). This equation is the

statment that the flux striking the target must be consistent with the temperature gradient at the

surface.

Equation (3.75) may be solved implicitly or explicitly. When solved implicitly, the target

temperature is added to the set of solution variables and equation (3.76) (not equation (3.75) is

47
added to the equation set solved by DASSL. When solved explicitly, equation (3.75) is solved as a

stand-alone equation advancing the temperature profile in time.

Convection

In general, convective heat transfer, q

00, across a surface of area AS, is defined as

q

00 = hAS(Tg −Ts) (3.77)

The convective heat transfer coefficient, h, is a function of the gas properties, temperature, and

velocity. The Nusselt number is defined as NuL = hL/k, which for natural convection is related to

the Rayleigh number, RaL = gβ(Ts −Tg)L

3/να where L is a characteristic length of the geometry,

g is the gravitational constant (m/s2), k is the thermal conductivity (W/m2 K), β is a volumetric

expansion coefficient (K-1), Ts and Tg are the temperatures of the surface and gas, respectively

(K), ν is the kinematic viscosity (m2/s), and α is the thermal diffusivity (m2/s). All properties

are evaluated at the film temperature, Tf = (Ts +Tg)/2. The typical correlations applicable to the

problem at hand are available in the literature [58]. The table below gives the correlations used in

CFAST.

Geometry Correlation Restrictions

Walls NuL =

 

0.825+

0.387Ra1/6

L



1+(0.492/Pr)

9/168/27!2

= 0.12

none

Ceilings and floors

(hot surface up or cold

surface down

NuL = 0.13Ra1/3

L

2x108 ≤ RaL ≤ 1011

Ceilings and floors

(cold surface up or hot

surface down

NuL = 0.16Ra1/3

L

2x108 ≤ RaL ≤ 1010

The Prandtl number, Pr, is the ratio of the kinematic viscosity and the thermal diffusivity. The

thermal diffusivity, α, and thermal conductivity, k, of air are defined as a function of the film

temperature from data in reference [58] as

α = 1.0x10−9T

7/4

f

(3.78)

k =

0.0209+1.33x10−5Tf

1−0.000267Tf

(3.79)

Conduction

Procedures for solving 1-d heat conduction problems are well known, (e.g., backward difference

(fully implicit), forward difference (fully explicit) or Crank-Nicolson [59]). A finite difference

48
approach using a non-uniform spatial mesh is used to advance the wall temperature solution. The

heat equation is discretized using a second order central difference for the spatial derivative and

a backward differences for the time derivative. The resulting tri-diagonal system of equations is

then solved to advance the temperature solution to time t +∆T. This process is repeated , using the

work of Moss and Forney [60], until the heat flux striking the wall (calculated from the convection

and radiation algorithms) is consistent with the flux conducted into the wall calculated via Fouriers

law

q

00 = −k

dT

dx

(3.80)

where k is the thermal conductivity. This solution strategy requires a differential algebraic (DAE)

solver that can simultaneously solve both differential and algebraic equations. With this method,

only one or two extra equations are required per wall segment (two if both the interior and exterior

wall segment surface temperatures are computed). This solution strategy is more efficient than

the method of lines since fewer equations need to be solved. Wall segment temperature profiles,

however, still have to be stored so there is no decrease in storage requirements. Conduction is

then coupled to the room conditions by temperatures supplied at the interior boundary by the

differential equation solver. The exterior boundary condition types (constant flux, insulated, or

constant temperature) are specified in the configuration of CFAST.

A non-uniform mesh scheme was chosen to allow breakpoints to cluster near the interior and

exterior wall segment surfaces. This is where the temperature gradients are the steepest. A breakpoint xb was defined by xb = MIN(xp,W/2) where xp = 2

√αtfinaler f c−1

(.05) and er f c−1 denotes

the inverse of the complementary error function. The value xp is the location in a semi-infinite wall

where the temperature rise is 5 % after tfinal seconds and is sometimes called the penetration depth.

Eighty percent of the breakpoints were placed on the interior side of xb and the remaining 20 %

were placed on the exterior side.

To illustrate the method, consider a one room case with one active wall. There are four gas

equations (pressure, upper layer volume, upper layer temperature, and lower layer temperature)

and one wall temperature equation. Implementation of the gradient matching method requires that

storage be allocated for the temperature profile at the previous time, t, and at the next time, t +δt.

Given the profile at time t and values for the five unknowns at time t + δt (initial guess by the

solver), the temperature profile is advanced from time t to time t + δt. The temperature profile

gradient at x = 0 is computed followed by the residuals for the five equations. The DAE solver

adjusts the solution variables and the time step until the residuals for all the equations are below an

error tolerance. Once the solver has completed the step, the array storing the temperature profile

for the previous time is updated, and the DAE solver is ready to take its next step.

Inter-compartment Heat Transfer

Heat transfer between vertically connected compartments is modeled by merging the connected

surfaces for the ceiling and floor compartments or for the connected horizontal compartments. A

heat conduction problem is solved for the merged walls using a temperature boundary condition

for both the near and far wall. As before, temperatures are determined by the DAE solver so that

the heat flux striking the wall surface (both interior and exterior) is consistent with the temperature

gradient at that surface.

49
For horizontal heat transfer between compartments, the connections may be between partial

wall surfaces, expressed as a fraction of the wall surface. CFAST first estimates conduction fractions analogous to radiative configuration factors. For example, if only one half of the rear wall

in one compartment is adjacent to the front wall in a second compartment, the conduction fraction

between the two compartments is 1/2. Once these fractions are determined, an average flux, q

00

avg,

is calculated using

q

00

avg = ∑

walls

Fi jq

00

wallj

(3.81)

where Fi j is the fraction of flux from wall i that contributes to wall j, q

00

wallj

is the flux striking wall

j.

3.4.6 Ceiling Jet

Relatively early in the development of a fire, fire-driven ceiling jets and gas-to-ceiling convective

heat transfer can play a significant role in room-to-room smoke spread and in the response of

near-ceiling mounted detection hardware. Cooper [61] details a model and computer algorithm

to predict the instantaneous rate of convective heat transfer from fire plume gases to the overhead

ceiling surface in a room of fire origin. The room is assumed to be a rectangular parallelepiped

and, at times of interest, ceiling temperatures are simulated as being uniform. Also presented is

an estimate of the convective heat transfer due to ceiling-jet driven wall flows. The effect on the

heat transfer of the location of the fire within the room is taken into account. This algorithm has

been incorporated into the CFAST model. In this section, we provide an overview of the model.

Complete details are available in reference [[61].

A schematic of a fire, fire plume, and ceiling jet is shown in figure 3.15. The buoyant fire

plume rises from the height Zfire toward the ceiling. When the fire is below the layer interface, its

mass and enthalpy flow are assumed to be deposited into the upper layer at height Zlayer. Having

penetrated the interface, a portion of the plume typically continues to rise toward the ceiling. As it

impinges on the ceiling surface, the plume gases turn and form a relatively high temperature, high

velocity, turbulent ceiling jet which flows radially outward along the ceiling and transfers heat

to the relatively cool ceiling surface. The convective heat transfer rate is a strong function of the

radial distance from the point of impingement, reducing rapidly with increasing radius. Eventually,

the relatively high temperature ceiling jet is blocked by the relatively cool wall surfaces [62]. The

ceiling jet then turns downward and outward in a complicated flow along the vertical wall surfaces

[63, 64] . The descent of the wall flows and the heat transfer from them are eventually stopped by

upward buoyant forces. They are then buoyed back upward and mix with the upper layer.

The average convective heat transfer from the ceiling jet gases to the ceiling surface, Qceil, can

be expressed in integral form as

Qceil =

Z Xwall

0

Z Ywall

0

q

00

ceil(x, y)dxdy (3.82)

The instantaneous convective heat flux, q

00

ceil(x, y) can be determined as derived by Cooper [61]

as

q

00

ceil(x, y) = h(Tad −Tceil) (3.83)

50
ZLayer

(Xre, Yre, Zre)

Figure 3.15: Convective heat transfer to ceiling and wall surfaces via the ceiling jet.

where Tad is a characteristic ceiling jet temperature that would be measured adjacent to an adiabatic

lower ceiling surface, and h is a heat transfer coefficient. h and Tad are given by

h

h ̃

=







8.82Re−1/2

H Pr−2/3



1−



5−0.284Re2/5

H



r

H



0 ≤

r

H < 0.2

0.283Re0.3

H Pr−2/3



r

H

−1.2

r

H −0.0771

r

H +0.279 0.2 ≤

r

H

(3.84)

Tad −Tu

TuQ∗

H

2/3

=







10.22−14.9

r

H

0 ≤

r

H < 0.2

8.39 f



r

H



0.2 ≤

r

H

(3.85)

where

f

 r

H



=

1−1.10

r

H

0.8 +0.808

r

H

1.6

1−1.10

r

H

0.8 +2.20

r

H

1.6 +0.690

r

H

2.4

(3.86)

r =

q

X −Xfire2 +



Y −Yfire2(3.87)

h ̃ = ρucpg

1/2H1/2Q∗

H

1/3

, ReH =

g

1/2H3/2Q∗

H

1/3

νu

, Q

∗

H =

Q

ρucpTug

1/2H5/2

(3.88)

51
Q =







Qf,C



σM ̇ ∗

1+σ



Zfire < Zlayer < Zceil

Qf,C

Zfire ≥ Zlayer

Zlayer = Zceil

, M ̇

∗ =







0 −1 < σ ≤ 0 < 0.2

1.04599σ+0.360391σ

2

1+1.37748σ+0.360391σ2 σ > 0

(3.89)

σ =

1−

Tu

Tl

+CTQ

∗

EQ

2/3

Tu

tl

, CT = 9.115 , Q

∗

EQ =



0.21Qf c

cpTlm ̇ p



(3.90)

In the above, H is the distance from the (presumed) point source fire and the ceiling, Xfire and

Yfire are the position of the fire in the room, Pr is the Prandtl number (taken to be 0.7) and ν is the

kinematic viscosity of the upper layer gas which is assumed to have the properties of air and can

be estimated from ν = 0.04128x107T

5/2

u /(Tu +110.4). Q

∗

H

and Q

∗

EQ are dimensionless numbers

and are measures of the strength of the plume at the ceiling and the layer interface, respectively.

When the ceiling jet is blocked by the wall surfaces, the rate of heat transfer to the surface

increases. Reference [61] provides details of the calculation of wall surface area and convective

heat flux for the wall surfaces.

3.5 Heat Detection

Heat detection is modeled using temperatures obtained from the ceiling jet [61]. Rooms without

fires do not have ceiling jets. Sensors in these types of rooms use gas layer temperatures instead of

ceiling jet temperatures. The characteristic detector temperature is simply the temperature of the

ceiling jet (at the location of the detector). The characteristic heat detector temperature is modeled

using the differential equation [65]

dTL

dt

=

p

v(t)

RT I

(Tg(t)−TL(t)) , TL(0) = Tg(0) (3.91)

where TL and Tg are the link and gas temperatures, v is the gas velocity, and RT I (response time

index) is a measure of the sensor’s sensitivity to temperature change (thermal inertia). The heat

detector differential eq (3.91) may be rewritten to

dT(t)

dt

= a(t)−b(t)T(t) , T(t0) = T0 (3.92)

where a(t) =

√

V(t)T(t)

RT I and b(t) =

√

V(t)

RT I . Equation (3.92) may be solved using the trapezoidal rule

to obtain

Ti+1 −Ti

∆T

=

1

2

((ai −biTi) + (ai+1 −bi+1Ti+1)) (3.93)

where the subscript i denotes time at ti. Equation (3.93) may be simplified to

Ti+1 = Ai+1 −bi+1Ti+1 (3.94)

52
where Ai+1 = Ti +

∆t

2

(ai −biTi +ai+1) and Bi+1 =

∆t

2

bi+1 which has a solution

Ti+1 =

Ai+1

1+Bi+1

=

1−

∆t

2

bi

1+

∆t

2

bi+1

Ti +

∆T

1+

∆t

2

bi+1



ai +ai+1

2



(3.95)

Equation (3.95) reduces to the trapezoidal rule for integration when b(t) = 0. When a(t) and

b(t) are constant (the gas temperature, Tg, and gas velocity, V are not changing), eq (3.91) has the

solution

T(t) = a

b

+

e

−b(t−t0)

(bT0 −a)

b

= Tg +e

−

√

V(t)(t−t0)

RT I (T0 −Tg) (3.96)

3.6 Sprinkler Activation and Fire Attenuation

For suppression, the sprinkler is modeled using a simple model [66] generalized for varying sprinkler spray densities [67]. It is then modeled by attenuating all fires in the room where the sensor

activated by a term of the form e

−(t−tact)/trate where tact is the time when the sensor activated and

trate is a constant determining how quickly the fire attenuates. The term trate can be related to

spray density of a sprinkler using a correlation developed in [67]. The suppression correlation was

developed by modifying the heat release rate of a fire. For t > tact the heat release is given by

Qf(t) = e

−(t−tact)/(3Q

−1.8

spray)Qf(tact) (3.97)

where Qspray is the spray density of a sprinkler. Note that decay rate can be formulated in terms of

either the attenuation rate or the spray density. trate can be expressed in terms of Qspray as trate =

3Q

−1.8

spray. A calculation is done to make sure that the fuel burned is consistent with the available

oxygen. Once detection has occurred, then the mass and energy release rates are attenuated by

m ̇ f(t) = e

−(t−tact)/tratem ̇ f(tact) (3.98)

Qf(t) = e

−(t−tact)/trateQf(tact) (3.99)

There are assumptions and limitations in this approach. Its main deficiency is that it assumes

that sufficient water is applied to the fire to cause a decrease in the rate of heat release. This

suppression model cannot handle the case when the fire overwhelms the sprinkler. The suppression

model as implemented does not include the effect of a second sprinkler. Detection of all sprinklers

are noted but their activation does not make the fire go out any faster. Further, multiple fires in a

room imply multiple ceiling jets. It is not clear how this should be handled, i.e.,how two ceiling jets

should interact. When there is more than one fire, the detection algorithm uses the fire that results

in the worst conditions (usually the closest fire) in order to calculate the fire sensor temperatures.

Finally, the ceiling jet algorithm that we use results in temperature predictions that are warmer

(for a given heat release rate) than those used in the correlation developed by Madrzykowski [68],

which will cause activation sooner than expected.

53
3.7 Species Concentration and Deposition

CFAST uses a combustion chemistry scheme based on a carbon-hydrogen-oxygen balance. The

scheme is applied in three places. The first is burning in the portion of the plume which is in

the lower layer of the compartment of fire origin. The second is the portion in the upper layer,

also in the compartment of origin. The third is in the vent flow which entrains air from a lower

layer into an upper layer in an adjacent compartment. Included in the combustion calculation is

the generation and transport of a number of species that may be produced by a fire. These species

include unburned fuel, nitrogen, oxygen, carbon monoxide, carbon dioxide, hydrogen, carbon

(assumed to be soot produced by the fire), hydrogen cyanide, hydrogen chloride, and an arbitrary

trace species.

3.7.1 Species Transport

The species transport in CFAST is primarily a matter of bookkeeping to track individual species

mass as it is generated by a fire, transported through vents, or mixed between layers in a compartment. When the layers are initialized at the start of the simulation, they are set to ambient

conditions. These are the initial temperature prescribed by the user, and 23 % by mass fraction (21

% by volume fraction) oxygen, 77 % by mass fraction (79 % by volume fraction) nitrogen, a mass

concentration of water prescribed by the user as a relative humidity, and a zero concentration of all

other species. As fuel is burned, the various species are produced in direct relation to the mass of

fuel burned (this relation is the species yields prescribed by the user for the fuel burning). Since

oxygen is consumed rather than produced by the burning, the yield of oxygen is negative, and is

set internally to correspond to the amount of oxygen used to burn the fuel (within the constraint

of available oxygen limits discussed in sec. 3.4.1). Two special separate species calculations are

included in the model, a time-integrated value for a generic toxic species, Ct, and an arbitrary trace

species, TS. Both are assumed not to be part of the overall mass balance, but are rather generated

by a fire and transported through a structure in a manner identical to other species.

Each unit mass of a species produced by a fire is carried in the flow to the various rooms and

accumulates in the layers. The model keeps track of the mass of each species in each layer, and

knows the volume of each layer as a function of time. The mass divided by the volume is the mass

concentration, which along with the relative molecular mass gives the concentration in volume

percent or parts per million as appropriate. Filters can be used in mechanical ventilation systems

to remove species. The phenomenon has been implemented in CFAST to remove trace species

and soot. It is implemented by modifying the source terms which describe gas flow. Mass that

is filtered remains on the filter and is removed from the air stream. Both the resulting species

density and total species removed can be analyzed. See reference [16] for an example on the use

of filtering.

The calculation of radiation exchange in CFAST also depends in part on the species concentrations calculated by the model (and thus the user inputs for species yields). There are two separate

radiation calculations done by CFAST. The first is for broadband radiation transfer for energy

balance. The way this calculation is done is discussed in section 3.4.5. The second is a visible

light calculation to answer the question of whether exit signs will be visible. The absorption of

broadband radiation depends on the concentration of water, carbon dioxide and soot. The visibility

calculation depends solely on the soot concentration For soot, the input for C/CO2 is used to cal54
culate a soot yield from the fire (assuming all the excess carbon goes to soot). This soot generation

is then transported as a species to yield a soot mass concentration to use in the optical density

calculation based originally on the work of Seader and Einhorn [69]. The most recent work is by

Mulholland and Croakin[70]. Based on their experimental measurements, the soot mass density

is multiplied by 3,817 m2

/kg (formerly 3,500 m2/kg) to obtain an optical density (in units of m-1)

which is the value reported by the model.

3.7.2 HCl Deposition

Hydrogen chloride produced in a fire can produce a strong irritant reaction that can impair escape

from the fire. It has been shown [71] that significant amounts of the substance may be removed by

adsorption by surfaces which contact smoke. In our model, HCl production is treated in a manner

similar to other species. However, an additional term is required to allow for deposition on, and

subsequent absorption into, material surfaces.

The physical configuration that we are modeling is a gas layer adjacent to a surface (figure

3.16). The gas layer is at some temperature Tg with a concomitant density of hydrogen chloride,

ρHCl. The mass transport coefficient is calculated based on the Reynolds analogy with mass and

heat transfer; that is, hydrogen chloride is mass being moved convectively in the boundary layer,

and some of it simply sticks to the wall surface rather than completing the journey during the

convective roll-up associated with eddy diffusion in the boundary layer. The boundary layer at

the wall is then in equilibrium with the wall. The latter is a statistical process and is determined

by evaporation from the wall and stickiness of the wall for HCl molecules. This latter is greatly

influenced by the concentration of water in the gas, in the boundary layer and on the wall itself.

Gas Layer at Tg and !HCl

k = transport coefcient (m/s)

Wall Boundary Layer at !bl-HCl

k = equilibrium coefcient (m)

Wall at Tw and mw-HCl

k = wall absorbtion coefcient (1/s)

Figure 3.16: Schematic of hydrogen chloride deposition region.

The rate of addition of mass of hydrogen chloride to the gas layer is given by

d

dt

mHCl = source−kc(ρHCl −ρbl−HCl)Aw (3.100)

where source is the production rate from the burning object plus flow from other compartments.

For the wall concentration, the rate of addition is

d

dt

dw−HCl = kc(ρHCl −ρbl−HCl)−ksmw−HCl (3.101)

where the concentration in the boundary layer, ρbl−HCl is related to the wall surface concentration

by the equilibrium constant ke, by the relation ρbl−HCl = dw−HCl/ke. We never actually solve for

55
Table 3.3: Transfer coefficients for HCl deposition

Surface b1 b2 b3 b4 b5 b6 b7

(m) (m3/kg) (s-1) (J/g mol) (m3/kg)b7 −b6(note a) (note b)

Painted Gypsum 0.0063 191.8 0.0587 7476 193 1.021 0.431

PMMA 9.6x10−5 0.0137 0.0205 7476 29 1.0 0.431

Ceiling Tile 4.0x10−3 0.0548 0.123 7476 30a 1.0 0.431

Cement Block 1.8x10−2 5.48 0.497 7476 30a 1.0 0.431

Calcium Silicate Board 1.9x10−2 0.137 0.030 7476 30a 1.0 0.431

a - very approximate value, insufficient data for high confidence value

b - non-dimensional

the concentration in the boundary layer, but it is available, as is a boundary layer temperature if it

were of interest. The transfer coefficients are

kc =

q ̇

∆Tρgcp

(3.102)

ke =

b1e

1500/Tw

1+b2e

1500/TwρHCl 

1+

b5(ρH2O)

b6



ρH2O,sat −ρH2O,g

b7

!

(3.103)

ks = b3e

−



b4

RTw



(3.104)

The only values currently available for these quantities are shown in table 3.3 [72]. The “b”

coefficients are parameters which are found by fitting experimental data to the above equations.

These coefficients reproduce the adsorption and absorption of HCl reasonably well. Note though

that error bars for these coefficients have not been reported in the literature.

The experimental basis for poly(methyl methacrylate) and gypsum cover a sufficiently wide

range of conditions that they should be usable in a variety of practical situations. The parameters

for the other surfaces do not have much experimental backing, and so their use should be limited

to comparison purposes.

3.8 Single Zone Approximation

A single zone approximation is appropriate for smoke flow far from a fire source where the twozone layer stratification is less pronounced than in compartments near the fire. In this situation,

a single zone approximation may be derived by using the normal two-zone source terms and the

substitutions:

m ̇

new

U = m ̇ L +m ̇U

m ̇

new

L = 0

Q

new

U = QL +QU

Q

new

L = 0

(3.105)

56
This is used in situations where the stratification does not occur. Examples are elevators shafts,

complex stairwells, natural venting ductwork, and compartments far from the fire.

3.9 Review of the Theoretical Development of the Model

Details of the software quality assurance process for CFAST is included in the Software and Model

Evaluation Guide [8]. This section provides a summary of this process The current version of

ASTM E 1355-04 includes provisions to guide in the assessment of the theoretical basis of the

model that includes a review of the model “by one or more recognized experts fully conversant

with the chemistry and physics of fire phenomenon, but not involved with the production of the

model. Publication of the theoretical basis of the model in a peer-reviewed journal article may be

sufficient to fulfill this review [1].

CFAST has been subjected to independent review in two ways, internal and external. First,

all documents issued by the National Institute of Standards and Technology receive three levels of

internal review by members of the staff not involved in the preparation of the report or underlying

research. The theoretical basis of CFAST is presented in this document, and is subject to internal

review by staff members who are not active participants in the development of the model, but

who are members of the Fire Research Division and are considered experts in the fields of fire

and combustion. The same was true of previous versions of the technical reference guide over

the last decade [2, 73, 74]. Externally, the theoretical basis for the model has been published

in peer reviewed journals [75, 76, 77] and conference proceedings [78]. In addition, CFAST is

used worldwide by fire protection engineering firms who review the technical details of the model

related to their particular application. Some of these firms also publish in the open literature reports

documenting internal efforts to validate the model for a particular use. Many of these studies are

discussed in more detail in the present document.

In addition to the formal review, procedures were in place during the development of CFAST

to assure the quality of the model. These procedures included several components:

• Review of proposed changes to the code by at least two others involved in the development

process to insure that a proposed change was consistent with the rest of the CFAST code and

was implemented correctly. These reviews, while informal in nature, provided a comprehensive review of the changes to the model during its development. Significant changes were

documented in internal memorandums covering such areas as the numerics and structure of

the model [79], improvements in the chemistry [80], convection [81], HCl deposition [82]

algorithms, and output formats for the model [83]. Comparisons of the impact of the changes

on the output results were often described in internal memorandums (see, for example, reference [79]).

• Internal review of the model prior to public release. In addition to the normal NIST document

review process, the CFAST software was circulated internally to Fire Research Division Staff

to allow interested staff members to test the model [84, 85, 86]. These memorandums detail

changes to the model since the last public release of the model and provide documentation

of the history of the model development.

57
• For each major release of CFAST, NIST has maintained a history of the source code which

goes back to March 1989. While it is not practical to reconstruct the programs for each

release for use with modern software tools and computer operating systems, the source code

history allows the developers to examine what changes were made at each release point. This

provides detailed documentation of the history of model development and is often useful to

understand the impact of changes to submodels over the development of the model.

• Once a release of CFAST was approved by NIST, it was announced with a letter to model

users which provided a summary of model changes and available documentation. In essence,

these were a condensation of the internal memorandums, without details or printout of specific code changes. These memorandums provide documentation of the history of the model

development [87, 88, 89, 90, 91].

Finally, CFAST has been reviewed and included in industry-standard handbooks such as the

SFPE Handbook [92] and referenced in specific standards, including NFPA 805 [93] and NFPA

551 [94].

3.9.1 Assessment of the Completeness of Documentation

There are three primary documents on CFAST, this Technical Reference Guide, the Users Guide

[7], and the Software Development and Model Evaluation Guide [8]. This document is the Technical Reference Guide and provides documentation of the governing equations, assumptions, and

approximations of the various submodels. It also includes a summary description of the model

structure, and numerics. The Model Users Guide includes a description of the model input data requirements and model results. The Software Development and Model Evaluation Guide describes

the software quality assurance process used in the development and maintenance of the model and

includes an extensive discussion of the validation of the model.

The extensive formal review process for all NIST publications in part insures the quality of the

CFAST Guides. In addition, the model developers routinely receive feedback from users on the

completeness of the documentation and add clarifications when needed. It is estimated that there

are several thousand users of CFAST. Before new versions of the model are released, there is a

“beta test” period in which the users test the new version using the updated documentation. This

process is similar, although less formal, to that which most computer software programs undergo.

Training courses for use of the model in fire hazard analysis have been developed from the model

documentation and presented at training courses worldwide [95].

3.9.2 Assessment of Justification of Approaches and Assumptions

The technical approach and assumptions of the model have been presented in the peer reviewed

scientific literature and at technical conferences. Also, all documents released by NIST are required to go through an internal editorial review and approval process. This process is designed

to ensure compliance with the technical requirements, policy, and editorial quality required by

NIST. The technical review includes a critical evaluation of the technical content and methodology, statistical treatment of data, uncertainty analysis, use of appropriate reference data and units,

and bibliographic references. CFAST manuals are always first reviewed by a member of the Fire

58
Research Division, then by the immediate supervisor of the author of the document, then by the

chief of the Fire Research Division, and finally by a reader from outside the division. Both the

immediate supervisor and the division chief are technical experts in the field. Once the document

has been reviewed, it is then brought before the Editorial Review Board (ERB), a body of representatives from all the NIST laboratories. At least one reader is designated by the Board for each

document that it accepts for review. This last reader is selected based on technical competence

and impartiality. The reader is usually from outside the division producing the document and is

responsible for checking that the document conforms with NIST policy on units, uncertainty and

scope. This reader does not need to be a technical expert in fire or combustion.

Besides formal internal and peer review, CFAST is subjected to continuous scrutiny because it

is available to the general public and is used internationally by those involved in fire safety design

and postfire reconstruction. The source code for CFAST is also released publicly, and has been

used at various universities worldwide, both in the classroom as a teaching tool as well as for

research. As a result, flaws in the theoretical development and the computer program itself have

been identified and fixed. The user base continues to serve as a means to evaluate the model, which

is as important to its development as the formal internal and external peer review processes.

3.9.3 Assessment of Constants and Default Values

A comprehensive assessment of the numerical parameters (such as default time step or solution

convergence criteria) and physical parameters (such as empirical constants for convective heat

transfer or plume entrainment) used in CFAST is not available in one document. Instead, specific

parameters have been tested in various verification and validation studies performed at NIST and

elsewhere. Numerical parameters are described in this Technical Reference Guide and are subject

to the internal review process at NIST, but many physical parameters are extracted from the literature and do not undergo a formal review. In addition, default values for the various model inputs

have been specifically reviewed by a professional fire protection engineering university professor

to insure appropriate default values and suggested limits for the various input values. The model

user is expected to assess the appropriateness of default values provided by CFAST and make

changes to the default values if need be.

59
60
Chapter 4

Mathematical and Numerical Robustness

The mathematical and numerical robustness of a deterministic computer model depends upon three

issues: the code must be transparent so that it can be understood and modified by visual inspection;

it must be possible to check and verify with automated tools; and there must be a method for

checking the correctness of the solution, at least for asymptotic (steady state) solutions (numerical

stability and agreement with known solutions).

In order to understand the meaning of accuracy and robustness, it is necessary to understand the

means by which the numerical routines are structured. In this chapter, details of the implementation

of the model are presented, including the tests used to assess the numerical aspects of the model.

These include

• the structure of the model, including the major routines implementing the various physical

phenomena included in the model,

• the organization of data initialization and data input used by the model,

• the structure of data used to formulate the differential equations solved by the model,

• a summary of the main control routines in the model that are used to control all input and

output, initialize the model and solve the appropriate differential equation set for the problem

to be solved,

• the means by which the computer code is checked for consistency and correctness,

• analysis of the numerical implementation for stability and error propagation, and

• comparison of the results of the system model with simple analytical or numerical solutions.

4.1 Structure of the Numerical Routines

A methodology which is critical to verification of the model is the schema used to incorporate physical phenomena. This is the subroutine structure discussed below. The method for incorporating

new phenomena and insuring the correctness of the code was adopted as part of the consolidation

of CCFM and FAST. This consolidation occurred in 1990 and has resulted in a more transparent,

61
Main Routine

Solve Auxiliary Equations

Solve Differential Equations

Calculate Right-hand Side of

Differential Equations and

Residuals

Input Routines

Output Routines

CFAST

SOLVE

DASSL

RESID

MFLOW HFLOW VFLOW FIRES DJET CJET CVHEAT CNHEAT RDHEAT HCL

Mechanical

Ventilation

Horizontal

Flow

Vertical

Flow

Fire

Chemistry

Door Jet

Fires

Ceiling

Jets

Conduction Convection Radiation HCl

Deposition

Output

Outputsmv

Outputspreadsheet

Readinputle

Keywordcases

Inputmainre

Inputobject

Inputtpp

Readcsvformat

Physical Interface Routines

Figure 4.1: Subroutine structure for the CFAST model.

transportable and verifiable numerical model. This transparency is crucial to a verifiable and robust

numerical implementation of the predictive model as discussed in the sections on code checking

and numerical analysis.

The model can be split into distinct parts. There are routines for reading data, calculating results

and reporting the results to a file or printer. The major routines for performing these functions are

identified in figure 4.1. These physical interface routines link the CFAST model to the actual

routines which calculate quantities such as mass or energy flow at one particular point in time for

a given environment.

The routines SOLVE, RESID and DASSL are the key to understanding how the physical equations are solved. SOLVE is the control program that oversees the general solution of the problem.

It invokes the differential equation solver DASSL [96] which in turn calls RESID to solve the

transport equations. Given a solution at time t, what is the solution at time t plus a small increment of time, ∆t, (where the time increment is determined dynamically by the program to insure

convergence of the solution at t +∆t)? The differential equations are of the form

dy

dt

= f(y,t) , y(t0) = y0 (4.1)

where y is a vector representing pressure, layer height, mass and such, and f is a vector function

that represents changes in these values with respect to time. The term y0 is an initial condition at

62
the initial time t0 . The subroutine RESID computes the right hand side of eq (4.1) and returns

a set of residuals of that calculation to be compared to the values expected by DASSL. DASSL

then checks for convergence. Once DASSL reaches an error limit (defined as convergence of the

equations) for the solution at t +∆t, SOLVE then advances the solution of species concentra- tion,

wall temperature profiles, and mechanical ventilation for the same time interval. Note that there are

several distinct time scales that are involved in the solution of this type of problem. The fastest will

be chemical kinetics. We avoid that scale by assuming that the chemistry is infinitely fast. The next

larger time scale is that associated with the flow field. These are the equations which are cast into

the form of ordinary differential equations. Then there is the time scale for mechanical ventilation,

and finally, heat conduction through objects. Chemical kinetic times are typically on the order of

milliseconds. The transport time scale are on the order of 0.1 s. The mechanical ventilation and

conduction time scales are typically several seconds, or even longer. The time step is dynamically

adjusted to a value appropriate for the solution of the currently defined equation set. In addition to

allowing a more correct solution to the pressure equation, very large time steps are possible if the

problem being solved approaches steady-state.

4.2 Code Checking

There are two means to automate checking the correctness of the language used by a numerical

model. The first is the use of standard methods for checking the structure and interface. Programs

such as Flint and Lint are standard tools to do such checking. They are applied to the whole

model. There are three aspects of the model checked by this procedure: correctness of the interface,

undefined or incorrectly defined (or used) variables and constants, and completeness of loops and

threads. It does not check for the correctness of the numerical use of constants or variables only

that they are used correctly in a syntactical sense. Lint is part of most C language distributions of

Unix. Flint is the equivalent for the FORTRAN language. Though it is not usually included with

FORTRAN distributions Flint is generally available 1. Both have been used with CFAST.

The second is to use a variety of computer platforms to compile and run the code. Since FORTRAN and C are implemented differently for various computers, this represents both a numerical

check as well as a syntactic check. CFAST has been compiled for the Sun (Solaris), SGI (Irix), the

windows-based PCs (Lahey, Digital, and Intel FORTRAN), and the Concurrent Computer platforms. Within the precision afforded by the various hardware implementations, the answers are

identical 2.

4.3 Numerical Tests

There are two components to testing the numerical solutions of CFAST. First, the DASSL solver

is well tested for a wide variety of differential equations, and is widely used and accepted [96].

Also, the radiation and conduction routines are tested with known solutions. These are not analytical tests, but physical limits, such as an object immersed in a fluid of constant temperature, to

1Cleanscape Software, 445 Sherman Ave, Ste. Q, Palo Alto, CA 94306

2Typically one part in 10-6, which is the error limit used for DASSL.

63
which the temperature must equilibrate. The solver(s) must show that the differential equations

asymptotically converge to these answers.

The second is to insure that the coupling between algorithms and the solver is correct. Most

errors are avoided because of the structure discussed in section 4.1. The error due to the numerical

solution is far less than that associated with the model assumptions. Two examples of this are

the coupling of mechanical ventilation with buoyant flow, and the Nusselt number assumption for

boundary layer convection. For the former, the coupling of a network of incompressible flow with

an ODE for compressible flow has to deal with disparate calculations of pressure. For the latter, a

very small time step occurs when a floor is heated and the thermal wave reaches the far (unexposed)

side. This is a limitation of the physical implementation of the heat flow algorithm (convection).

The solver arrives at the correct solution, but the time step becomes very small in order to achieve

this.

Numerical error can be divided into three categories: roundoff, truncation and discretization

error. Roundoff error occurs because computers represent real numbers using a finite number of

digits. Truncation error occurs when an infinite process is replaced by a finite one. This can happen,

for example, when an in finite series is truncated after a finite number of terms or when an iteration

is terminated after a convergence criterion has been satisfied. Discretization error occurs when a

continuous process such as a derivative is approximated by a discrete analog such as a divided

difference. CFAST is designed to use 64-bit precision for real number calculations to minimize

these effects.

Implicit in solving the equations discussed in chapter 3, is that the solver will arrive at a solution. Inherent in the DASSL solver are convergence criteria for the mass and energy balance within

CFAST to insure mass and energy conservation within 1 part in 106. There are, however, limitations introduced by the algorithmic realization of physical models, that can produce errors and

instabilities. Using the example above, if a mechanical ventilation system injects or removes mass

and enthalpy from a small duct, then there can be a stability issue with the layer interface bobbing

up and down over the duct. These are annoyances to the user community and shortcomings of the

implementation of algorithm rather than failure of the system model.

Problems of this sort are noted in the frequently asked questions on the CFAST web site

(http://cfast.nist.gov).

4.4 Comparison with Analytic Solutions

There do not exist general analytic solutions for fire problems, even for the simplest cases. That

is, there are no closed form solutions to this type of problem. However, it is possible to do two

kinds of checking. The first type is discussed in the section on the theoretical basis of the model,

for which individual algorithms are validated against experimental work. The second is simple

experiments, especially for conduction and radiation, for which the results are asymptotic. For

example, for a simple, single compartment test case with no fire, all temperatures should equilibrate

asymptotically to a single value. Such comparisons are common and not usually published.

64
Chapter 5

Sensitivity of the Model

A sensitivity analysis considers the extent to which uncertainty in model inputs influences model

output. For a sensitivity analysis, this uncertainty includes not only that inherent in the input of

data for specific scenarios by the model user, but also uncertainty in empirical data or numerical

parameters in the model such as the time step size used by the model to obtain a solution.

Among the purposes for conducting a sensitivity analysis are to determine

• the important variables in the models,

• the computationally valid range of values for each input variable, and

• the sensitivity of output variables to variations in input data.

Conducting a sensitivity analysis of a complex model is not a simple task and it will differ

depending on the application. CFAST typically requires the user to provide numerous input parameters that describe the building geometry, compartment connections, construction materials,

and description of one or more fires.

Iman and Helton [97] studied the sensitivity of complex computer models developed to simulate the risk of severe nuclear accidents which may include fire and other risks. Consistent with

the work of Iman and Helton [97], ASTM E1355 [1] provides overall guidance on typical areas of

evaluation of the sensitivity of deterministic fire models. These areas may involve one or more of

the following techniques: finite difference or direct analysis methods that provide an explicit solution of the sensitivity equations associated with the governing equations of the model, factorial

design or Latin hypercube sampling studies that investigate the effect of varying the input parameters and consequential interactions between parameters that may be deemed important, and global

or response surface methods that investigate the overall behavior of model outputs for a desired

range of inputs.

This chapter provides a review of the sensitivity studies that have been conducted using CFAST

with an emphasis on uncertainty in the input. Other sensitivity investigations of CFAST are also

available [98, 99, 100].

5.1 Factorial Design Studies

Khoudja [[101] has studied the sensitivity of an early version of the FAST [2] (predecessor to

CFAST) model with a fractional factorial design involving two levels of 16 different input pa65
rameters. The statistical design, taken from the texts by Box and Hunter [102], and Daniel [103]

reduced the necessary model runs from more than 65 000 to 256 by studying the interactions of

input parameters simultaneously. The choice of values for each input parameter represented a

range for each parameter. The analysis of the FAST model showed sensitivity to heat loss to the

compartment walls and to the number of compartments in the simulation. Without the inclusion of

surface thermophysical properties, this model treats surfaces as adiabatic for conductive heat transfer. Thus, consistent sensitivity should be expected. Sensitivity to changes in thermal properties of

the surfaces were not explored.

Walker [104] discussed the uncertainties in components of zone models and showed how uncertainty within user-supplied data affects the results of calculations using CFAST as an example.

The study systematically varied inputs related to the fire (heat release rate, heat of combustion,

mass loss rate, radiative fraction, and species yields) and compartment geometry (vent size and

ceiling height) ranging from ± 1 % to ± 20 % of base values for a one- compartment scenario.

Heat release rate and ceiling height are seen to be the dominant input variables in the simulations.

Upper layer temperature changed ± 10 % for a ± 10 % change in heat release rate. Typical variation of ± 10 s in time to untenable conditions for a 20 % variation in the inputs was noted for the

scenarios studied.

Peacock et al. [98] studied the sensitivity of CFAST for a range of input parameters. They

used simple factorial designs for model inputs deemed important to investigate local behavior of

important model outputs along with response surface methods to evaluate overall model behavior.

Results of the parametric investigations are discussed below and the application of response surface

methods is summarized in section 5.2. Both are discussed in more detail in reference [98].

5.1.1 Model Inputs and Outputs

Most studies of modeling related to fire hazard and fire reconstruction present a consistent set of

variables of interest to the model user [105, 106, 99, 107] : upper and lower gas layer temperatures,

gas species concentrations, and layer interface position. Other variables of interest include

• mass pyrolysis and heat release rate,

• room pressure, and

• vent flow.

Although there are certainly other comparisons of interest, these will provide evidence of the

sensitivity of the model to most model inputs. Tables 4 and 5 show typical inputs and outputs for

the CFAST model.

Consider the following fire scenario: The building geometry (figure 5.1) includes four rooms

on two floors with horizontal, vertical, and mechanical vents connecting the rooms and venting

to the outdoors. The fire source in one of the rooms on the lower floor is a medium growth rate

t-squared fire [108] chosen to simulate a mattress fire [23].

Sensitivity to Small Changes in Model Inputs

To investigate the sensitivity of the model, a number of simulations were conducted varying the

input parameters about the base scenario discussed in the previous section. Both small (±10 %) and

66
Table 5.1: Typical Inputs for a Two-Zone Fire Model

Ambient Conditions Inside temperature and pressure.

Outside temperature and pressure.

Wind speed.

Relative humidity

Building Geometry Compartment width, depth, height, and surface material properties

(conductivity, heat capacity, density, thickness).

Horizontal Flow Vents: Height of soffit above floor, height of sill

above floor, width of vent, angle of wind to vent, time history of

vent openings and closings.

Vertical Flow Vents: Area of vent, shape of vent.

Mechanical Ventilation, Orientation of vent, Center height of vent,

area of vent, length of ducts, diameter of ducts, duct roughness, duct

flow coefficients, fan flow characteristics.

Fire Specification Fire room, X, Y, Z position in room, fire area. Fire Chemistry: Molar

Weight, Lower oxygen limit, heat of combustion, initial fuel temperature, gaseous ignition temperature, radiative fraction. Fire History:

Mass loss rate, heat release rate, species yields for HCN, HCl, H/C,

O2/C, C/CO2, CO/CO2.

Table 5.2: Typical Outputs for a Two-Zone Fire Model

Environment for each compartment: Compartment pressure and layer interface

height.

for each layer and compartment: Temperature, layer mass density,

layer volume, heat release rate, gas concentrations (N2, O2, CO2,

CO, H2O, HCl, HCN, soot optical density), radiative heat into layer,

convective heat into layer, heat release rate in layer.

for each vent and layer: Mass flow, entrainment, vent jet fire.

for each fire: Heat release rate of fire, mass flow from plume to upper

layer, plume entrainment, pyrolysis rate of fire.

for each compartment surface: Surface temperatures.

Tenability Temperature.

Fractional Effective Dose (FED).

67
Room 1

4 m x 3 m x 2.4 m

Room 2

10 m x 3 m x 2.4 m

Room 3 and Room 4

4 m x 3 m x 2.4 m

All doorways 1 m wide x 2 m high

All surfaces gypsum

Vertical ow vent

1 m2

!

Figure 5.1: Building Geometry for base case scenario.

larger (up to an order of magnitude) variations for selected inputs were studied. Varying most of

the inputs by small amounts had little effect on the model outputs. Figure 5.2 presents an example

of the time dependent sensitivity of several outputs to a 10 % change in room volume for the fire

compartment in the scenario described above. For example, the pair of dotted-line curves labeled

Upper Layer Volume were created by comparing the base case scenario with a scenario whose

compartment volume was increased and decreased by 10 %. The resulting curves presented on the

graph are the relative difference between the variant cases and the base case defined by (Variant

value - Base value) / Base value for each time point. The graph shows that temperature and pressure

are insensitive to changes in the volume of the fire room since a 10 % change in room volume led to

smaller relative changes in layer temperature and room pressure for all times. Upper layer volume

can be considered neutrally sensitive (a 10 % change in room volume led to about a 10 % change

in layer volume). Further, this implies that there is negligible effect on the average layer interface

height. This is consistent with both experimental observations in open compartment room fires

and analytical solutions for single compartment steady-state fires . For transient conditions early

in the fire or when the fire burns out (illustrated in the figure at 300 s when the gas burner fire heat

release rate goes to zero) higher uncertainties are noted. While these are transient effects, the early

phases of the fire, in particular, may be important in calculating tenability for occupants during

egress. While an uncertainty in the compartment volumes results in an equivalent uncertainty in

calculated outputs, accurate specification of compartment dimensions within 5 % is often easily

obtained.

In addition, figure 5.2 shows a somewhat constant relative difference for the changes as a

function of time. As suggested by Iman and Helton [[97], an average relative difference could thus

be used to characterize the model sensitivity for comparing individual inputs and outputs.

68
Figure 5.2: An example of time dependent sensitivity of fire model outputs to a 10 % change in

room volume for a single room fire scenario.

5.1.2 Sensitivity to Larger Changes in Model Inputs

To investigate the effects of much larger changes in the inputs, a series of simulations was conducted where the inputs were varied from 10 % to 400 % of base values. Simulations changing the

heat release rate inputs from the base peak heat release rate of 750 kW are shown in figure 5.3.

Each set appears as families of curves with similar functional forms. This indicates that the heat

release rate has a monotonic effect on the layer temperatures, with not as clear an effect on upper

layer volume due to compartment filling and flow between compartments. Like the sensitivity to

compartment volume in the previous section, changing the heat release rate by a factor of two

results in a factor of two change in the upper layer temperature. Thus, in absolute terms, heat

release rate and compartment volume are equally sensitive. However, compartment volume is

easily determined accurately while heat release rate is typically estimated with far less accuracy

and may be uncertain to within an order of magnitude or larger.

In the majority of fire cases, the most crucial question that can be asked by the person responsible for fire protection is: How big is the fire? Put in quantitative terms, this translates to: What is the

heat release rate of this fire? Recently the National Institute of Standards and Technology (NIST)

examined the pivotal nature of heat release rate measurements in detail [109]. Not only is heat

release rate seen as the key indicator of real-scale fire performance of a material or construction,

heat release rate is, in fact, the single most important variable in characterizing the flammability

of products and their consequent fire hazard. Much of the remainder of this paper focuses on heat

release rate as an example for examining sensitivity analysis.

69
Figure 5.3: Layer temperatures and volumes in several rooms resulting from variation in heat

release rate for a four-room growing fire scenario.

70
Figure 5.4: Comparison of the time dependent heat release rate and layer temperatures in several

rooms for a four-room growing fire scenario.

5.2 Response Surface Studies

A next step beyond the simple plots presented in figure 5.3 is a cross-plot of outputs of interest

against heat release rate. Figure 5.4 presents plots of the upper layer temperature (presented in

figure 5.3) plotted against the heat release rate for all the simulations. The temperature curves for

upper layer temperature in all four rooms (figure 5.4) show a strong functional dependence on heat

release rate. Even for the wide variation in inputs, the heat release rate provides a simple predictor

of the temperature in the rooms. In addition, this relationship allows calculation of the sensitivity of

the temperature outputs to the heat release rate inputs as a simple slope of the resulting correlation

between heat release rate and temperature.

Figure 5.5, simply a plot of the slope of the regression curves in figure 5.4, shows this sensitivity, ∂(T)/∂(heatreleaserate), for the four-room scenarios studied and represents all time points

in all he simulations in which the peak heat release rate was varied from 0.1 to 4.0 times the base

value. Except for relatively low heat release rate, the upper layer temperature sensitivity is less

than 1 K/kW and usually below 0.2 K/kW. Not surprisingly, the layer that the fire feeds directly is

most sensitive to changes. The lower layer in the fire room and all layers in other rooms have sensitivities less than 0.2 K/kW. This implies, for example, that if the heat release rate for a 1 MW fire

is known to within 100 kW, the resulting uncertainty in the calculation of upper layer temperature

in the fire room is about ± 30 K.

For upper layer volumes (figure 5.6) of both rooms 1 and 2, it is again a simple correlation

between heat release rate and volume fraction (upper layer volume expressed as a fraction of the

71
Figure 5.5: Sensitivity of temperature to heat release rate for a four-room growing fire scenario.

72
Figure 5.6: Sensitivity of temperature to heat release rate for a four-room growing fire scenario.

total room volume). The shaded gray area on the graph shows the locus of all individual time point

values of temperature and volume in the four compartments of the simulation. The correlations

for the upper layer volumes of room 1 and room 2 could also be differentiated as was done for the

temperature correlations to obtain sensitivities for the upper layer volume. For rooms 3 and 4, the

relationship is not as clear. The flow into the layers of these rooms is more complicated than for

rooms 1 and 2, resulting from flow from the first floor through a vent in the floor of room 3 and

from a vent to the outside in room 4. However, even these rooms approach a constant value for

higher heat release rate values, implying near zero sensitivity for high heat release rate.

Figure 5.7 presents the effect of both peak heat release rate and vent opening (in the fire room)

on the peak upper layer temperature. In this figure, actual model calculations, normalized to the

base scenario values are indicated by circles overlaid on a surface grid generated by a spline interpolation between the data points. At high heat release rate and small vent openings, the fire

becomes oxygen limited and the temperature trails off accordingly, but for the most part, the behavior of the model is monotonic in nature. Although more laborious, the approaches used to

calculate sensitivities for single variable dependencies illustrated earlier are thus equally applicable to multivariate analyses.

From the surface, it is clear that heat release rate has more of an effect on the peak temperature

than does the vent width. Until the fire becomes oxygen limited, the trends evident in the surface

are consistent with expectations temperature goes up with rising heat release rate and down with

rising vent width. The effects are not, of course, linear with either heat release rate or vent opening.

73
Figure 5.7: Sensitivity of temperature to heat release rate for a four-room growing fire scenario.

Plume theory and typically used algorithms for estimating upper layer temperature in a single room

with a fire suggest that the dependence is on the order of Q

2/3

f

for heat release rate and A

√

h for

the vent opening where A is the area of the vent and h is the height of the vent. Although these

correlations are based on a simple analysis of a single room fire, the dependence suggested is

similar to that illustrated in figure 5.7.

5.3 Latin Hypercube Sampling Studies

Notarianni [100] developed an iterative methodology for the treatment of uncertainty in fire-safety

engineering calculations to identify important model parameters for detailed study of uncertainty.

She defines a nine-step process to identify crucial model inputs and parameters, select sampling

methods appropriate for the important parameters, and evaluate the sensitivity of the model to chosen outcomes. Both factorial designs and Latin hypercube sampling are included in a case study

involving the CFAST model. In a performance-based design of a 16 story residential structure,

the impact of model uncertainty on a chosen design and inclusion of residential sprinklers in the

design would effect the resulting safety of the design. For a seven-compartment scenario representing one living unit in the structure, distributions of input variables based on Latin hypercube

sampling of selected ranges of the inputs were developed and used as input for a series of 500

CFAST simulations for the scenario. The results of the calculations are presented in a series of

cumulative distribution functions which show the probability that a chosen criterion of the design

is exceeded within a given time. Depending on the evaluation criterion chosen, times to unaccept74
able designs varied by as little as 10 s to as much as 470 s. To determine important input variables,

Notarianni used a multivariate correlation of the input and output variables to determine statistical

significance at a 95 % confidence level. Input variables deemed important in the analysis included

fire-related inputs (growth rate, heat of combustion, position of the base of the fire, and generation

rates of products of combustion) and door opening sizes. Other inputs were determined to be less

important.

5.4 Summary

Many of the outputs of the CFAST model are quite insensitive to uncertainty in the input parameters

for a broad range of scenarios. Not surprisingly, heat release rate was consistently seen as the most

important variable in a range of simulations. Heat release rate and related variables such as heat of

combustion or generation rates of products of combustion provide the driving force for fire-driven

flows. For CFAST, all of these are user inputs. Thus, careful selection of these fire related variables

are necessary for accurate predictions. Other variables related to compartment geometry such as

compartment height or vent sizes, while deemed important for the model outputs, are typically

more easily defined for specific design scenarios than fire related inputs. For some scenarios, such

as typical building performance design, these vents may need to include the effects of leakage to

insure accurate predictions. For other scenarios, such as shipboard use or nuclear power facilities,

leakage (or lack thereof) may be easily defined and may not be an issue in the calculations.

75
76
Chapter 6

Survey of Past Validation Work

CFAST has been subjected to extensive validation studies by NIST and others. There are two ways

of comparing predictive capability with actual events. The first is simply graphing the time series

curves of model results with measured values of variables such as temperature. Another approach

is to consider the time to critical conditions such as flashover. Making such direct comparisons between theory and experiment provides a sense of whether predictions are reasonable. This chapter

provides a review of CFAST validation efforts by NIST and others to better understand the quality

of the predictions by the model.

Some of the work has been performed at NIST, some by its grantees and some by engineering

firms using the model. Because each organization has its own reasons for validating the model, the

referenced papers and reports do not follow any particular guidelines. Some of the works only provide a qualitative assessment of the model, concluding that the model agreement with a particular

experiment is “good” or “reasonable.” Sometimes, the conclusion is that the model works well in

certain cases, not as well in others. These studies are included in the survey because the references

are useful to other model users who may have a similar application and are interested in qualitative

assessment. It is important to note that some of the papers point out flaws in early releases of

CFAST that have been corrected or improved in more recent releases. Some of the issues raised,

however, are still subjects of active research. Continued updates for CFAST are greatly influenced

by the feedback provided by users, often through publication of validation efforts.

6.1 Comparisons with Full-Scale Tests Conducted Specifically

for the Chosen Evaluation

Several studies have been conducted specifically to validate the use of CFAST in building performance design. Dembsey [110] used CFAST version 3.1 to predict the ceiling jet temperatures,

surface heat fluxes and heat transfer coefficients for twenty compartment fire experiments in a

compartment that is similar in size, geometry, and construction to the standard fire test compartment specified in the Uniform Building Code [111]1

. Results from 330 kW, 630 kW, and 980 kW

fires were used. In general, CFAST made predictions which were higher than the experimental

1The 1997 Uniform Building Code has been superceded by the International Building Code, 2003 Edition, International Code Council, Country Club Hills, Illinois.

77
results. In these cases, the temperature prediction is typically 20 % to 30 % higher than measured

values. Much of this can be attributed to not knowing the species production (soot) and relative

absorption of radiation by the gas layers which highlights the importance of scenario specification.

This is the most common cause of over prediction of temperature by CFAST. A secondary source

of discrepancy is correcting for radiation from thermocouple beads. The authors provide for this

correction, but the corrections cited are not as large as has been reported in other fire experiments

[112].

He et al. [113] describe a series of full-scale fire experiments that were designed to investigate

the validity of two zone models including CFAST version 3.1. The experiments, involving steady

state burning rates and a number of ventilation conditions, were conducted in a four-story building.

Temperature, pressure, flow velocity, smoke density and species concentrations were measured in

various parts of the building. The stack effect and its influence on temperature distribution in a

stair shaft were observed. Comparisons were then made between the experimental results and the

model predictions. Early in the fire there is a few percent difference2 between the predictions and

measurements; beyond 10 min, there are significant variations. Both the experiment and the model

are internally consistent; that is, higher flow leads to a higher interface height (figure 13 in the

paper). Once again, the difference is about 25 %. The authors discuss the effect of fuel composition

and correction for radiation from thermocouple beads but did not draw firm conclusions based on

their measurements of fuel products.

A series of experimental results for flaming fires, obtained using realistic fires in a prototype

apartment building were performed by Luo et al. [114]. Fuel configurations in the fire test included a horizontal plain polyurethane slab, mock-up chair (polyurethane slabs plus a cotton linen

cover), and a commercial chair. CFAST version 3.1 typically over-predicted upper layer temperatures by 10 % to 50 % depending on the test conditions and measurement location in that test.

The predicted and experimental time dependent upper layer temperatures were similar in shape.

The time to obtain peak upper layer temperatures was typically predicted to within 15 % of the

experimental measurements. The authors concluded that CFAST was conservative in terms of life

safety calculations.

In order to optimize fire service training facilities, the best use of resources is imperative. The

work reported by Poole et al. [115] represents one aspect of a cooperative project between the

city of Kitchener Fire Department (Canada) and the University of Waterloo aimed at developing

design criteria for the construction of a fire fighter training facility. One particular criterion is that

realistic training with respect to temperature, heat release and stratification be provided in such a

facility. The purpose of this paper was to compare existing analytical heat release and upper and

lower gas temperature rise correlations and models with data from actual structures which were

instrumented and burned in collaboration with the Kitchener Fire Department. According to the

authors, the CFAST model was used ‘successfully’ to predict these conditions and will be used in

future design of such facilities.

A report by Bailey et al. [116] compares predictions by CFAST version 3.1 to data from

real scale fire tests conducted onboard ex-USS SHADWELL, the Navy’s R&D damage control

platform. The phenomenon of particular interest in this validation series was the conduction of

heat in the vertical direction through compartment ceilings and floors. As part of this work, Bailey et al. [117] compared CFAST temperature predictions on the unexposed walls of large metal

2Unless otherwise noted, percent differences are defined as (model-experiment)/experiment x100.

78
boxes, driven by steady state fires. This tested the models prediction of radiation and conduction

in both the vertical and horizontal directions. Indirectly it quantifies the quality of the conduction/convection/radiation models. The model and experiment compared well within measurement

error bounds of each. The comparison was particularly good for measurements in the fire compartment as well as for the compartment and deck directly above it, with predictions typically agreeing

with experiments within measurement uncertainty. The model under-predicted the temperatures of

the compartments and decks not directly adjacent to the fire compartment early in the tests. Most

of the error arose due to uncertainty in modeling the details of the experiment. The size of the vent

openings between decks and to the outside must be included, but these were not always known.

Cracks formed in the deck between the fire compartment and the compartment above due to the

intense fire in the room of origin, but a time dependent record was not kept. The total size of the

openings to the outside of warped doors in both compartments was not recorded. As can be seen

in figures 7 and 8 of reference [116], the steady state predictions are identical (within error bounds

of the experiment and prediction). The largest error is after ignition (uncertainty in the initial

fire) and during development of the cracks between the compartments. While this does not affect

the agreement in the room of origin, it does lead to an uncertainty of about 30 % in the adjacent

compartment.

6.2 Comparisons with Previously Published Test Data

A number of researchers have studied the level of agreement between computer fire models and

real-scale fires. These comparisons fall into two broad categories: fire reconstruction and comparison with laboratory experiments. Both categories provide a level of verification for the models

used. Fire reconstruction, although often more qualitative, provides a higher degree of confidence

for the user when the models successfully simulate real-life conditions. Comparisons with laboratory experiments, however, can yield detailed comparisons that can point out weaknesses in the

individual phenomena included in the models.

Deal [118] reviewed four computer fire models (CCFM [3], FIRST [119], FPETOOL [120]

and FAST [2] version 18.5 (the immediate predecessor to CFAST)) to ascertain the relative performance of the models in simulating fire experiments in a small room (about 12 m3

in volume)

in which the vent and fuel effects were varied. Peak fire size in the experiments ranged up to

800 kW. According to the author, all the models simulated the experimental conditions including

temperature, species generation, and vent flows ‘quite satisfactorily.’ With a variety of conditions,

including narrow and normal vent widths, plastic and wood fuels, and flashover and sub-flashover

fire temperatures, competence of the models at these room geometries was ‘demonstrated.’

NIST has studied the predictive capability of CFAST in detail for several scenarios where experimental data were available. Peacock et al. [121] compared the performance of the CFAST

model with experimental measurements for the variables presented above. Using a range of laboratory tests, they presented comparisons of peak values, average values, and overall curve shape

for a number of variables of interest to model users. A total of five different real-scale fire tests

were selected for the comparisons to represent a range of challenges for the CFAST model. Details

of the experimental measurements and procedure for model calculations are available in the original paper [121]. Typical agreement between model predictions and experimental values ranged

from about 2 % to 25 %. Careful specification of a simulation and building leakage were seen as

79
important factors in assuring an accurate prediction.

6.2.1 NIST/NRC Verification and Validation

The U.S. Nuclear Regulatory Commission performed an extensive verification and validation of

several fire models commonly used in nuclear power plant applications [9]. These models included

simple spreadsheet calculations, zone models (including CFAST), and CFD models. The results

of this study are presented in the form of relative differences between fire model predictions and

experimental data for fire modeling attributes such as plume temperature that are important to NPP

fire modeling applications. While the relative differences sometimes show agreement, they also

show both under-prediction and over-prediction in some circumstances. These relative differences

are affected by the capabilities of the models, the availability of accurate applicable experimental data, and the experimental uncertainty of these data. The two-zone models performed well

when compared with the experiments considered. Evaluation of the two-zone models showed

that the models simulated the experimental results within experimental uncertainty for most of the

parameters of interest The reason for this may be that the relatively simple experimental configurations selected for this study conform well to the simple two-layer assumption that is the basis of

these models. However, users must remain cautious when applying these models to more complex

scenarios, or when predicting certain phenomena, like heat fluxes. The results and comparisons

included the the NRC study are included in the CFAST Software Development and Experimental

Evaluation Guide for the current version of CFAST [8].

6.2.2 Fire Plumes

Davis compared predictions by CFAST version 5 (and other models) for high ceiling spaces [36].

In this paper, the predictive capability of two algorithms designed to calculate plume centerline

temperature and maximum ceiling jet temperature in the presence of a hot upper layer were compared to measurements from experiments and to predictions using CFASTs ceiling jet algorithm.

The experiments included ceiling heights of 0.58 m to 22 m and heat release rates of 0.62 kW

to 33 MW. When compared to the experimental results CFASTs ceiling jet algorithm tended to

over-predict the upper layer temperature by 20 %. With proper adjustment for radiation effects in

the thermocouple measurements, some of this difference disappears. The effect of entrainment of

the upper layer gases was identified for improvement.

CFAST includes the calculation of plume centerline temperature from Davis’ work.

6.2.3 Multiple Compartments

Jones and Peacock [122] presented a limited set of comparisons between the FAST model (version

18.5) and a multi-room fire test. The experiment involved a constant fire of about 100 kW in a

three-compartment configuration of about 100 m3. They observed that the model predicted an

upper layer temperature that was too high by about 20 % with satisfactory prediction of the layer

interface position. These observations were made before the work of Pitts et al. [112] showed that

the thermocouple measurements need to be corrected for radiation effects. Convective heating and

plume entrainment were seen to limit the accuracy of the predictions. A comparison of predicted

80
and measured pressures in the rooms showed within 20 %. Since pressure is the driving force for

flow between compartments, this agreement was seen as important.

Levine and Nelson [123] used a combination of full-scale fire testing and modeling to simulate a fire in a residence. The 1987 fire in a first-floor kitchen resulted in the deaths of three

persons in an upstairs bedroom, one with a reported blood carboxyhemoglobin content of 91 %.

Considerable physical evidence remained. The fire was successfully simulated at full scale in a

fully-instrumented seven-room two-story test structure. The data collected during the test have

been used to test the predictive abilities of two multiroom computer fire models: FAST and HARVARD VI. A coherent ceiling layer flow occurred during the full-scale test and quickly carried high

concentrations of carbon monoxide to remote compartments. Such flow is not directly accounted

for in either computer code. However, both codes predicted the carbon monoxide buildup in the

room most remote from the fire. Prediction of the pre-flashover temperature rise was also ‘good’

according to the authors. Prediction of temperatures after flashover that occurred in the room of

fire origin was seen as ‘less good.’ Other predictions of conditions throughout the seven test rooms

varied from ‘good approximations’ to ‘significant deviations’ from test data. Some of these deviations are believed to be due to combustion chemistry in the not upper layer not considered in detail

in either of the two models.

6.2.4 Large Compartments

Duong [124] studied the predictions of several computer fire models (CCFM, FAST, FIRST, and

BRI [125]), comparing the models with one another and with large fires (4 MW to 36 MW) in an

aircraft hanger (60 000 m3). For the 4 MW fire size, he concluded that all the models are ‘reasonably accurate.’ At 36 MW, however, ‘none of the models did well.’ Limitations of the heat

conduction and plume entrainment algorithms were thought to account for some of the inaccuracies.

6.2.5 Prediction of Flashover

A chaotic event that can be predicted by mathematical modeling is that of flashover. Flashover is

the common term used for the transition a fire makes from a few objects pyrolyzing to full room

involvement. It is of interest to the fire service because of the danger to fire fighters and to building

designers because of life safety and the attendant impact on occupants. Several papers have looked

at the capability of CFAST to predict the conditions under which flashover can occur.

Chow [126] concluded that FAST correctly predicted the onset of flashover if the appropriate

criteria were used. The criteria were gas temperature near the ceiling, heat flux at the floor level

and flames coming out of the openings. This analysis was based on a series of compartment fires.

A paper by Luo et al. [127] presents a comparison of the results from CFAST version 3 against

a comprehensive set of data obtained from one flashover fire experiment. The experimental results

were obtained from a full-scale prototype apartment building under flashover conditions. Three

polyurethane mattresses were used as fuel. It was found that the predicted temperatures from

the CFAST fire model agreed well with the experimental results in most areas, once radiation

corrections are applied to the thermocouple data.

Collier [128] makes an attempt to quantify the fire hazards associated with a typical New

Zealand dwelling with a series of experiments. These tests, done in a three-bedroom dwelling,

81
included both non-flashover and flashover fires. The predictions by CFAST version 2 were seen

by the author as consistent with the experiments within the uncertainty of each.

Post-flashover fires in shipboard spaces have a pronounced effects on adjacent spaces due to

highly conductive boundaries. CFAST (version 3.1) predictions for the gas temperature and the

cold wall temperature were compared with shipboard fires [129]. The comparisons between the

model and experimental data show ‘conservative predictions’ according to the authors. The authors

attribute this to an overestimation of the average hot wall temperature and an underestimation of

external convective losses due to wind effects.

Finally, a comparison of CFAST with a number of simple correlations was used by Peacock

and Babrauskas [130, 131] to simulate a range of geometries and fire conditions to predict the

development of the fire up to the point of flashover. The simulations represent a range of compartment sizes and ceiling heights. Both the correlations and CFAST predictions were seen to provide

a lower bound to observed occurrence of flashover. For very small or very large compartment

openings, the differences between the correlations, experimental data, and CFAST predictions was

more pronounced.

The important test of all these prediction methods is in the comparison of the predictions with

actual fire observations. Figure 6.1 (reference [131]) presents estimates of the energy required to

achieve flashover for a range of room and vent sizes. This figure is an extension of the earlier work

of Babrauskas [132] and includes additional experimental measurements from a variety of sources,

most notably the work of Deal and Beyler [133]. For a number of the experimental observations,

values are included that were not explicitly identied as being a minimum value at flashover. In

addition, figure 6.1 includes predictions from the CFAST model (version 5).

As with some of the experimental data defining flashover as an upper layer temperature reaching 600 ◦C, many experimental measures were reported as peak values rather than minimum values

necessary to achieve flashover. Thus, ideally all the predictions should provide a lower bound for

the experimental data. Indeed, this is consistent with the graph the vast majority of the experimental observations lie above the correlations and model predictions. For a considerable range

in the ratio AT /A

√

h, the correlations of Babrauskas [132] Thomas [134], and the MQH correlation of McCaffrey et al. [135] provide similar estimates of the minimum energy required to

produce flashover. The estimates of Hagglund [136] yields somewhat higher estimates for values  ̈

of AT /A

√

h greater than 20 m−1/2.

The results from the CFAST model for this single compartment scenario provide similar results

to the experiments and the correlations for most of the range of AT /A

√

h. For small values of

AT /A

√

h, the CFAST values rise somewhat above the values from the correlations. These small

values of AT /A

√

h result from either very small compartments (small AT ) or very large openings

(large AT /A

√

h), both of which stretch the limits of the assumptions inherent in the model. For very

small compartments, radiation from the fire to the compartment surfaces becomes more important,

enhancing the conductive heat losses through the walls. However, the basic two-zone assumption

may break down as the room becomes very small. For very large openings, the calculation of

vent flow via an orifice flow coefficient approach is likely inaccurate. Indeed, for such openings,

this limitation has been observed experimentally [132]. The estimates are close to the range of

uncertainty shown by the correlations which also diverge at very small values of AT /A

√

h.

Perhaps most significant in these comparisons is that all the simple correlations provide estimates similar to the CFAST model and all the models are consistent with a wide range of experimental data. For this simple scenario, little is gained with the use of the more complex models.

82
t

Figure 6.1: Comparison of correlations, CFAST predictions, and experimental data for the prediction of flashover in a compartment fire.

For more complicated scenarios, the comparison may not be as simple.

6.3 Comparison with Documented Fire Experience

There are numerous cases of CFAST being used to adjudicate legal disputes. Since these are

discussed in courts of law, there is a great deal of scrutiny of the modeling, assumptions, and

results. Most of these simulations and comparisons are not available in the public literature. A few

of the cases which are available are discussed below. The metric for how well the model performed

is its ability to reproduce the time-line as observed by witnesses and the death of occupants or the

destruction of property as was used in evidence in legal proceedings.

As mentioned in section 6.2.3, Levine and Nelson describe the use of FAST for understanding

the deaths of two adults in a residence in Sharon, Pennsylvania in 1987 [123]. The paper compared

the evidence of the actual fire, a full scale mockup done at NIST and the results from FAST (version

18) [76] and Harvard VI [137]. The most notable shortcoming of the models was the lower than

actual temperatures in the bedrooms, caused by loss of heat through the fire barriers. This led to

the improvement in CFAST in the mid-90s to couple compartments together so that both horizontal

and vertical heat transfer can occur to adjacent compartments.

Bukowski used CFAST version 3.1 to analyze a fire in New York City [138] in 1994 which

resulted in the death of three fire fighters. The CFAST model was able to reproduce the observed

conditions and supported the theory as to how the fire began and the cause of death of the three fire

83
fighters.

Chow describes the use and comparison of CFAST simulations with a 1996 high rise building

fire in Hong Kong [139]. CFAST simulations were performed to help understand the probable

fire environment under different conditions. Three simulations were performed to study the consequences of a fire starting in the lift shaft. Smoke flow in the simulations qualitatively matched

those observed during the incident.

In the early morning hours of March 25,1990 a tragic fire took the lives of 87 persons at a

neighborhood club in the Bronx, New York [140]. The New York City Fire Department requested

the assistance of the NIST Center for Fire Research (CFR) in understanding the factors which

contributed to this high death toll and to develop a strategy that might reduce the risk of a similar

occurrence in the many similar clubs operating in the city. The simulation showed the potential for

development of untenable conditions within the club and particularly in the single exit stairway.

6.4 Comparison with Experiments Which Cover Special Situations

There are several sets of comparisons used in the development of the model or specific applications

beyond those discussed more generally above.

6.4.1 Nuclear Facilities

Floyd validated CFAST version 3.1 by comparing the modeling results with measurements from

fire tests at the Heiss-Dampf Reaktor (HDR) facility [141]. The structure was originally the containment building for a nuclear power reactor in Germany. The cylindrical structure was 20 m in

diameter and 50 m in height topped by a hemispherical dome 10 m in radius. The building was

divided into eight levels. The total volume of the building was approximately 11 000 m3. From

1984 to 1991, four fire test series were performed within the HDR facility. The T51 test series

consisted of 11 propane gas tests and three wood crib tests. To avoid permanent damage to the

test facility, a special set of test rooms were constructed, consisting of a fire room with a narrow

door, a long corridor wrapping around the reactor vessel shield wall, and a curtained area centered

beneath a maintenance hatch. The fire room walls were lined with fire brick. The doorway and

corridor walls had the same construction as the test chamber. Six gas burners were mounted in the

fire room. The fuel source was propane gas mixed with 10 % air fed at a constant rate to one of the

six burners.

In general, the comparison between CFAST and the HDR results was seen as ‘good’ by the

author, with two exceptions. The first is the over estimate of the temperature of the upper layer,

typically within about 15 % of the experimental measurements. This is common and generally

results from using too low a value for the production of soot, water (hydrogen) and carbon monoxide. The other exception consists of predictions in spaces where the zone model concept breaks

down, for example in the stairways between levels. In this case, CFAST has to treat the space

either in the filling mode (two layer approximation) or as a fully mixed zone (using the SHAFT

option). Neither is quite correct, and in order to understand the condition in such spaces in detail

(beyond the transfer of mass and energy), a more detailed CFD model must be used, for example,

84
FDS [47].

The U.S. Nuclear Regulatory Commission performed an extensive verification and validation

of several fire models commonly used in nuclear power plant applications [142]. These models

included simple spreadsheet calculations, zone models (including CFAST [9]), and CFD models.

The results of this study are presented in the form of relative differences between fire model predictions and experimental data for fire modeling attributes such as temperature or heat flux that are

important to NPP fire modeling applications. These relative differences are affected by the capabilities of the models, the availability of accurate applicable experimental data, and the experimental

uncertainty of these data. Evaluation of the two-zone models showed that the models simulated the

experimental results within experimental uncertainty for many of the parameters of interest. The

reason for this may be that the relatively simple experimental configurations selected for this study

conform well to the simple two-layer assumption that is the basis of these models.

While the relative differences sometimes show agreement for many parameters, they also show

both under-prediction and over-prediction in some circumstances, most notably when conditions

vary within a compartment or detailed local conditions are important to accurate prediction (for

example, plume temperature or heat flux near to the fire source). The results and comparisons

included the the NRC study are included in this report for the current version of CFAST.

6.4.2 Small Scale Testing

As an implementation of the zone model concept, CFAST is applicable to a wide range of scenarios. One end of this spectrum are small compartments, one to two meters on a side. Several research efforts have looked at small scale validation. There are three papers by Chow

[143, 144, 145] which examine this issue. The first is the use of an electric heater with adjustable

thermal power output was to verify temperature predictions by CFAST version 3.1. The second was

closed chamber fires studied by burning four types of organic liquids, namely ethanol, N-heptane,

and kerosene. The burning behavior of the liquids was observed, and the hot gas temperature

measured. These behaviors along with the transient variations of the temperature were then compared with those predicted by the CFAST model. Finally, in another series of experiments, three

zone models, one of which was CFAST, were evaluated experimentally using a small fire chamber.

Once again, liquid fires were chosen for having better control on the mass loss rate. The results on

the development of smoke layer and the hot gas temperature predicted by the three models were

compared with those measured experimentally. According to Chow, ‘fairly good agreement’ was

found if the input parameters were carefully chosen.

6.4.3 Unusual Geometry and Specific Algorithms

A zone model is inherently a volume calculation. There is an assumption in the derivation of

the equations that gas layers are strongly stratified. This allows for the usual interpretation that

a volume can then be thought of as a rectangular parallelepiped, which allows the developers

to express the volume in terms of a floor area and height of a compartment, saying simply that

the height times the floor area is the volume. However, there are other geometries which can be

adequately described by zone models. Tunnels, ships, and attics are the most common areas of

application which fall outside of the usual scope.

85
Railway and Vehicle Tunnels

Altinakar et al. [146] used a modified version of CFAST for predicting fire development and smoke

propagation in vehicle or railroad tunnels. The two major modifications made to the model dealt

with mixing between the upper and lower layers and friction losses along the tunnel. The model

was tested by simulating several full-scale tests carried out at memorial Tunnel Ventilation Test

Program in West Virginia, and the Offeneg Tunnel in Switzerland. His article compares simulated

values of temperature, opacity and similar sensible quantities with measured values and discusses

the limits of the applicability of zone models for simulating fire and smoke propagation in vehicle

and railroad tunnels.

Peacock et al. [147] compared times to untenable conditions determined from tests in a passenger rail car with those predicted by CFAST for the same car geometry and fire scenarios. For a

range of fire sizes and growth rates, they found agreement that averaged approximately 13 %.

Non-Uniform Compartments

In January 1996, the U.S. Navy began testing how the CFAST model would perform when tasked

with predicting shipboard fires. These conditions include mass transport through vertical vents

(representing hatches and scuttles), energy transport via conduction through decks, improvement

to the radiation transport sub-model, and geometry peculiar to combat ships. The purpose of this

study was to identify CFAST limitations and develop methods for circumnavigating these problems

[148]. A retired ship representing the forward half of a USS Los Angeles class submarine was

used during this test. Compartments in combat ships are not square in floor area, nor do they have

parallel sides.

Application of CFAST to these scenarios required a direct integration of compartment crosssectional area as a function of height to correctly interpret the layer interface position and provide

correct predictions for flow through doors and windows (vertical vents). This required user specification of the area as a function of height (ROOMA and ROOMH inputs) to provide a description

for the model to use. For most applications of CFAST, the effort required for the input outweighs

any additional precision in the calculated results gained by use of the ROOMA and ROOMH inputs

in the model.

Long Corridors

Prior to development of the corridor flow model, the implementation of flow in compartments

assumed that smoke traveled instantly from one side of a compartment to another. The work

of Bailey et al. [149] provided the basis for the corridor flow model in CFAST. According to

the author, it shows ‘good agreement’ for the delay time calculated using CFAST version 5 and

measured flow along high aspect ratio passageways.

Mechanical Ventilation

There have been two papers which have looked at the effectiveness of the mechanical ventilation

system. The first considered a fire chamber of length 4.0 m, width 3.0 m and height 2.8 m with

adjustable ventilation rates [150]. Burning tests were carried out with wood cribs and methanol to

study the preflashover stage of a compartmental fire and the effect of ventilation. The mass loss rate

86
of fuel, temperature distribution of the compartment and the air intake rate were measured. The

heat release rates of the fuel were calculated and the smoke temperature was used as a validation

parameter. A scoring system was proposed to compare the results predicted by the three models.

According to the author, CFAST does ‘particularly well,’ though there are some differences which

can be attributed to the zone model approach.

A second series of experiments by Luo [151] indicate that the CFAST model (version 3.1) generally over predicts the upper layer temperature in the burn room because the two-zone assumption

is likely to break down in the burn room. It was found that the room averaged temperatures obtained

from CFAST were in ‘good overall agreement’ with the experimental results. The discrepancies

can be attributed to the correction needed for thermocouple measurements. The CO concentration,

however, was inconsistent. CFAST tended to overestimate CO concentration when the air handling system was in operation. This was seen due to inconsistencies in what is measured (point

measurements) and predicted (global measurements).

Sprinkler Activation

A suppression algorithm [66] was incorporated into CFAST. Chow [152] evaluates the predictive

capability for a sprinkler installed in an atrium roof. There were three main points being considered: the possibility of activating the sprinkler, thermal response, and water requirement. The zone

model CFAST was used to analyze the possibility of activation of a sprinkler head. Results derived from CFAST were seen to be ‘accurate, that is, providing good agreement with experimental

measurements.’

t

2 Fires

Matsuyama conducted a series of full-scale experiments [153] using t2 fires. Fire room and corridor

smoke filling processes were measured. The size of the corridors and arrangements of smoke

curtains were varied in several patterns. Comparisons were then made between the experimental

results and those predicted by CFAST. The author concludes that while the model does a ‘good

job’ of predicting experimental results, there are systematic differences which could be reduced

with some revision to zone model formulation to include the impact of smoke curtains.

6.5 Summary

How to best quantify the comparisons between model predictions and experiments is not obvious.

The necessary and perceived level of agreement for any variable is dependent upon both the typical

use of the variable in a given simulation, the nature of the experiment, and the context of the

comparison in relation to other comparisons being made. For instance, the user may be interested

in the time it takes to reach a certain temperature in the room, but have little or no interest in peak

temperature for experiments that quickly reach a steady-state value. Insufficient experimental data

and understanding of how to compare the numerous variables in a complex fire model prevent a

complete validation of the model.

A true validation of a model would involve proper statistical treatment of all the inputs and

outputs of the model with appropriate experimental data to allow comparisons over the full range of

87
the model. Thus, the comparisons of the differences between model predictions and experimental

data discussed here are intentionally simple and vary from test to test and from variable to variable

due to the changing nature of the tests and typical use of different variables. Table 6.1 summarizes

the Validation comparisons included for the current version of the model detailed in the Software

Development and Experimental Evaluation Guide for CFAST [8].

Table 6.1: Summary of Model Comparisons

Quantity Average Median Within 90th

Differencea Differenceb Experimental Percentiled

Uncertaintyc

(%) (%) (%) (%)

HGL Temperature 6 14 52 30

HGL Depth 3 15 40 28

Plume Temperature 17 11 39 29

Ceiling Jet Temperature 16 5 70 61

Oxygen Concentration -6 18 12 32

Carbon Dioxide Concentration -16 16 21 52

Smoke Obscuratione 272/22 227/18 0/82 499/40

Pressure 43 13 77 206f

Target Flux (Total) -23 27 42 51

Target Temperature 0 18 38 34

Surface Flux (Total) 5 25 40 61

Surface Temperature 24 35 17 76

a - average difference includes both the sign and magnitude of the relative differences in order to

show any general trend to over- or under-prediction.

b - median difference is based only on the magnitude of the relative differences and ignores the

sign of the relative differences so that values with opposing signs do not cancel and make the

comparison appear closer than individual magnitudes would indicate.

c - the percentage of model predictions that are within experimental uncertainty.

d - 90 % of the model predictions are within the stated percentage of experimental values. For

reference, a difference of 100 % is a factor of 2 larger or smaller than experimental values.

e - the first number is for the closed door NIST/NRC tests and the second number if for the open

door NIST/NRC tests.

f - high magnitude of the 90th percentile value driven in large part by two tests where underprediction was approximately 2 Pa.

CFAST predictions in this validation study were consistent with numerous earlier studies,

which show that the use of the model is appropriate in a range of fire scenarios. The CFAST

model has been subjected to extensive evaluation studies by NIST and others. Although differences between the model and the experiments were evident in these studies, most differences can

be explained by limitations of the model as well as of the experiments. Like all predictive models,

the best predictions come with a clear understanding of the limitations of the model and the inputs

provided to perform the calculations.

88
Chapter 7

Conclusion

CFAST is a collection of data, computer programs, and documentation which are used to simulate

the important time-dependent phenomena describing the character of a compartment fire. The

major functions provided include calculation of the buoyancy-driven as well as forced transport of

energy and mass through a series of specified compartments and connections (e.g., doors, windows,

cracks, ducts), and the resulting temperatures, smoke optical densities, and gas concentrations after

accounting for heat transfer to surfaces and dilution by mixing with clean air.

CFAST is a zone model. The basic assumption of all zone fire models is that each compartment

can be divided into a small number of control volumes, each of which is internally uniform in

temperature and composition. Beyond these basic assumptions, the model typically involves a

mixture of established theory (e.g., conservation equations), empirical correlations where there are

data but no theory (e.g., flow and entrainment coefficients), and approximations where there are

neither (e.g., post-flashover combustion chemistry) or where their effect is considered secondary

compared to the “cost” of inclusion (e.g., temperature dependent material properties)..

The predictive equations are based on the fundamental laws of conservation of mass and energy.

Empirical correlations are employed to bridge gaps in existing knowledge. Since the necessary

approximations required by operational practicality result in the introduction of uncertainties in

the results, the user should understand the inherent assumptions and limitations of the programs,

and use these programs judiciously - including sensitivity analyses for the ranges of values for key

parameters in order to make estimates of these uncertainties.

As discussed in this report, the CFAST model has been subjected to extensive evaluation studies

by NIST and others. Although differences between the model and the experiments were evident

in these studies, most differences can be explained by limitations of the model as well as of the

experiments. Like all predictive models, the best predictions come with a clear understanding of

the limitations of the model and of the inputs provided to do the calculations.

CFAST has proven to be fast, robust and reliable. While the focus of the development of the

model has been whole building simulations for assessing the effect of fire on a building environment, principally to calculate threats to life safety of occupants and insults to the building structure,

it has been used for a wide variety of building and fire scenarios. The simplest use has been to ascertain the sufficiency of an air handling system to extract smoke. The most complex has been

an assessment of fire propagation in a high-rise complex. It is also widely used as the fire model

in egress calculations and is described as the basis for hazard estimates in the Simulex [154] and

Exodus [155] egress models.

89
Because of the speed of the model, it is possible to do real parameter studies of the building

environment. It is reasonable to do actual parameter studies including the tens of thousands of

variations needed for a proper hazard and risk calculation. Even in those cases where more detailed predictions are needed (e.g., smoke detector and sprinkler head siting), CFAST provides the

capability to scope the problem, in essence doing parameter studies to determine what specific

scenario should be addressed by more detailed calculations.

90
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