NIST Technical Note 1889v1
 

CFAST – Consolidated Fire 

and Smoke Transport 

(Version 7) 

Volume 1: Technical Reference Guide 

Richard D. Peacock 

Kevin B. McGrattan 

Glenn P. Forney 

Paul A. Reneke 

This publication is available free of charge from: 

http://dx.doi.org/10.6028/NIST.TN.1889v1 

NIST Technical Note 1889v1 

CFAST – Consolidated Fire 

And Smoke Transport 

(Version 7) 

Volume 1: Technical Reference Guide 

Richard D. Peacock 

Kevin B. McGrattan 

Glenn P. Forney 

Paul A. Reneke 

This publication is available free of charge from: 

http://dx.doi.org/10.6028/NIST.TN.1889v1 

October 2015 

CFAST Version 7.0.0 

GIT Revision: Gitv7-0-g4eddb8a 

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

U.S. Department of Commerce 

Penny Pritzker, Secretary 

National Institute of Standards and Technology 

Willie May, Acting Under Secretary of Commerce for Standards and Technology and Acting Director 

Preface 

This document provides the theoretical basis for the Consolidated Fire And Smoke Transport 

(CFAST) model, following the general framework set forth in the “Standard Guide for Evaluating 

the Predictive Capability of Deterministic Fire Models,” ASTM E 1355 [1]. Instructions for using 

CFAST are contained in a separate user’s guide, and model assessment information is contained in 

a separate verification and validation guide. 

iii
 
iv
 
 

Disclaimer 

The US Department of Commerce makes no warranty, expressed or implied, to users of CFAST, 

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 analysis 

performed using these tools. 

Users are warned that CFAST is intended for use only by those competent in the fields of 

fluid dynamics, thermodynamics, heat transfer, combustion, and fire science, and is intended only 

to supplement the informed judgment of the qualified user. The software package is a computer 

model that may or may not have predictive capability 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. 

Throughout this document, the mention of computer hardware or commercial software does not 

constitute endorsement by the National Institute of Standards and Technology, nor does it indicate 

that the products are necessarily those best suited for the intended purpose. 

National Institute of Standards and Technology Technical Note 1889v1 

Natl. Inst. Stand. Technol. Tech. Note 1889v1, 40 pages (October 2015) 

CODEN: NTNOEF 

This publication is available free of charge from: 

http://dx.doi.org/10.6028/NIST.TN.1889v1 

v
 
vi
 
Acknowledgments 

CFAST was originally developed by Walter Jones, formerly of NIST. 

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 (NRC) and the Department of Energy (DOE). The 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 David Stroup for their support. Support to refine the software development and 

quality assurance process for CFAST has been provided by the DOE. The assistance of Subir Sen 

and Debra Sparkman is gratefully acknowledged. 

Doug Carpenter, Combustion Sciences and Engineering, has contributed numerous corrections, 

clarifications, and updates to the guides and the model through his detailed review of the model 

and documentation. Allan Coutts, Washington Safety Management Solutions, provided insight 

into the application of fire models to nuclear safety applications and detailed review of the CFAST 

document updates for DOE. 

Colleen Wade of the Building Research Association of New Zealand (BRANZ), Fred Mowrer 

of the California Polytechnic State University, and David Sheppard of the U.S. Bureau of Alcohol, 

Tobacco and Firearms (ATF) provided useful comments on the development of CFAST version 7. 

vii
 
viii
 
Contents 

Preface iii
 

Disclaimer v
 

Acknowledgments vii
 

1 Overview 1
 

1.1 Model Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 

1.2 Model Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 

1.3 Model Developers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
 

1.4 Relevant Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
 

1.5 Governing Equations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 2
 

1.6 Input Data Required to Run the Model . . . . . . . . . . . . . . . . . . . . . . . . 4
 

1.7 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
 

1.8 Model Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 

1.9 Organization of this Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 

2 The Basic Transport Equations 7
 

2.1 Equation Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
 

2.2 Species Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 

3 The Fire Plume 11
 

3.1 Combustion Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 

3.2 Heat Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
 

3.3 Plume Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
 

3.3.1 Unobstructed Plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
 

3.3.2 Wall and Corner Plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
 

3.3.3 Weak Plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
 

3.4 Plume Temperature and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
 

4 Ventilation 17
 

4.1 Vertically-Oriented Vents (Doors and Windows) . . . . . . . . . . . . . . . . . . . 17
 

4.2 Horizontally-Oriented Vents (Ceiling Vents) . . . . . . . . . . . . . . . . . . . . . 19
 

4.3 Forced Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 

ix
 
5 Heat Transfer 21
 

5.1 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 

5.1.1 Solving the Radiation Equations . . . . . . . . . . . . . . . . . . . . . . . 23
 

5.1.2 Configuration Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 

5.1.3 Solid Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
 

5.1.4 Transmittance and Absorptance . . . . . . . . . . . . . . . . . . . . . . . 26
 

5.1.5 Radiation Heat Flux to a Target . . . . . . . . . . . . . . . . . . . . . . . 26
 

5.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
 

5.2.1 Walls and Floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
 

5.2.2 Ceiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
 

5.3 Heat Conduction within Solid Walls or Targets . . . . . . . . . . . . . . . . . . . 28
 

5.4 Coupling the Gas and Solid Phase Calculations . . . . . . . . . . . . . . . . . . . 29
 

6 Fire Protection Devices 31
 

6.1 Sprinkler and Heat Detector Activation . . . . . . . . . . . . . . . . . . . . . . . . 31
 

6.2 Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
 

6.3 Smoke Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 

6.4 Fire Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 

Nomenclature 35
 

References 37
 

x
 
List of Figures 

2.1	 Schematic of control volumes in a two-layer zone model. . . . . . . . . . . . . . . 8
 

4.1	 Flow patterns for horizontal flow through a vertical vent. . . . . . . . . . . . . . . 18
 

4.2	 Relative fraction of upper layer gases extracted via a ceiling vent . . . . . . . . . . 20
 

5.1	 Schematic diagram of a compartment with indices for computing configuration
 

factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
 

5.2	 Solid angle formed by the vectors v1, v2, and v3. . . . . . . . . . . . . . . . . . . 25
 

xi
 
xii
 
Chapter 1 

Overview 

This chapter provides a general description of the Consolidated Fire And Smoke Transport (CFAST) 

model following the general guidance put forth in ASTM E1355 [1]. 

1.1 Model Type 

CFAST is a two-zone fire model that predicts the thermal environment caused by a fire within a 

compartmented structure. Each compartment is divided into an upper and lower gas layer. The 

fire drives combustion products from the lower to the upper layer via the plume. The temperature 

within each layer is uniform, and its evolution in time is described by a set of ordinary differential 

equations derived from the fundamental laws of mass and energy conservation. The transport of 

smoke and heat from zone to zone is dictated by empirical correlations. Because the governing 

equations are relatively simple, CFAST simulations typically require a few tens of seconds of CPU 

time on typical personal computers. 

1.2 Model Version 

The first public release of CFAST was version 1.0 in June, 1990. This version was restructured 

from FAST [2] to incorporate the “lessons learned” from the zone model CCFM developed by 

Cooper and Forney [3]. 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 non-uniform heat loss to the ceiling, spot targets, and heating and burning of multiple 

objects (ignition by heat 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 current version of CFAST, version 7, was released in 2015. 

1
 
1.3 Model Developers 

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

Standards and Technology. The developers are Richard Peacock, Glenn Forney, and Paul Reneke. 

Kevin McGrattan has participated in the changes leading to CFAST, version 7. 

1.4 Relevant Publications 

The manuals for CFAST consist of the Technical Reference Guide [6], User’s Guide [7], and a 

Software and Validation Guide [8]. The Technical Reference Guide describes the underlying physical principles. The User’s Guide describes how to use the model. The Software and Validation 

Guide documents sensitivity analyses, model verification, model validation, and model limitations 

consistent with ASTM E1355 [1]. 

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 computational fluid dynamics codes such as the Fire Dynamics Simulator (FDS) 

developed by the National Institute of Standards and Technology (NIST) [10]. 

1.5 Governing Equations and Assumptions 

The governing equations of CFAST are for conservation of mass and energy within the lower and 

upper layers of connected compartments within a building. 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 a layer, the equations are a set of ordinary differential equations that 

can be solved in seconds on a typical personal computer. Other features of CFAST include: 

• Compartment geometry: CFAST is generally limited to fire scenarios where the compartment volumes are strongly stratified. The empirical correlations contained in CFAST were 

developed for relatively uncluttered, flat ceilings in compartments that can be characterized 

as “rooms” as opposed to corridors or vertical shafts. There are no hard limits on what kind 

of compartment can or cannot be modeled in CFAST. The CFAST Validation Guide indicates 

the accuracy of its predictions for compartments of various aspect ratios. 

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

rate is specified by the user for one or more fires. There is a simple sub-model to limit the 

heat release based on available oxygen. 

• Radiation from fires is modeled with a simple point source approximation. This limits the 

accuracy of the model within a few diameters of the fire. Calculation of radiative exchange 

between compartments is not modeled. 

• Mechanical ventilation is modeled by specifying volumetric flow rates into or out of compartments. The overall HVAC (heating, ventilation, air conditioning) system is not modeled. 

2
 
• Natural Ventilation and Leakage: The flow through vertical openings, like doors and windows, is modeled using the Bernoulli equation for the pressure difference between two compartments. Horizontal openings, like hatches, are treated with a single empirical correlation 

based on pressure and density differences between upper and lower compartments. Leakage 

is modeled by explicitly creating a small vertical or horizontal opening. 

• Suppression: CFAST predicts sprinkler activation based on an empirical ceiling jet correlation and activation model. A simple suppression model decreases the specified heat release 

rate. 

The technical approach and assumptions of the model have been presented in the peer-reviewed 

scientific literature [11, 12, 13] and conference proceedings [14]. CFAST has been reviewed and 

included in industry-standard handbooks such as the SFPE Handbook [15] and referenced in specific standards, including NFPA 805 [16] and NFPA 551 [17]. 

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 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. These reviewers 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. 

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 sub-models as the model continues to evolve. 

3
 
1.6 Input Data Required to Run the Model 

All of the data required to run the CFAST model reside in a single input file that the user generates. 

The file consists of the following information: 

• compartment dimensions (height, width, length) 

• lining materials of the floor, walls, and ceiling of each compartment, respectively 

• 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 

The input files are provided for the validation exercises described in the Validation Guide [8]. A 

complete description of the input parameters can be found in the CFAST User’s Guide [7]. 

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. The model user is expected to assess the appropriateness 

of default values provided by CFAST and make changes to the default values, if needed. 

1.7 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, radiative, and 

total heat fluxes) 

4
 
• fire intensity and flame height 

• flow velocities through vents and openings 

• detector and sprinkler activation times 

1.8 Model Scenarios 

While the governing transport equations are based on the fundamental conservation laws of mass 

and energy, the fire-specific algorithms within CFAST are based on empirical correlations. These 

correlations include fire plume and ceiling jet temperatures and velocities, vent flow rates, sprinkler 

activation, and so on. These sub-models were developed independently of each other under ideal 

conditions. CFAST combines these sub-models in such a way that there are no hard limits on when 

a particular sub-model is appropriate or not. The decision as to whether CFAST is appropriate for 

a given fire scenario is based primarily on the hundreds of experiments and thousands of pointto-point comparisons between CFAST and measured quantities that are included in the CFAST 

Validation Guide [8]. This document includes a list of the experiments and their important physical 

attributes such as the nature of the fire, the aspect ratio of the compartment, the ventilation rate, 

and the relative location of targets. For each quantity of interest, such as upper layer temperature or 

target heat flux, there is a calculated bias factor and standard deviation that indicates the accuracy 

of the model for the particular quantity of interest which is based on measurement uncertainty. 

Thus, the CFAST Validation Guide indicates what fire scenarios are appropriate for CFAST, and 

the degree of accuracy that can be expected for a particular type of prediction. 

In addition to what is included in the CFAST Validation Guide, validation studies have been 

performed by NIST grantees, students at universities, and 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. 

1.9 Organization of this Document 

The rest of this document is broken into several chapters as follows: 

Chapter 2 presents the basic transport equations used by CFAST derived from the conservation 

laws of mass and energy, along with the ideal gas law. 

5
 
Chapter 3 describes the algorithms and empirical correlations used to represent fires in the model, 

including combustion chemistry and heat release rate, plume entrainment, plume temperature, and plume velocity. 

Chapter 4 includes the empirical correlations used to estimate natural flow through doors and 

windows (vertically-oriented vents), floor and ceiling vents (horizontally-oriented vents), 

and mechanical ventilation systems. 

Chapter 5 documents the calculation of heat transfer including radiation exchange between fires, 

walls, gas layers, and objects within compartments, convection between gases and compartment bounding surfaces or objects within compartments, and conduction in compartment 

bounding surfaces and object within compartments. 

Chapter 6 details calculations for fire sprinkler and heat detector activation, smoke detection, fire 

suppression, and visibility through smoke. 

6
 
Chapter 2 

The Basic Transport Equations 

The equations used in CFAST take the form of an initial value problem for a system of ordinary 

differential equations. These equations are derived from the conservation laws of mass and energy 

(equivalently the first law of thermodynamics) and the ideal gas law. These equations predict the 

evolution in time of the compartment pressure, layer height, and layer temperatures due to the 

gains and losses of mass and energy. The assumption of a zone model is that properties such as 

temperature can be approximated throughout a control volume by a representative average value. 

Many formulations based upon these assumptions can be derived [18]. Though equivalent mathematically, these formulations differ in their numerical solution. 

The exchange of mass and enthalpy between zones is due to physical phenomena such as 

fire 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. The 

momentum equation is explicitly included since conditions within a control volumes are assumed 

to be uniform. Of course, included plume entrainment, ceiling jet, and vent flow correlations are 

applications of momentum principles used for specific purposes within the model. 

2.1 Equation Derivation 

This section includes the derivation of the governing transport equations. All of the mathematical 

symbols are defined in Appendix 6.4. 

It is assumed that each compartment is divided into two control volumes, a relatively hot upper 

layer and a relatively cool lower layer, as illustrated in Fig. 2.1. The gas temperature and density 

are assumed constant in each layer. The compartment as a whole is assumed to have a single value 

of pressure, P. It is also assumed that all thermodynamic parameters are constant. The specific 

heat at constant volume and at constant pressure, cv and cp, the specific1 gas constant, R, and the 

ratio of specific heats, γ, are related by γ = cp/cv and R = cp −cv. Regardless of the composition 

1The specific gas constant is the universal gas constant divided by the molar mass of the gas mixture. In CFAST, it 

is assumed that the molar mass of the mixture is that of air; thus, the specific gas constant is indeed constant. 

7
 
of the gas mixture, cp = 1012 J/(kg·K) and γ = 1.4; thus, 

γ−1 J

R = cp ≈ 289.14 (2.1) γ kg ·K 

Lower Layer

Upper Layer

Plume

Natural Vent

Components of mass and

enthalpy entering or leaving

a control volume

Layer Interface

Figure 2.1: Schematic of control volumes in a two-layer zone model. 

The set of governing equations start with the conservation of mass. The change of mass in each 

layer, i, is expressed as 

dmi = m ̇i (2.2) dt 

where  ̇mi represents the sum of all mass flow terms, such as plume mass entrainment and ventilation, entering and leaving layer i. Conservation of energy takes the form of the first law of 

thermodynamics, which 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: 

d(cvmiTi) dVi +P = h ̇i (2.3) dt dt 

The enthalpy source term, h ̇i, consists of the fire’s heat release rate, conduction losses to walls, and 

radiation exchange. The layer temperature and mass are related to the layer volume and compartment pressure via the ideal gas law: 

PVi = mi RTi (2.4) 

8
 
A system of ordinary differential equations for the compartment pressure, upper layer volume, and 

layer temperatures can be derived from these three basic principles ( Eqs. 2.2-2.4): 

dP 

dt = γ−1 V

 

h ̇l +h ̇u

 (2.5) 

dVu 

dt = 1 Pγ

 

(γ−1)h ̇u −Vu 

dP 

dt

 

(2.6) 

dTu 

dt = 1 cp mu

 

h ̇u −cp m ̇ u Tu +Vu 

dP 

dt

 

(2.7) 

dTl 

dt = 1 cp ml

 

h ̇l −cp m ̇l Tl +Vl 

dP 

dt

 

(2.8) 

Equation (2.5) is derived by summing Eq. (2.3) for the upper and lower layer, applying the equation 

of state (2.4), and noting that the time derivative of the total room volume is zero. Equation (2.6) 

is derived by rearranging Eq. (2.3) applied to the upper layer and applying the equation of state to 

replace the internal energy term. Equations (2.7) and (2.8) are derived from Eq. (2.3) applied to 

each layer, along with substitutions of the equation of state and mass conservation equation. 

As discussed in Refs. [18] and [19], Eqs. (2.5) through (2.8) are stiff, meaning that the pressure 

adjusts to changing conditions far more quickly than the other variables. 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 (see, for example, [20]). 

2.2 Species Transport 

The equation of state, (2.4), assumes that the molecular weight of the gaseous mixture throughout 

the domain is approximately that of air, 29 g/mol. However, CFAST does track the products of 

combustion and the depletion of oxygen in each zone. At the start of the simulation, the composition of each layer is set to ambient conditions. The initial temperature is specified by the user. 

The oxygen mass fraction is 23 % (21 % volume fraction) and the nitrogen mass fraction is 77 % 

(79 % volume fraction). The mass fraction of water vapor is specified by the user in terms of a relative humidity and the oxygen and nitrogen mass fractions are adjusted accordingly. All other gas 

species are initially zero. As fuel is burned, product species are produced in direct proportion to the 

rate of fuel consumption (the major products of combustion are determined from the specified fuel 

molecule and the minor species yields are specified by the user). The mass fraction of oxygen can 

limit the combustion rate as discussed in Section 3.2). Two special separate species are included 

in the model – a generic toxic species and an arbitrary trace species. Both are excluded from the 

overall mass balance, but they are generated by the fire and transported in a manner identical to the 

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 species mass divided by the layer volume is the mass concentration. 

Filters can be used in mechanical ventilation systems to remove species. The phenomenon has 

9
 
been implemented in CFAST to remove trace species and soot. It is implemented by modifying 

the source terms which describe gas flow. See Ref. [21] 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 performed by CFAST. The first is for thermal radiation as part of the overall 

heat transfer calculation, discussed in Section 5.1. The second is for visible light extinction to 

determine visibility, discussed in Section 6.2. 

10
 
Chapter 3 

The Fire Plume 

Fires in CFAST are specified by the user in terms of a time-dependent heat release rate (HRR), an 

effective fuel molecule, and the yields of the products of incomplete combustion like soot and CO. 

Fires can be specified in multiple compartments and 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, as opposed to specify, the growth and 

spread of the fire. Rather, the transient pyrolysis rates for each fire are prescribed by the user. 

Whereas this approach does not directly calculate the increased pyrolysis due to radiative feedback 

from the flame or compartment, in theory these effects could be prescribed by the user. For larger 

fires, this can be an important consideration, and the specification used should reflect the actual 

conditions as closely as possible. 

3.1 Combustion Chemistry 

The HRR of the fire is specified by the user, but it may be constrained by the availability of oxygen 

in the compartment. The combustion of a hydrocarbon fuel is described by the following singlestep reaction: 

CnCHnHOnONnNClnCl + νO2 O2 →

νCO2 CO2 + νH2O H2O + νCO CO + νS Soot + νHClHCl + νHCNHCN (3.1) 

The user specifies the composition of the fuel molecule and the yields of soot and CO, yS and yCO, 

which are related to their stoichiometric coefficients as follows: 

MF

νS = yS (3.2) MS 

MF

νCO = yCO (3.3) MCO 

11
 
Under the assumption that all of the nitrogen and chlorine in the fuel are converted to HCN and 

HCl, the other stoichiometric coefficients are: 

νCO2 = nC −(νCO +νHCN +νS) (3.4) 

nH −(νHCl +νHCN) νH2O = (3.5) 2 

νH2O +νCO −nO

νO2 = νCO2 + (3.6) 2 

νHCl = nCl (3.7) 

νHCN = nN (3.8) 

Note that the nitrogen in the air acts only as a diluent. The yields of hydrogen cyanide and hydrogen 

chloride are based solely on the composition of the fuel molecule. 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 [21], but may be useful for any 

trace amounts released by a fire. 

3.2 Heat Release Rate 

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: 

Q ̇r = χr Q ̇ (3.9) 

Q ̇ c = (1 −χr)Q ̇ (3.10) 

(3.11) 

where Q ̇ is the heat released by the fire. The parameters Q ̇r and Q ̇ c are the heat released by radiation 

and convection, respectively, and χr is the fraction of the fire’s heat release rate emitted as radiation. 

The user specifies the heat release rate, Q ̇, as the actual heat released, accounting for combustion efficiency, along with a characteristic base diameter, D, which is used in the plume temperature 

and mass entrainment correlations. The combustion efficiency, χa, is the fraction of the theoretical 

energy that is actually released during combustion [22]. χa is a function of fuel type, scale, and vitiation. For small fires, Tewarson provides measured values for specific fuels [23]. Within CFAST, 

the user also specifies a radiative fraction which takes a default value of 0.35 ; i.e., 35 % of the fire’s 

energy is released via radiation. For specific fuels, the work of Tewarson [23], McCaffrey [24], 

or Koseki [25] is available for reference. The effects of scale, fuel type, vitiation, and combustion 

efficiency are all important to the radiation released by a fire [26, 27]. The typical range for the 

radiative fraction is from about 0.05 to 0.4. The assumed and constant values for the combustion 

efficiency and radiative fraction may add uncertainty to the calculated results, so the heat release 

rate and radiative fraction should be chosen carefully to best model the scenario of interest. 

Using the specified heat release rate of the fire, Q ̇, and a user-specified the heat of combustion, 

12
 
 

 

ΔH, the model calculates the pyrolysis rate of fuel,  ̇mf: 

Q ̇ m ̇f = (3.12) ΔH 

In the event that the HRR is constrained by the availability of oxygen, it is assumed that the 

pyrolysis rate does not change. However, only part of the pyrolyzed fuel burns and the HRR 

becomes: 

 

Q ̇ = min m ̇f ΔH , m ̇ e YO2 CLOL ΔHO2 (3.13) 

where  ̇me is the entrainment rate, YO2 is the mass fraction of oxygen in the layer containing the fire, 

ΔHO2 is the heat of combustion based on oxygen consumption1, and CLOL (Lower Oxygen Limit) 

is the smoothing function ranging from 0 to 1: 

 

tanh 800(YO2 −YO2,l)−4 +1 

CLOL ≈ (3.14) 2 

The limiting oxygen mass fraction, YO2,l, is 0.15, by default. This value is not a function of temperature. 

Any unburned fuel is tracked by the model, and transported to the upper layer via entrainment 

in the fire plume or to other compartments through any user-specified vents. Unburned fuel may 

burn in the upper layer or at vents if sufficiently hot and if additional oxygen is available. 

3.3 Plume Entrainment 

A fire pumps mass and energy from the lower layer into the upper layer. The vertical flow of mass 

through a horizontal plane at height z above the base of the fire is called the mass entrainment rate, 

m ̇ e(z). The vertical flow of energy through this horizontal plane is given by Q ̇ c +m ̇ e(z) cp Tl. The 

empirical correlation for the mass entrainment rate depends on whether the plume is unobstructed, 

against a wall, or in a corner. 

3.3.1 Unobstructed Plumes 

The plume mass entrainment, m ̇ e(z), at a height z above the base of the fire is estimated using 

Heskestad’s correlation [29]: 

 1/3 Q2/3 

gρ2 2.9  ̇ ∞ Q ̇ 1/3 5/3 c m ̇ e(z) = 0.196 c (z −z0) 1 +

√

(3.15)

cp T∞ gcp ρ∞ T∞ 2/3 (z −z0)5/3

where z0 is a virtual origin defined as 

z0 

D = −1.02 +1.4 Q ̇ ∗2/5 ; Q ̇ ∗ = Q  ̇

ρ∞ cp T∞

√gDD2 (3.16) 

1The heat of combustion based on oxygen consumption is taken to be 13.1 MJ/kg, representative of typical hydrocarbon fuels [28]. 

13
 
Note that the virtual origin is defined in terms of the total heat release rate of the fire, Q ̇. Equation (3.15) is recommended above the mean flame height, L. Below the flame height, Heskestad [29] recommends the following: 

m ̇ e(z) = m ̇ e(L) z 

; 

L = −1.02 +3.7Q ̇ ∗2/5 (3.17) L D 

where 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. 

3.3.2 Wall and Corner Plumes 

If the fire is located in a corner or against a wall2, the unobstructed plume entrainment correlation, 

Eq. (3.15), is modified. For a corner, it is assumed that the fire is mirrored in each wall face, 

quadrupling the convective HRR, Q ̇ c, and doubling the base diameter, D. The entrainment rate, 

m ̇ e(z), of this hypothetical larger fire is then divided by a factor of 4. The effective entrainment 

rate is approximately 41/3/4 ≈ 0.40 of its unobstructed value. Of course, the virtual origin, z0, in 

Eq. (3.15) is affected by the change in effective diameter as well. 

√ If the fire is against a wall, the convective HRR is doubled, the diameter is multiplied by 2, 

and the result of Eq. (3.15) is divided by 2. The effective entrainment rate is 21/3/2 ≈ 0.63 of its 

unobstructed value. 

3.3.3 Weak Plumes 

In CFAST, there is a constraint on the mass entrainment rate because the plume can rise only 

so high for a given HRR. Early in a fire, the plume may not have sufficient energy to reach the 

compartment ceiling. Therefore, a limit is placed on the entrainment rate. For the plume to be able 

to penetrate the hot upper layer, the density of the gas in the plume must be less than the density 

of the gas in the upper layer. This implies that the upper layer temperature must be less than the 

plume temperature: 

Q ̇ c +m ̇ e cp Tl Tu < Tp ≈ (3.18)

m ̇ e cp 

Rearranging terms yields a limit on the mass entrainment: 

Q ̇ c m ̇ e < (3.19)

cp(Tu −Tl)

2CFAST assumes that the fire is in a corner or against a wall only when the user specifies the exact coordinates of 

the corner or wall. 

14
 
 

 

 

 

 

 

 

3.4 Plume Temperature and Velocity 

The centerline plume temperature rise, ΔT0(z), and velocity, u0(z), at a height z above the base of 

the fire is estimated using Heskestad’s correlations [29]: 

⎡ ⎤ 1/3 

T∞ Q2/3 ΔT0(z) = min⎣900 K , 9.1  ̇ c (z −z0)

−5/3

⎦ (3.20)

gc2 

p ρ2∞

 

g 

1/3 

Q ̇ 1/3 (z −z0)

−1/3 u0(z) = min u0,max , 3.4 c (3.21)

cp ρ∞ T∞

where the virtual origin, z0, is defined in Eq. (3.16). It is assumed that the temperature and velocity 

decrease following a Gaussian profile about the centerline: 

ΔT (r,z) = ΔT0(z) exp 

− r 

σΔT 

2

 

; σΔT = 0.14 

T0(z)

T∞

1/2 

(z −z0) (3.22) 

u(r,z) = u0(z) exp 

− r 

2

 

; σu ≈ 1.1σΔT (3.23)

σu 

Note that the maximum plume temperature of 900 K is suggested by Heskestad [29]. It is also 

suggested that the maximum centerline plume velocity is reached when the plume temperature rise 

is 650 K, and is a weak function of the convective HRR: 

2/5 

u0,max = 2.2 g 650Q ̇ c 1/5 (3.24) 

T 2/5 ∞ (cp ρ∞)1/5 

Note that if the velocity is expressed in units of m/s and the convective HRR in kW, then u0,max ≈

Q1/5 2  ̇ c . 

Above the interface between the hot upper layer and the cooler lower layer, the temperature 

and density of the entrained air is significantly different than that of the lower layer. To account for 

this, the plume temperature correlation [29] is modified for values of z greater than the interface 

height, zI: 

1/3 

Tu Q ̇ 2/3 �

0)

−5/3 ΔT0(z) = 9.1 c (z −z ; z > zI (3.25)

gc2 

p ρ2u 

where the modified virtual origin is given by: 

3/5 

� Tu z = zI − (zI −z0) (3.26) 0 Tl 

Equations (3.25) and (3.26) are obtained by asserting that ΔT0 is continuous across the layer interface and that Tuρu ≈ Tlρl. 

The mass entrainment correlation, Eq. (3.15), is modified the same way above the layer interface. 

15
 
16
 
Chapter 4
 

Ventilation
 

CFAST models three types of vent flow: natural flow through vertical vents (such as doors or 

windows), natural flow through horizontal vents (such as ceiling holes or hatches), and forced flow 

via mechanical ventilation. Forced flow can occur through either vertical or horizontal vents. 

4.1 Vertically-Oriented Vents (Doors and Windows) 

Natural flow through windows and doors is governed by the vertical stratification of the pressure 

difference across the opening [30]. The mass flow is calculated by dividing the opening into 

discrete horizontal segments, each of which is bounded by the top or bottom of the opening, the 

zone interface of either compartment, or the neutral plane, which is where the velocity changes 

direction. This is shown schematically in Fig. 4.1. 

Let z = b and z = t denote the height of the bottom and top of the segment, and ΔPb and ΔPt 

denote the pressure differences at these heights. Because a given segment is either completely 

above or completely below the neutral plane, the two pressure differences will have the same sign. 

The mass flow through the segment can then be computed by integrating Bernoulli’s equation from 

b to t: 

 t 

 m  ̇ = b 

C 2ρΔP(z)w dz (4.1) 

= C

 2ρw

 t 

b

 |(t −z)ΔPb +(z −b)ΔPt |

t −b dz (4.2) 

= 2 

3 

C

 2ρw(t −b)

|ΔPt |

3/2 −|ΔPb|3/2 

|ΔPt |−|ΔPb| (4.3) 

Here, C is the orifice coefficient taken to be 0.7 [31], ρ is the gas density of the upwind compartment, w is the width of the opening, and ΔP(z) is the pressure across the interface at elevation z. 

Note the use of the integral formula

 √

A +Bz dz = 2 (A +Bz)

3/2 +constant (4.4) 3B

17
 
 

 

 

 

�

Compartment 1 Compartment 2 

m ̇ u→u -

// 

Layer Interface m ̇ u→l 

Neutral Plane 

m ̇l→u 

Layer Interface ///

m ̇l→l 

Figure 4.1: Flow patterns for horizontal flow through a vertical vent. 

where A = (|t ΔPt | −b|ΔPb|)/(t −b) and B = (|ΔPt | −|ΔPb|)/(t −b). Equation 4.3 can be written: 

2 |ΔPt |+ |ΔPt ΔPb|+|ΔPb|

m ̇ = C 2ρw(t −b) (4.5) 3 |ΔPt |+ |ΔPb|

This is the way it is written in Ref. [30]. 

Figure 4.1 indicates schematically how the various mass flows through the opening are distributed. For the flow originating in the upper layer of the upstream compartment flowing into the 

upper layer of the downstream compartment,  ̇mu→u, or the flow from the lower layer to the lower 

layer,  ̇ml→l, the mass is applied directly to the downstream layer. 

The mass flow from the upper layer of the upstream compartment to the lower layer of the 

downstream,  ̇mu→l, is assumed to rise into the upper layer via a spill plume. The enthalpy flow rate 

of the plume is: 

h ̇u→l = cp Tu,1 −Tl,2 m ̇ u→l (4.6) 

Assuming that Tu,1 > Tl,2, the mass entrainment of the spill plume is given by Poreh et al. [32]: 

2/3 g ρ2 1/3 

Tl,2 l,2 h ̇1/3 

m ̇ e,p = m ̇ u→l + Cm u→l w2/3 (zI −zN) (4.7) Tu,1 cp Tl,2 

where Cm is an empirical constant equal to 0.44, w is the width of the opening, zI is the height of 

the layer interface, and zN is the height of the neutral plane. Suggested values for Cm vary between 

18
 
 

 

 

�

0.44 and 0.66, all of which were determined empirically in a number of different experimental 

configurations. 

For the mass flow leaving the lower layer of the upstream compartment and entering the upper 

layer of the downstream compartment,  ̇ml→u, the shear flow causes vortex shedding that entrains 

upper layer gas and deposits it in the lower layer. It is assumed that the incoming cold plume 

behaves like the inverse of the usual door jet between adjacent hot layers; forming a descending 

plume. The same equations are used to calculate this inverse plume as are used for the upright door 

mixing, above. 

4.2 Horizontally-Oriented Vents (Ceiling Vents) 

Cooper [33] developed a semi-empirical correlation for ceiling vents in which the volumetric flow 

rates upwards and downwards are functions of both pressure and density differences: 

V ̇

up = 0.68 Av 2 |max(ΔP,0)|/ρbot +V ̇ex (4.8) 

V ̇

down = 0.68 Av 2 |min(ΔP,0)|/ρtop +V ̇

ex (4.9) 

Here, ΔP is the lower compartment pressure minus the upper, Av is the area of the vent, and ρtop 

and ρbot are the layer densities adjacent to the vent. The term, V ̇

ex, represents the density-driven 

exchange flow; that is, when Δρ = ρtop −ρbot > 0 and |ΔP| is relatively small, there is an exchange 

of mass and enthalpy at the vent interface where hot gas from below mixes with cooler gases above: 

1/2 

gΔρA5

v 

/2 |ΔP| C2 gΔρD5 s V ̇

ex = 0.10 1 − ; |ΔP| < |ΔPflood| ≡ (4.10) 2A2 ρavg |ΔPflood| v 

Here, ρavg is the average of the densities above and below the vent, D = 2 Av/π, and the shape 

factor, Cs, is 0.754 for round and 0.942 for square openings. 

As with doors and windows, mass and enthalpy are exchanged between the layers of the upper 

and lower compartments. Take the typical case where the lower compartment has a hot gas layer 

venting into the upper compartment. Consider first the relative amounts of mass and enthalpy extracted from the layers of the lower compartment, and then consider where this mass and enthalpy 

are deposited in the upper compartment. First, the mass and enthalpy are extracted from the lower 

and upper layers of the lower compartment according to the following weighted average: 

m ̇ up = α ρu V ̇

up + (1 −α) ρl V ̇

up (4.11) 

h ̇up = α cp ρu V ̇

up Tu + (1 −α) cp ρl V ̇up Tl (4.12) 

where ρu and ρl refer to the upper and lower layers of the lower compartment. 

The weighting factor, α, indicates the relative amounts of mass extracted from the upper and 

lower layers of the lower compartment. If the hot gas layer of the lower compartment is sufficiently 

deep, all of the gas passing into the upper compartment is extracted from the upper layer. If the 

layer is not sufficiently deep, gases from the lower layer are also drawn through the vent, which 

reduces the amount of smoke that is exhausted from the compartment. This phenomenon is referred 

19
 
 

to as plug-holing. Cooper [34] suggests that the degree of plug-holing is a function of the following 

form of the Froude number: 

 −1/2 Tu −T∞ Fr = V ̇ g(H −zI)

5 (4.13) T∞

where V ̇ is the volume flow through the vent and (H −zI) is the depth of the hot gas layer. Using 

the CFD model Fire Dynamics Simulator (FDS), a correlation was developed that defines α as a 

function of Fr: −(Fr/2)2

α = e (4.14) 

Figure 4.2 displays the FDS results and correlation. 

Froude Number

0 1 2 3 4 5

Fraction of Upper Layer Mass

0

0.2

0.4

0.6

0.8

1

FDS

Best Fit

Figure 4.2: Relative fraction of upper layer gases extracted via a ceiling vent. 

The previous paragraph explains from where the gas flowing through the vent is extracted. 

Where the gas goes is more simple. If the temperature of the hot gas layer of the lower compartment 

is greater than the temperature of the lower layer of the upper compartment, then the mass and 

enthalpy are deposited into the upper layer of the upper compartment. Otherwise, the mass and 

enthalpy are deposited into the lower layer of the upper compartment. 

4.3 Forced Flow 

CFAST models mechanical ventilation in terms of user-specified volume flows at various points in 

the compartment. The model does not include duct work or fan curves. These equations are highorder, non-linear and in some cases ill-posed, which caused a great deal of difficulty in reaching a 

numerical solution. 

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 the flow of these two species. 

20
 
Chapter 5 

Heat Transfer 

This section discusses thermal radiation, convection, and conduction, the three mechanisms by 

which heat is transferred between the gas layers and the enclosing compartment walls. Hot gases 

exchange heat with solid surfaces via convection and radiation. Heat is transferred through solids 

via conduction. 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, each 

surface can 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. 

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. 

5.1 Radiation 

Radiation heat transfer is calculated between the ceiling, floor, wall layers, and fire, with the inclusion of emission and absorption by the hot gas layer [35]. The following assumptions are made: 

• Each gas layer and each wall segment is assumed to be at a uniform temperature. 

• The wall and gas layer temperatures are assumed to change slowly over the duration of the 

time step of the governing equations. 

• The fire is assumed to radiate uniformly in all directions emitting a fraction, χr, of the total 

heat 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. 

• The radiation emitted is assumed to be diffuse and gray.	 In other words, the radiant fluxes 

emitted are independent of direction and wavelength. At a solid surface, the emittance, ε, 

absorptance, α, and reflectance, ρ, are related via ε = α = 1 − ρ. In the gas phase, the 

emittance, ε, absorptance, α, and transmittance, τ, are related via ε = α = 1 −τ. 

21
 
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�� ��

�� 

�� ��

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• 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. 

The compartment lining is divided into four surfaces: the ceiling, the floor, and the wall sections above and below the layer interface. The outgoing radiant flux at surface i consists of an 

emissive power and reflectance term given by 

qout,i = σεiT 4 + (1 −εi)qin,i	 (5.1) i 

where qin,i is a weighted average of all outgoing radiant fluxes from sources such as wall segments, 

gas layers and fires. This incoming radiant flux is given by 

4 

qin,i =	 1 ∑ Ajqout, j Fj−i τ j−i +ci (5.2) Ai j=1 

which simplifies to 

4 

qin,i = ∑ qout, j Fi−j τ j−i +ci (5.3) 

j=1 

after noting that AiFi−j = AjFj−i. The term Fj−i is the configuration factor (fraction of radiant 

energy emitted by surface j that is intercepted by surface i), τ j−i is the transmittance, σ is the 

Stefan-Boltzman constant, εi is the emissivity, Ti is the temperature and ci is the radiative flux 

from gas layers and fire of surface i . 

The problem then is to determine the net radiative flux,  ̇qr,i

, at each of these surfaces. This flux 

is defined as the difference between the incoming and outgoing radiative flux1: 

q ̇r,i = qin,i −qout,i	 (5.4) 

Following Eq. (17-20) in Siegel and Howell [36], Eq. (5.4) is written as a system of four linear 

equations (after using Eqs. (5.1) and (5.3) to eliminate qout,i and qin,i

): 

q 4	 4  ̇r,i 1 −ε j − + ∑ Fi−j τ j−i q ̇r, j = σTi 

4 − ∑ Fi−j τ j−i σTj 4 −ci (5.5) εi j=1 ε j j=1 

The radiation from the gases in the upper and lower layers and the fire is included in the last term: 

2 ωi−f χr Q ̇ ci = ∑ ε j Fi−j σTj 

4 +	 (5.6) Ai 4π j=1 

where εj is the emittance (absorptance) of the layer, Fi−j is the view factor between the layer and 

solid surface, ωi−f is the normalized solid angle between the fire and wall2, and Q ̇ is the heat 

1The sign for the net radiative flux,  ̇qr,i

, is defined so that positive radiant fluxes heat up wall segments, consistent 

with how convective fluxes are defined. Note that this is opposite to the sign convention used in Siegel and Howell [36]. 

2Note that as the area of surface i shrinks to zero, ωi−f/Ai → 1/R2, yielding the classic point source radiation 

model. 

22
 
��

��

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release rate of the fire. If the solid surface, i, is the floor or the lower wall and the gas layer, j, 

is the upper layer, the view factor, Fi−j, refers to the layer interface. If the solid surface is the 

upper wall or ceiling, the view factor is 1. Eq. (5.5) is a set of linear equations (4 equations and 4 

unknowns) whose solution is discussed in the next section. 

The net radiative flux,  ̇qr,i found in Eq. (5.5), along with the convective flux discussed in Section 5.2 is used as a boundary condition for computing heat conduction within the solid walls. The 

outgoing radiative flux, qout,i is used in Section 5.1.5 for computing radiative flux to a target. It 

may be rewritten in terms of known quantities, in particular the net radiative flux found in Eq. 

(5.5), by eliminating qin,i in Eqs. 5.1 and 5.4 and is given by 

1 −εi 

qout,i = σTi 

4 + q ̇r,i (5.7) εi 

5.1.1 Solving the Radiation Equations 

Equation (5.5) is a system of linear equations of the form 

Aq = Be −c (5.8) 

where A and B are 4 × 4 matrices, q is the unknown vector of net radiative fluxes, and e is the 

vector of emission terms: 

ai j = δi j −Fi−j τi−j (1 −ε j) (5.9) 

bi j = δi j −Fi−j τi−j (5.10) 

qj = −q ̇r, j/ε j (5.11) 

e j = σTj 

4 (5.12) 

The components of the source term vector, c j, are given by Eq. (5.6). The term δi j is the usual 

Kronecker delta function: 

 1 i = j δi j = 0 i = j 

5.1.2 Configuration Factors 

The configuration factor, Fi−j, is the fraction of radiant energy emitted by surface i that is intercepted by surface j. Sixteen configuration factors are required for each compartment. Figure 5.1 

depicts the system of indices for the wall surfaces and layer interface. In general, these factors may 

be computed using 

1 cosθi cosθj Fi−j = dAj dAi (5.13) Ai Ai Aj πL2 

where L is the distance along the line of integration, θi and θj are the angles for surface i and j 

between the respective normal vectors and the line of integration, and Ai and Aj are the areas of 

the two surfaces. 

23
 
 

 

�

4 - Floor 

d - Layer Interface 

1 - Ceiling 

3 - Lower Wall 

2 - Upper Wall 

Figure 5.1: Schematic diagram of a compartment with indices for computing configuration factors. 

A more efficient procedure is to compute configuration factors between the ceiling and floor, 

F1−4, between the ceiling and layer interface, F1−d, and between the floor and layer interface, F4−d, 

using a formula for the configuration factor between parallel plates (Appendix C of Ref. [36]): 

2 (1 +X2)(1 +Y 2) 1/2 X

Fi−j = ln +X 1 +Y 2 tan−1 

√ +

πX Y 1 +X2 +Y 2 1 +Y 2 

 

Y 1 +X2 tan−1 

√

Y −X tan−1 X −Y tan−1 Y (5.14) 1 +X2 

where a and b are the rectangle dimensions, c is the separation distance, X = a/c and Y = b/c. 

Algebraic relations in terms of F1−4, F1−d and F4−d may then be used to determine the 16 required configuration factors. Two properties are used to derive these relations. First, the symmetry 

relation, 

Ai Fi−j = Aj Fj−i (5.15) 

follows from Eq. (5.13). Second, configuration factors for surfaces forming an enclosure satisfy 

4 

∑ Fi−j = 1 (5.16) 

j=1 

The configuration factor F1−4 does not change during a simulation since its value depends only 

on the compartment height and floor/ceiling area. Therefore, F1−4 need only be computed once. 

Configuration factors F1−d and F4−d depend on the layer interface height so need to be determined 

each time the radiation exchange is calculated. 

F1−4, F1−d and F4−d are determined using Eq. (5.14). Since A1 = A4 it follows that F4−1 =

F1−4. The other 14 configuration factors can be calculated using simple algebraic formulas. Since 

the floor and the ceiling are assumed to be flat rectangular surfaces, it follows that F1−1 = F4−4 = 0. 

From Eq. (5.16) and the symmetry condition, F2−1 = F2−d, it follows that 

F1−2 +F1−d = 1 (5.17) 

A1 F2−1 +F2−2 +F2−d = 2F2−1 +F2−2 = 2 F1−2 +F2−2 = 1 (5.18) A2 

24
 
 

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���

� �

� � � �

� �

� � � �

� �

� � � �

��

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Solving equations (5.17) and (5.18) for F1−2 and F2−2 gives 

F1−2 = 1 −F1−d
 

F2−2 = 1 −2F2−1
 

Similarly, 

F4−3 = 1 −F4−d
 

F3−3 = 1 −2F3−4
 

Using the above configuration factors and Eq. (5.16), it follows that 

F1−3 = 1 −F1−4 −F1−2 

F3−2 = 1 −F3−1 −F3−3 −F3−4 

F2−4 = 1 −F2−1 −F2−2 −F2−3 

5.1.3 Solid Angles 

The normalized solid angle ω in Eq. (5.6) is the view factor between a given point, say the fire, 

and a solid wall. In other words, it is the fraction of radiant energy emitted by the fire that is 

intercepted by a particular wall surface. Normalized solid angles are also used to determine the 

fraction of radiation from each wall surface that strikes a target as in Eq. (5.21). 

v3 

v2� �

v1 � �

Figure 5.2: Solid angle formed by the vectors v1, v2, and v3. 

To compute a solid angle, consider the triangle defined by the vertices v1, v2, and v3 as illustrated in Fig. 5.2. The solid angle of this triangle with respect to the origin is computed 

1 |vˆ1 ×vˆ2 · vˆ3| ω = tan−1 

2π 1 +vˆ1 · vˆ2 +vˆ1 · vˆ3 +vˆ2 · vˆ3 

where vˆ denotes the normalized vector [37]. The normalized solid angle for a rectangular wall 

surface is computed by splitting the surface into two triangles, computing the solid angle for each 

triangle and summing. 

25
 
�� 

�� 

��

5.1.4 Transmittance and Absorptance 

The transmittance is the fraction of radiant energy that will pass through a volume filled with an 

absorbing medium. It is usually expressed in the form: 

τ = e−aL	 (5.19) 

where a is the absorption coefficient and L is the path length. The absorptance is the fraction of 

radiant energy absorbed by that volume. For a gray gas, α+τ = 1. 

In general, the transmittance and absorptance are functions of wavelength. This is an important 

factor to consider for the major gaseous products, CO2 and H2O. However, soot has a continuous 

absorption spectrum that allows the transmittance and absorptance to be approximated as “gray” 

[36] across the entire spectrum. The total transmittance over a path length L through a volume of 

combustion products is taken as the product of the transmittance of the soot and major gas phase 

products: 

τ = e−asL(1 −αH2O −0.5αCO2 )	 (5.20) 

The factor of 0.5 applied to the absorptance of CO2 accounts for the overlap of the wavelength 

bands of the two gases. Tien et al. [38] suggest that the absorption coefficient for soot may be 

approximated as as = 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 (absolute) temperature. Values of k have been 

found to be about constant for a wide range of fuels [39]. 

Absorptance data for H2O and CO2 are reported in Ref. [40]. For each gas, these data are tabulated and implemented as a two-dimensional array based on temperature and gas concentration. 

The effective path length, L, is calculated between the center-points of the exchanging surfaces 

(i.e., from the center height of a fire to the center of a wall surface or between the center points of 

two compartment surfaces). Path lengths are calculated separately for the upper and lower layers 

in a compartment. 

5.1.5 Radiation Heat Flux to a Target 

Objects, often referred to as “targets,” may be added to any compartment to represent measurement 

devices or equipment that might be damaged in a fire. Targets absorb thermal radiation from the 

fire, walls and hot gas layer, but targets do not affect the fire simulation. That is, there is a one-way 

coupling between the target and the fires, 10 compartment surfaces (ceiling, floor, 4 walls–upper 

and lower) and 2 gas layers. The net radiative heat flux to a target is given by: 

nfires  ̇ 10	 2 χr Qf 

q ̇r = εt	 ∑ cos(θf)

4πR2 + ∑ τ j Fj ε j qout, j + ∑ τ j Fj ε jσTj 

4 −σTt4 (5.21) 

f=1 f j=1 j=1 

where εt is the emissivity of the target, εj is the emissivity of the wall surface, θf is the angle 

formed by the line of sight segment (between the fire f and the target) of length Rf and the normal 

direction to the target surface, Fj and Fu are view factors for the walls and upper layer, τ j and τu 

are the values of transmittance for the path between the target and the walls and upper layer, Tt is 

the temperature of the target. The term qout, j

, given by Eq. (5.7), is the outgoing radiative flux at 

the j’th wall surface. 

26
 
�� 

�� 

�� ��

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5.2 Convection 

The transfer of heat between the gas and solid surfaces is handled slightly differently at the ceiling, 

floor and walls, due to the difference in orientation and the presence of a relatively thin hot flow 

near the ceiling known as the ceiling jet. The following two sections describe how the convective 

heat transfer is done for these different surfaces. 

5.2.1 Walls and Floor 

In general, the convective heat flux to a solid surface is given by: 

q ̇c = h Tg −Ts (5.22) 

The convective heat transfer coefficient, h, is a function of the gas properties, temperature, and 

velocity. In CFAST, simple correlations for natural convection are used, since the gas velocity is 

unknown: 

h = C|Tg −Ts|

1/3 (5.23) 

where C is an empirical coefficient (1.52 for the floor and ceiling (in the absence of a ceiling jet) 

and 1.31 for the walls [41]), Tg is the average gas layer temperature adjacent to the surface, and Ts 

is the surface temperature. 

5.2.2 Ceiling 

During the early stage of a fire before a hot gas layer has formed, the convective heat transfer to the 

ceiling is governed by the temperature and velocity of the ceiling jet. Alpert’s chapter in the SFPE 

Handbook [42] presents an empirical correlation for the convective heat flux from the ceiling jet to 

a relatively cool surface: 

Q ̇ c r −1.36 

q ̇c = 1.323 f (5.24) H2 H 

where f is a friction factor estimated to be 0.03, r is the radial distance to the plume centerline, H is 

the ceiling height, and Q ̇ c is the convective fraction of the heat release rate. The average convective 

heat flux to the ceiling can be obtained by integrating this expression over the entire ceiling: 

1 2π R 0.27Q ̇ c q ̇c,avg = q ̇c r dr dθ = (LW )0.68 H0.64 (5.25) LW 0 0 

Note that the integration is carried out over a circle whose area, πR2, is taken as equal to the area 

of the ceiling, LW . 

Equation (5.25) applies to the early stage of the fire; thus, a modified heat transfer coefficient 

is used so that there is a transition from the early to later stages when a layer has formed: 

q ̇c,avg h = max , C|Tu −Ts|

1/3 (5.26) Tu −Ts 

Here, Tu is the average temperature of the upper layer and Ts is the ceiling surface temperature. 

Notice that the rightmost term is simply the correlation used for the walls and floor. 

27
 
�� �� ��

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5.3 Heat Conduction within Solid Walls or Targets 

The heat conduction equation is solved in the direction normal to solid target or wall surfaces 

using non-uniformly spaced nodes and a second-order accurate central difference scheme for the 

spatial derivatives and a semi-implicit time marching scheme. At each time step, the internal solid 

temperatures are updated in time until the net convective and radiative heat flux striking the wall 

equals with the heat flux into the solid [43]: 

 dT 

 q ̇ ≡ q ̇r +q ̇c = −k 

 (5.27) dx x=0 

where k is the thermal conductivity of the solid. This solution strategy requires a differential algebraic equation (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. Conduction is 

then coupled to the gas phase energy exchange. 

A non-uniform array of internal nodes is used to capture steep gradients in temperature near 

the surface. Define a penetration depth of 

√ xp = 2 αtend erfc−1(0.05) (5.28) 

where erfc−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 tend seconds. Eighty percent of the 

nodes are placed on the interior side of xp and the remaining 20 % are placed on the exterior side. 

The heat conduction equation normal to the solid surface is: 

∂T k ∂2T = (5.29) ∂t ρc ∂x2 

where k, ρ and c are the thermal conductivity, density and heat capacity of the target. At the 

surface, x = 0, the boundary condition is: 

dT 

q ̇ = −k (5.30) dx 

where  ̇q is the net convective and radiative heat flux. 

The 1-D heat conduction equation can be solved in either Cartesian or cylindrical coordinates. 

The solution methodology shall be presented for cylindrical coordinates: 

∂T k 1 ∂ ∂T = r (5.31) ∂t ρc r ∂r ∂r 

Dividing the cylinder into N uniformly spaced concentric control volumes, this equation can be 

written in discretized form: 

Tn+1 −Tn+1 Tn+1 −Tn+1 Δt k

Tn+1 i+1 i ri i i−1 ri−1 −Tn i i = − Δr ρc Δr ri−1/2 Δr ri−1/2 

Δt α i i −1

Tn+1 −Tn+1 Tn+1 −Tn+1 = − (5.32) i+1 i i i−1 Δr2 i −0.5 i −0.5 

28
 
 

 

 �� 

��

��

�� ��

��

where α = k/(ρc). Defining Ci and Di as 

αΔt i −1 αΔt i

Ci = ; Di = (5.33) Δr2 i −0.5 Δr2 i −0.5 

Equation (5.32) can be written: 

Tn+1 −Ci T n+1 + 1 +2

αΔt −Di T n+1 = Tn i = 1...N −1 (5.34) i−1 i i+1 i Δr2 

The boundary condition is applied at control volume N: 

αΔt Δr q ̇ N N −1

Tn+1 −Tn −(Tn+1 −Tn+1 

N N = N N−1) Δr2 k N −0.5 N −0.5 

or 

−CN TN

n

−

+

1

1 + (1 +CN )Tn+1 = TN

n +DN 

Δr

q ̇ (5.35) N k 

The internal temperature profile, Ti, is then obtained with a tri-diagonal linear solver. 

5.4 Coupling the Gas and Solid Phase Calculations 

To illustrate the method, consider a one room case with one active wall. There are four gas phase 

equations (pressure, upper layer volume, upper and lower layer temperatures) and one wall temperature equation. Implementation of the gradient matching method requires that storage be allocated 

for the temperature profiles at the current time step, t, and at the next time step, 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 t +Δt. The temperature gradient at the wall surface 

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. 

Heat transfer between connected compartments is modeled by merging the back surfaces of the 

connected ceiling and floor of the compartments or the back wall surfaces of 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. 

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 radiation 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 0.5. Once these fractions are determined, an average flux,  ̇qavg, 

is calculated using 

q ̇avg = ∑ Fi−j q ̇ j (5.36) 

walls 

where Fi j is the fraction of flux from wall i that contributes to wall j,  ̇qj is the flux striking wall j. 

29
 
30
 
 

 

 

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Chapter 6 

Fire Protection Devices 

6.1 Sprinkler and Heat Detector Activation 

The link temperature of a sprinkler or heat detector is modeled using the differential equation [44]: 

√ dTL v = Tg −TL (6.1) dt RTI 

where TL and Tg are the link and gas temperatures, v is the gas speed, and RTI (Response Time 

Index) is a measure of the sensor’s thermal inertia. The gas temperature and speed are obtained 

from the plume algorithm (Section 3.4) and the ceiling jet algorithm, described below. Rooms 

without fires do not have ceiling jets, in which case the upper layer temperature is used, along with 

a fixed speed of 0.1 m/s. This value is simply an order of magnitude estimate. The link and gas 

temperatures and the speed are functions of time; the RTI is a constant for a given detector type. 

The detector equation is solved numerically using the semi-implicit updating scheme: 

√ √ Tn+1 −Tn 1 vn vn+1 

L L Tn Tn+1 −Tn+1 = g −TL 

n + g L (6.2) Δt 2 RTI RTI 

where the superscript n denotes the value at the current time, and Δt is the time step. 

The temperature and velocity just below the ceiling is typically greater than that of the upper layer due to the presence of a ceiling jet. From the work of Alpert and Heskestad [42], the 

temperature of an unconfined ceiling jet is given by: 

6.3 ; r/H ≤ 0.2 

Q ∗2/3 Tcj(r)−T∞ = T∞  ̇

H (6.3) 

(0.225 +0.27r/H)

−4/3 ; 0.2 < r/H < 4.0 

To be consistent with the calculated plume temperature, this is adjusted so that the calculated plume 

and ceiling jet temperatures are the same for r/H ≤ 0.2: 

ΔT0(H) ; r/H ≤ 0.2 

Tcj(r)−T∞ = (6.4) 

0.182 ΔT0(H) (0.225 +0.27r/H)

−4/3 ; 0.2 < r/H < 4.0 

where ΔT0(H) is the calculated centerline plume temperature rise at the compartment ceiling from 

4/3 Eq. (3.20) and the constant 0.182 is (0.225 +0.27 · 0.2) . 

31
 
 �

 

 

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The radial velocity is given by: 

3.61 ; r/H ≤ 0.17 

Q ∗1/3 

vcj(r) = gH  ̇ (6.5) H 1.06 (r/H)

−0.69 ; 0.17 < r/H < 4.0 

where the heat release rate of the fire is contained within the non-dimensional expression: 

Q ̇ Q ̇ ∗

H = √ (6.6)

ρ∞ cp T∞ gH5/2 

To account for the presence of an upper layer, the background temperature, T∞, in Eq. (6.4) is 

replaced with the upper layer temperature, Tu. No adjustment need be made to Eq. (6.6) because 

ρ∞ T∞ is assumed constant. 

An estimate of the ceiling jet thickness, δ, where the excess temperature drops to 1/e of its 

maximum value, is given by Motevalli and Marks [42]: 

 δ r r = 0.112 1 −exp −2.24 ; 0.26 ≤ ≤ 2.0 (6.7) H H H 

If the detector or sprinkler is below the ceiling jet layer or in a compartment without a fire, the gas 

temperature is taken to be upper layer temperature with a default speed of 0.1 m/s. The basis of 

this last value is simply an order of magnitude estimate. 

If the sprinkler or detector is in a corridor of width, W , Delichatsios [42] suggests the following 

alternative to the ceiling jet correlation, Eqs. (6.4) and (6.5). For a given distance down the corridor, 

x, from the plume centerline, the excess temperature and velocity are estimated as: 

1/3 1/3 ΔTcj(x) H x W = 0.37 exp −0.16 (6.8) ΔTp W H H 

1/6 H 

vcj(x) = 0.114 H ΔTcj(x) W (6.9) 

where ΔTp is the excess plume temperature at the ceiling. This correlation is applicable once the 

plume has reached the full width of the corridor and the ceiling jet flow is parallel to the corridor 

walls so that x > W /2. For x < W /2, the normal correlations apply. 

6.2 Visibility 

The visibility calculation depends solely on the soot concentration. Soot production is specified by 

the user in terms of a yield. Soot is transported like all other gas species and its concentration is 

used in the optical density calculation. The optical density is given by the expression: 

Km ms D = (6.10) ln 10 

where ms is the mass of soot per unit volume and Km is the specific extinction coefficient. The 

default value is 8700 m2/kg based on the recommendation of Mulholland [45]. 

32
 
6.3 Smoke Detection 

CFAST does not contain an algorithm that accounts for the time delay owing to smoke transport 

from the fire to the detector location, nor to the penetration of smoke into the detector chamber 

itself. Instead, CFAST treats a smoke detector as a very sensitive heat detector with an activation 

temperature rise of 5 ◦C and RTI of 5 (m·s)1/2. The temperature rise estimate is based on a study 

performed by Bukowski and Averill [46]. The RTI is not based on experiment; it simply provides 

a slight time delay in activation. 

Users are cautioned that this model is very crude and the uncertainty in its predictions are 

substantial. Consult the CFAST Validation Guide for more details on its validation. 

6.4 Fire Suppression 

Fire suppression by water is predicted using a simple empirical model developed by Madrzykowski 

[47] and Evans [48]. After activation of the sprinkler, t > tact, the heat release rate is assumed to 

decrease exponentially: 

 ̇ −(t−tact)/τ −1.8 Q(t) = Q ̇(tact) e ; τ = 3 uw (6.11) 

where uw is the water spray density, expressed in units1 of mm/s. The product species mass production rates are reduced by the same amount as the heat release rate. 

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 the two ceiling jets should interact. When 

there is more than one fire, the detection algorithm uses the fire that results in the highest ceiling 

jet temperature in order to calculate the sprinkler link temperature. 

1Note the CFAST graphical interface might use other units, but the program does convert the user-specified value 

into mm/s. 

33
 
34
 
Nomenclature 

Note that the units associated with a given symbol are sometimes changed upon input to and output 

from the program. In particular, temperatures are typically input in degrees Celsius, converted to 

Kelvin, and then converted back to Celsius on output. Energy units involving Joules or Watts are 

typically input as kJ or kW, converted to J or W, then converted back to kJ or kW. 

A area, m2
 

Av cross-sectional area of a vent, m2
 

a absorption coefficient, m−1
 

C vent constriction (or flow) coefficient, dimensionless
 

CLOL lower oxygen limit coefficient, dimensionless
 

cp heat capacity of air at constant pressure, J/(kg·K) or kJ/(kg·K)
 

cv heat capacity of air at constant volume, J/(kg·K) or kJ/(kg·K)
 

c heat capacity of a solid, kJ/(kg·K)
 

D fire diameter, m
 

optical density, m−1
 

D∗ characteristic fire diameter, m
 

Fk−j configuration factor, surface k intercepted by surface j, dimensionless
 

fv soot volume fraction, 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, kW
 

H height of a compartment, m
 

k thermal conductivity of air, W/(m·K)
 

L length of a compartment, m
 

mean flame height, m 

characteristic length for radiation calculation, m
 

Mα molar mass of gas species α, g/mol
 

mi total mass in gas layer i, kg
 

m ̇i rate of mass addition, gas layer i, kg
 

m ̇ e plume mass entrainment rate, kg/s
 

m ̇f pyrolysis rate, kg/s
 

nα number of atoms of element α
 

P pressure at floor level of a compartment, Pa
 

Q ̇ total heat release rate of the fire, kW
 

Q ̇ c heat release rate of the fire released as convective energy, kW
 

35
 
�� q ̇	 heat flux, kW/m2 

R	 specific gas constant, J/(kg·K) 

RTI	 response time index of a sprinkler or heat detector, (m·s)1/2 

r radial distance from the fire, m 

radial coordinate of cylindrical solid, m 

S vent coefficient for vertical flow vents, dimensionless 

T∞ ambient gas temperature, K 

Ti gas temperature of layer i, K 

Tp gas temperature in the plume, K 

t time, s 

uw water spray density, m/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 

v gas velocity, m/s 

W width of a compartment, m; wall thickness, m 

w door or window width, m 

x length variable, m 

YLOL lower oxygen limit, by mass, dimensionless 

YO2 mass fraction of oxygen in a gas layer, dimensionless 

yCO carbon monoxide yield, kg/kg 

ys soot yield, kg/kg 

z height variable, m 

z ̄ mid-range height of a vertical vent segment, m 

z0 height of the virtual origin of fire, m 

α	 gas absorptance, dimensionless 

thermal diffusivity in conduction, (m2/s) 

γ	 ratio of specific heats, 1.4, dimensionless 

ΔH	 heat of combustion of the fuel, kJ/kg 

ΔHO2	 energy released per unit mass of oxygen consumed, kJ/kg 

ΔT	 temperature rise, K 

ε	 emissivity 

ν stoichiometric coefficient, dimensionless 

kinematic viscosity, m2/s 

ρ density, kg/m3 

ρ∞ ambient density of air, 1.2 kg/m3 

σ Stefan-Boltzman constant (5.67 ×10−8 W/(m2·K4) 

τ transmittance, dimensionless 

χr radiative fraction, dimensionless 

36
 
References 

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40