NOAA Technical Memorandum NOS OR&R 43

ALOHA ® 

(AREAL LOCATIONS OF HAZARDOUS

ATMOSPHERES) 5.4.4 

TECHNICAL DOCUMENTATION

Seattle, Washington

November 2013

DEPARTMENT OF COMMERCE • National Oceanic and Atmospheric Administration (NOAA)

National Ocean Service • Office of Response and Restoration
NOAA’s Office of Response and Restoration

NOAA’s Office of Response and Restoration (OR&R) is a center of expertise in preparing for, 

evaluating, and responding to threats to coastal environments, including oil and chemical spills, 

releases from hazardous waste sites, and marine debris.

To fulfill its mission of protecting and restoring NOAA trust resources, the Office of Response and 

Restoration:

•

•

•

•

Provides scientific and technical support to prepare for and respond to oil and chemical 

releases.

Determines damage to natural resources from these releases.

Protects and restores marine and coastal ecosystems, including coral reefs.

Works with communities to address critical local and regional coastal challenges.

OR&R is comprised of three divisions: Emergency Response, Assessment and Restoration, and 

Marine Debris. Collectively, the Office of Response and Restoration provides comprehensive 

solutions to environmental hazards caused by oil, chemicals, and marine debris.

For comments, questions, or copies of this document, please contact: ORR.CAMEO@NOAA.GOV

Cite as:

Jones, R., W. Lehr, D. Simecek-Beatty, R. Michael Reynolds. 2013. ALOHA® (Areal Locations of

Hazardous Atmospheres) 5.4.4: Technical Documentation. U. S. Dept. of Commerce, NOAA 

Technical Memorandum NOS OR&R 43. Seattle, WA: Emergency Response Division, NOAA. 96 

pp.
NOAA Technical Memorandum NOS OR&R 43

ALOHA® (Areal Locations of Hazardous 

Atmospheres) 5.4.4 

Technical Documentation

Robert Jones,1 William Lehr,1 Debra Simecek-Beatty,1 R. Michael Reynolds2

1 National Oceanic and Atmospheric Administration

Office of Response and Restoration

Emergency Response Division

7600 Sand Point Way

Seattle, WA 98115

2 National Oceanic and Atmospheric Administration (retired)

November 2013

Seattle, WA

U.S. DEPARTMENT OF COMMERCE

Penny Pritzker

Secretary

National Oceanic and 

Atmospheric Administration

Kathryn D. Sullivan, Ph.D.

Acting Under Secretary of Commerce for 

Oceans and Atmosphere, and Acting

Administrator

National Ocean Service

Holly A. Bamford, Ph.D.

Assistant Administrator for Ocean 

Services and Coastal Zone 

Management
DEPARTMENT OF COMMERCE

National Oceanic and Atmospheric Administration

National Ocean Service

Office of Response and Restoration

Emergency Response Division

Seattle, Washington, United States of America

NOTICE

This report has been reviewed by the Office of Response and Restoration and approved for 

publication. Such approval does not signify that the contents of this report necessarily represent the 

official position of NOAA of the government of the United States, nor does mention of trade names 

or commercial products constitute endorsement or recommendation for their use.
TABLE OF CONTENTS

Executive Summary .............................................................................................................................................. 1 

1 Introduction......................................................................................................................................................... 2

1.1 About ALOHA................................................................................................................................................................ 2

1.2 Design criteria and limitations .............................................................................................................................. 3

1.3 About this document.................................................................................................................................................. 4

2 The Integral Databases .................................................................................................................................... 6

2.1 Toxicological Data....................................................................................................................................................... 6

2.2 Flammability ................................................................................................................................................................. 7

2.3 Physical Properties..................................................................................................................................................... 7

2.4 Warnings.......................................................................................................................................................................10

2.5 Geographical data......................................................................................................................................................10

3 Source Strength Models ................................................................................................................................ 11

3.1 Background..................................................................................................................................................................11

3.2 Direct Source...............................................................................................................................................................12

3.3 Puddle Source .............................................................................................................................................................12

3.3.1 Evaporation from Non-boiling Puddles...................................................................................................12

3.3.2 Evaporation from Boiling Puddles ............................................................................................................16

3.3.3 Puddle Energy Balance...................................................................................................................................16

3.4 Chemicals released from a tank...........................................................................................................................21

3.4.1 Internal Temperature of the Tank.............................................................................................................22

3.4.2 Internal Pressure of the Tank......................................................................................................................23

3.4.3 Phase of the effluent from the Tank..........................................................................................................23

3.4.4 Release of a liquid stored below its ambient boiling point.............................................................25

3.4.5 Release of a 2-phase mixture or a liquid above its ambient boiling point................................27

3.4.6 Gas release through a hole or short pipe................................................................................................28

3.5 Gas pipeline..................................................................................................................................................................30

3.5.1 General Comments...........................................................................................................................................30

3.5.2 Theory ...................................................................................................................................................................31

3.5.3 Computational Notes.......................................................................................................................................34

4 Air Dispersion Models................................................................................................................................... 35

4.1 Introduction................................................................................................................................................................. 35

4.2 Methods for characterizing wind fields and turbulence...........................................................................36

4.2.1 Manual data entry of atmospheric conditions......................................................................................36

4.2.2 Automatic wind data entry using a SAM................................................................................................. 38

4.2.3 Wind Profile........................................................................................................................................................39
4.3 Gaussian model for neutrally buoyant gases .................................................................................................41

4.3.1 Determination of Dispersion Parameters...............................................................................................43

4.4 Heavy Gas Model for dense gases.......................................................................................................................45

4.4.1 Criteria for use of the Heavy Gas model..................................................................................................46

4.4.2 Conceptual Model.............................................................................................................................................48

4.4.3 Secondary Source in ALOHA........................................................................................................................48

4.4.4 Stably Stratified Shear Flow and Passive Diffusion Stages..............................................................51

4.5 Threat zone..................................................................................................................................................................56

4.5.1 Estimating maximum concentration........................................................................................................56

4.5.2 Drawing the threat zone ................................................................................................................................56

4.5.3 confidence contours ........................................................................................................................................57

4.6 Indoor air concentration........................................................................................................................................57

5 Models for Calculating Blast Effects from Vapor Cloud Explosions .............................................. 60

5.1 Background..................................................................................................................................................................60

5.2 Method...........................................................................................................................................................................61

6 Thermal Radiation and Flammable Area Models................................................................................ 65

6.1 Background..................................................................................................................................................................65

6.2 Levels of Concern for thermal radiation..........................................................................................................66

6.3 BLEVE-FIREBALL......................................................................................................................................................66

6.3.1 Emissivity.............................................................................................................................................................67

6.3.2 View Factor .........................................................................................................................................................68

6.3.3 Transmissivity ...................................................................................................................................................69

6.3.4 Duration................................................................................................................................................................69

6.4 Jet Fires..........................................................................................................................................................................70

6.4.1 Emissivity.............................................................................................................................................................72

6.4.2 View Factor .........................................................................................................................................................73

6.5 Pool Fires ......................................................................................................................................................................77

6.5.1 Emissivity.............................................................................................................................................................77

6.5.2 View Factor .........................................................................................................................................................78

6.5.3 Pool Dynamics....................................................................................................................................................80

6.6 Flash Fires and flammable area...........................................................................................................................81

7 Bibliography ..................................................................................................................................................... 83
Frequently-used Symbols

g acceleration of gravity (9.8 m s-2) 

Pr Prandtl number

R gas constant (8.314 J mol-1 K-1) 

Re Reynolds number

Sc Schmidt number

T Sc Turbulent Schmidt number

T Temperature (K)

t time (s)

x position in downwind direction (m)

y position in horizontal crosswind direction (m)

z position in vertical direction (m)

U wind speed (m s-1) 

EXECUTIVE SUMMARY

A description of the conceptual and mathematical models that form the basis of the ALOHA 

computer application are described. ALOHA is designed for use during accidental chemical spills to 

help spill response professionals assess the risk to human populations associated with toxic air 

hazards, thermal radiation from fires, and blast effects. 

ALOHA is designed to provide a close upper bound to the threat distances associated with chemical 

spills of a scale typical of transportation accidents, with typical threat zones in the range of 100 to 

10,000 meters. ALOHA is limited to hazards associated with chemical vapors or chemicals that 

become airborne. ALOHA includes an extensive library of chemical property data, and models to 

assess the rate at which a chemical is released from containment and vaporizes. ALOHA links 

source strength models to air dispersion models to estimate the spatial extent of airborne hazards. 

ALOHA uses a graphical interface for data entry, and display of results. Exposures to toxic chemical 

vapors, overpressure, thermal radiation, or areas where flammable gases are present are 

represented graphically and with a text summary.

Executive Summary 1 
1 INTRODUCTION

1.1 ABOUT ALOHA

ALOHA is a stand-alone software application developed for the Windows and Macintosh operating 

systems. It was developed and is supported by the Emergency Response Division1 (ERD), a division 

within the National Oceanic and Atmospheric Administration (NOAA) in collaboration with the 

Office of Emergency Management of the Environmental Protection Agency (EPA). Its primary 

purpose is to provide emergency response personnel estimates of the spatial extent of some 

common hazards associated with chemical spills. The ALOHA development team also recognizes 

that ALOHA can be an appropriate tool for training and contingency planning, but users should 

remain aware of its primary purpose in spill response. ALOHA provides estimates of the spatial 

extent of some of the hazards associated with the short-term accidental release of volatile and 

flammable chemicals. ALOHA deals specifically with human health hazards associated with inhalation of 

toxic chemical vapors, thermal radiation from chemical fires, and the effects of the pressure wave from 

vapor-cloud explosions.

Since ALOHA is limited to chemicals that become airborne, it includes models to assess the rate at 

which a chemical is released from containment and vaporizes. These “source strength” models can 

be critical components in the process of assessing hazards. ALOHA links source strength models to 

a dispersion model to estimate the spatial extent of toxic clouds, flammable vapors, and explosive 

vapor clouds. However, ALOHA does not model all combinations of source strength, scenario, and 

hazard category for combustion scenarios. The user must choose a specific combination from a 

limited selection. Table 1 shows the combination of source strength models, scenarios, and hazard 

categories allowed in ALOHA.

ALOHA uses a graphical interface for data entry and display of results. The area where there is a 

possibility of exposure to toxic vapors, a flammable atmosphere, overpressure from a vapor cloud 

explosion, or thermal radiation from a fire are represented graphically as threat zones. Threat 

zones represent the area within which the ground-level exposure exceeds the user-specified level of 

concern at some time after the beginning of a release. All points within the threat zone experience a 

transient exposure exceeding the level of concern at some time following the release; it is a record 

of the predicted peak exposure over time. In some scenarios, the user can also view the time 

dependence of the exposure at specified points. 

1 The Emergency Response Division (ERD), formerly Hazardous Materials Response Division (HAZMAT), 

supports emergency response and restoration activities for oil and hazardous chemical spills, provides 

preparedness aids for response communities, and offers training on the scientific aspects of oil and chemical 

spill response.

2 Introduction

 
Table 1. Hazard categories modeled in ALOHA. 

Scenario\Source Direct source Tank Puddle Gas Pipeline

Vapor cloud Toxic vapors Toxic vapors Toxic vapors Toxic vapors 

Vapor cloud (flash 

fire) Flammable area Flammable area Flammable area Flammable area 

Vapor cloud 

(explosion) Overpressure Overpressure Overpressure Overpressure

Pool fire NA Thermal radiation Thermal radiation NA

BLEVE (fireball) NA Thermal radiation NA NA

Jet fire NA Thermal radiation NA Thermal radiation

1.2 DESIGN CRITERIA AND LIMITATIONS

To achieve ALOHA’s function as a tool for emergency response, the following design criteria were 

given priority:

ALOHA was designed to provide an assessment of threat zones using information that is commonly 

available to responders during an emergency. The user is required to provide data on local 

atmospheric conditions, the identity of the chemical, and details about the spill scenario. To 

minimize input data requirements, an extensive database of chemical properties and geographical 

data are included in ALOHA.

ALOHA is designed to provide a close upper bound to the threat distances associated with chemical 

spills. Wherever uncertainty is unavoidable, ALOHA will err in favor of overestimating rather than 

underestimating threat distances. In some cases, ALOHA will significantly overestimate threat 

zones.

ALOHA was designed to be easy to use so that responders can use it during a spill event. ALOHA’s 

user interface is designed to minimize operator error. Navigation through the model input screens 

is designed to be intuitive and quick. User input data is checked for consistency and 

reasonableness. Results are presented graphically.

ALOHA provides methods for estimating the release and volatilization rate of chemicals for many 

common accident scenarios.

ALOHA runs quickly on small computers (PC or Macintosh) that are easily transportable and 

affordable for most users.

ALOHA is designed to predict hazards associated with spills of a scale that are typical of 

transportation accidents. Typical scales for threat zones are in the range of 102 to 105 meters, with 

durations of up to an hour.

Introduction 3 
ALOHA is limited in its ability to account for the effects associated with terrain and buildings. 

ALOHA uses a wind field that does not vary with time or horizontal position, but does vary with 

elevation. As such, it cannot resolve the steering effects from features in the terrain or buildings. 

ALOHA is applicable to cases where the wind speed is greater than 1 meter per second (at 10 

meters); it should not be used for very low wind speeds or calm conditions.

1.3 ABOUT THIS DOCUMENT

ALOHA includes data compilations, source strength models, and models to estimate hazard 

associated with airborne chemicals. This document describes the theoretical basis for these models, 

but does not describe the computer algorithms used to solve the associated numerical problems. It 

is organized into 6 chapters, each corresponding to a set of related functions performed by ALOHA.

Chapter 2 describes the databases contained within ALOHA. ALOHA includes physical, chemical, 

and toxicological properties for over 500 pure chemicals and some common chemical solutions. 

ALOHA also includes latitudes, longitudes, altitudes, and time zone data for many cities in the USA.

Chapter 3 encompasses the source strength models used to estimate the amount of material 

becoming airborne, thereby contributing to an airborne toxic or flammable cloud. ALOHA includes 

models that estimate the rate at which a chemical is released from a tank or gas pipeline, or 

evaporates from a puddle. In addition, the user can directly specify the amount of chemical 

entering the atmosphere.

Chapter 4 describes the air dispersion models that are at the core of ALOHA’s methodology for 

estimating the inhalation hazard associated with toxic airborne chemicals, and the extent of a 

flammable cloud. These air dispersion models are used to predict how the concentration of a 

pollutant, once released into the air, varies with time and position. ALOHA incorporates two semiempirical air dispersion models: the Gaussian model is used to predict the wind-dominated 

transport of pollutant clouds that are not significantly affected by gravity; the Heavy Gas model is 

used for pollutant clouds that are significantly affected by gravity due to their densities.

Chapter 5 deals with the models employed to estimate the effects associated with a vapor cloud 

explosion. In rare events, volatile chemicals released into the air have formed clouds with proper 

ratios of fuel to air, ignited, and burned sufficiently quickly to produce damaging shockwaves. The 

speed at which the combustion progresses through the cloud determines whether the combustion 

of a vapor cloud produces a shock wave with damaging overpressures extending beyond the 

bounds of the fire. ALOHA integrates air dispersion models and vapor cloud explosion models to 

estimate the extent of an area affected by a damaging overpressure.

Chapter 6 describes the models used to estimate the thermal radiation hazard associated with fires. 

Fires of sufficient size and duration can produce intense thermal radiation that poses a threat to 

human health and safety far beyond the bounds of the fire. Many chemicals found in the ALOHA 

database can be ignited and sustain a combustion when mixed with air. ALOHA can estimate the 

thermal radiation from pool fires, jet fires, and fireballs associated with BLEVEs. The user can view 

the peak thermal energy flux from a combustion process or thermal energy flux as a function of 

time. The threat zone associated with flash fires is also described in this chapter, but the thermal 

4 Introduction
radiation associated with flash fires is not explicitly modeled. ALOHA computes the area within 

which the chemical vapors might exceed the lower flammability limits.

Introduction 5 
2 THE INTEGRAL DATABASES

ALOHA includes data files with physical, chemical, and toxicological properties for hundreds of pure 

chemicals and some common chemical solutions. The chemical data files include chemicals that 

have the potential to be involved in accidental releases and generate toxic air hazards or pose the 

threat of fire or explosion. The chemicals included in the ALOHA chemical library are a subset of 

those found in CAMEO CHEMICALS, a database of hazardous chemicals compiled and maintained by 

the Emergency Response Division of NOAA and the Office of Emergency Management in EPA (see 

http://cameochemicals.noaa.gov/).

2.1 TOXICOLOGICAL DATA

ALOHA employs Levels of Concern (LOC) to address the impact of toxic air plumes, fires, and 

explosions on human populations. For inhalation hazards, ALOHA’s LOCs are concentrations of 

airborne chemicals associated with adverse health effects. Since ALOHA is used primarily for 

situations where the goal is to assess the threat posed to the general public by a chemical release, it 

includes LOCs that are specifically designed to predict how the general public will respond to a 

short-term release. In limited cases, exposure guidelines developed for worker safety are also 

compiled and presented to users as an option.

LOCs for toxic inhalation hazards are specific to the chemicals. Thresholds for inhalation toxicity 

are extracted from CAMEO Chemicals. AEGLs (Acute Exposure Guideline Levels), ERPGs 

(Emergency Response Planning Guidelines), PACs (Protective Action Criteria), and IDLH 

(Immediate Danger to Life and Health) limits are stored within data files integrated into ALOHA.

AEGLs, ERPGs, and PACs, are tiered public exposure guidelines developed for accidental chemical 

release events. (See http://orise.orau.gov/emi/scapa/ for a more complete description.) Though 

developed by different organizations, all three exposure guidelines share a similar approach in 

appraising the effects of chemical exposure in their use of levels to describe the increasing severity 

associated with increasing concentration. In addition, these exposure levels are differentiated by 

exposure times. AEGLs have been developed for a range of exposure times; ALOHA only includes 

60-minute AEGLs. ERPGs are defined for 60-minute exposures. 

IDLH (Immediate Danger to Life and Health) limits (Barsan, United States. Dept. of Health and 

Human Services., and National Institute for Occupational Safety and Health. 2010) are included in 

the list of LOCs available to users. They were developed primarily for making decisions regarding 

respirator use. In the 1980s, before public exposure guidelines were available for most common 

chemicals, the IDLH limit was used in public exposure situations. Unlike the three-tiered public 

exposure guidelines, only a single IDLH value is defined for applicable chemicals.

6 The Integral Databases
2.2 FLAMMABILITY

ALOHA includes data on chemical flammability. The lower flammability and upper flammability 

limits are included in the chemical library. These data were extracted from DIPPR (American 

Institute of Chemical Engineers). 

2.3 PHYSICAL PROPERTIES

A chemical name, CAS registry number, molecular weight, and toxicological data are included for 

every chemical in the data file. These are the minimum data required for the Direct Source option 

and the Gaussian dispersion model in ALOHA.

The other options in ALOHA, such as Source Strength modeling, Heavy Gas modeling, and modeling 

for fires and explosions require more extensive datasets. Full datasets are available for about half 

the chemicals in the data file; the additional physical property data required were extracted from 

the DIPPR data compilation, a proprietary database containing physical constants and formulas for 

temperature dependent properties (American Institute of Chemical Engineers). Data are available 

for the following properties: critical temperature, critical pressure, critical volume, freezing point, 

normal boiling point, vapor pressure, liquid density, gas density, heat of vaporization, heat of 

combustion, liquid heat capacity, and vapor heat capacity. 

Data required to model the evaporation of solutions is stored in a separate data file. ALOHA allows 

the user to model solutions within limited concentration ranges. In most cases, the upper limit is 

determined by the concentration range that is commonly available for shipment commercially. The 

lower limit was set such that the expected downwind air plume would be below toxic levels; the 

lower limit is often the concentration of the azeotropic mixture. Partial pressure, liquid density, 

liquid heat capacity, and heat of vaporization are functions of both temperature and concentration. 

Tabulated partial pressure data for a range of concentration and temperature are stored in a data 

file; ALOHA interpolates between data points using a linear interpolation in log(Pressure) and 

(1/Temperature). Formulas for liquid density (kg/m3), liquid heat capacity (J/kg K), and heat of 

vaporization (J/kg) were developed by fitting tabulated data to a polynomial equation of the form

( )

2

Value C C Temperature C MassFraction C MassFrac 1 2 3 4 =+× +× +× tion . 

Hydrochloric acid solutions with concentrations between 20% by mass and 42% by mass can be 

modeled in ALOHA. Vapor pressure tables were extracted from Perry’s Chemical Engineer’s 

Handbook, 6th edition (Perry, Green, and Maloney 1984). The coefficients for the heat capacity and 

density formulas were based on a least squares polynomial fit to data found in Perry’s Chemical 

Engineer’s Handbook, 6th edition (Perry, Green, and Maloney 1984). The coefficients for the heat of 

vaporization were derived from heat of formation data found in the NBS Technical Note 270-1 

(Wagman and Rossini 1965). 

The Integral Databases 7 
Table 2. Coefficients for polynomials describing physical properties of hydrochloric acid.

C1 C2 C3 C4

Density (kg/m3) 1152.8 -0.5 502.0 0

Heat capacity (J/kg K) 2470.8 4.0 -3390.8 0

Heat of vaporization (J/kg) 2023600 0 -476800 -1776900

Ammonia solutions with concentrations less or equal to 30% by mass can be modeled in ALOHA. 

Vapor pressure tables were extracted from Perry’s Chemical Engineer’s Handbook, 6th edition 

(Perry, Green, and Maloney 1984). The coefficients for the heat capacity and density formulae were 

based on a least squares polynomial fit to data found in Perry’s Chemical Engineer’s Handbook, 6th

edition (Perry, Green, and Maloney 1984). The coefficients for the heat of vaporization were 

derived from heat of formation data found in the NBS Technical Note 270-1 (Wagman and Rossini 

1965). 

Table 3. Coefficients for polynomials describing physical properties of ammonia solutions.

C1 C2 C3 C4

Density (kg/m3) 1101.7 -0.4 -315.4 0

Heat capacity (J/kg K) 2625.8 5.3 -26.9 0

Heat of vaporization (J/kg) 2003900 0 -241700 -715500

Nitric acid solutions with concentrations between 69% by mass and 100% by mass can be modeled 

in ALOHA. Vapor pressure tables were extracted from Perry’s Chemical Engineer’s Handbook, 6th

edition (Perry, Green, and Maloney 1984). The coefficients for the heat capacity formulae were 

based on a polynomial fit to the data found in the Kirk-Othmer Encyclopedia of Chemical 

Technology, 4th edition (Kirk et al. 1991). The coefficients for the density formulae were based on a 

least squares polynomial fit to data found in Perry’s Chemical Engineer’s Handbook, 6th edition 

(Perry, Green, and Maloney 1984). The coefficients for the heat of vaporization were derived from 

heat of formation data found in the NBS Technical Note 270-1 (Wagman and Rossini 1965). 

Table 4. Coefficients for polynomials describing physical properties of nitric acid.

C1 C2 C3 C4

Density (kg/m3) 1672 -1.7 331.2 0

Heat capacity (J/kg K) 4016 0 -2260 0

Heat of vaporization (J/kg) 1181100 0 -41300 -520200

8 The Integral Databases
Hydrofluoric acid solutions with concentrations between 37% by mass and 70% by mass can be 

modeled in ALOHA. Though solutions with concentrations exceeding 70% are commercially 

available, ALOHA does not model these due to the complexity of the chemistry. Vapor pressure 

tables were extracted from Kirk-Othmer Encyclopedia of Chemical Technology, 3rd edition (Kirk et 

al. 1978). The coefficients for the heat capacity formulas were based on a linear fit to data in the 

Kirk-Othmer Encyclopedia of Chemical Technology, 4th edition (Kirk et al. 1991). The coefficients 

for the density formulas were based on a least squares polynomial fit to data found in Perry’s 

Chemical Engineer’s Handbook, 6th edition (Perry, Green, and Maloney 1984). The coefficients for 

the heat of vaporization were derived from heat of formation data found in the NBS Technical Note 

270-1 (Wagman and Rossini 1965). 

Table 5. Coefficients for polynomials describing physical properties of hydrofluoric acid.

C1 C2 C3 C4

Density (kg/m3) 477.5 1.67 609.3 -296.0

Heat capacity (J/kg K) 4148 0 -1942.6 0

Heat of vaporization (J/kg) 2424600 0 -324200 0

Oleum is a mixture of sulfur trioxide and anhydrous sulfuric acid. Mixtures with concentrations of 

free sulfur trioxide between 4% by weight and 65% by weight can be modeled in ALOHA.

Coefficients for the density formula were based on 3 values, 20%, 25%, and 30%, reported in the 

Enviro TIPS manual (Environmental Protection Service. 1984) and a graph in Ullmann’s 

Encyclopedia of Industrial Chemistry (Ullmann 2000) which showed the dependence on 

temperature. Coefficients for the heat capacity formula were based on a graph in the Kirk-Othmer 

Encyclopedia of Chemical Technology, 4th edition (Kirk et al. 1991). Heat of Vaporization is an 

average of the values presented by Brand and Rutherford (Brand and Rutherford 1952); there 

appears to be no discernible dependence on concentration. The Vapor pressures for up to 34% can 

be found in Brand and Rutherford, and are complemented by measurement by Schrage (Schrage 

1991). 

Table 6. Coefficients for polynomials describing physical properties of oleum.

C1 C2 C3 C4

Density (kg/m3) 2162 -1.1 360 0

Heat capacity (J/kg K) 1333 0 485.7 0

Heat of vaporization (J/kg) 712756 0 0 0

The Integral Databases 9 
2.4 WARNINGS

In addition to data required to model release and downwind dispersion, ALOHA also includes data 

used to trigger warning statements while running the model. One option in ALOHA allows the user 

to model releases on water. Since ALOHA does not have the methods for modeling soluble chemical 

spills on water, the user is given a warning that the chemical is soluble. Information about which 

chemicals in ALOHA have solubilities exceeding 50 kg/m3 was based on data extracted from the 

CHEMWATCH database (Chemwatch). 

Some chemicals are sufficiently reactive that they undergo chemical reactions when in contact with 

air or moisture. The ALOHA development team examined every chemical in ALOHA to determine 

which would likely react sufficiently to have significant impacts on the accuracy of the dispersion 

models. Information about these reactive chemicals is included in a data file. 

2.5 GEOGRAPHICAL DATA

ALOHA includes latitudes, longitudes, altitudes, and time zone data for many cities in the USA. 

These data are used to compute solar radiation and local ambient pressure.

10 The Integral Databases
3 SOURCE STRENGTH MODELS

3.1 BACKGROUND

The rate at which a chemical becomes airborne is critical to the size and duration of a toxic or 

flammable cloud. ALOHA employs a variety of models to estimate the rate at which a chemical is 

released from confinement and enters the atmosphere; these are referred to as source strength 

models. ALOHA can predict source strength for four general classes of chemical releases, or 

sources:

•

•

•

•

Direct. An instantaneous or continuous release of chemical vapors into the air from a single 

point. This is the only option that allows for an elevated release.

Puddle. A puddle of constant area, containing either a non-boiling or boiling liquid.

Tank. A cylindrical or a spherical tank at ground level with a single hole or leaking valve. 

The tank may contain a liquid, pressurized gas, or gas liquefied under pressure. Tank 

contents may escape directly into the atmosphere or first form a spreading evaporating 

pool.

Gas pipeline. A pressurized pipe containing gas, either connected to a very large reservoir 

or unconnected to any storage vessel.

ALOHA limits the duration of any source to one hour, and the shortest source duration allowed in 

ALOHA is one minute (this is termed an instantaneous release in ALOHA). In most cases, source

strength changes continuously over the duration of the release. ALOHA approximates continuously

variable releases with a series of very short steady-state releases. Up to 150 time-steps are used to

model time-varying releases. These timesteps are reduced to five or fewer steady-state timesteps 

which are linked to the dispersion models.

Figure 1. An example of ALOHA’s timestep averaging scheme. ALOHA originally estimated source 

strength for this release in a series of 100 timesteps; these were averaged to four timesteps for use 

in predicting dispersion.

Source Strength Models 11
3.2 DIRECT SOURCE

A direct source option allows the user to directly specify the amount of chemical vapors introduced 

into the air from a point in space. The user can specify an instantaneous release, or a steady-state 

release of finite duration. This option can be used with gases that are denser than air and are 

affected by gravity, or gases that behave as neutrally buoyant. ALOHA allows for a release above 

ground level for gases that behave as neutrally buoyant.

3.3 PUDDLE SOURCE

Using the Puddle Option, users can model the evaporation of volatile chemicals from a puddle of 

fixed area. A limited number of mixtures containing a toxic chemical mixed with water or a nonvolatile solvent can be modeled. In these cases the evaporation and transport of only the toxic 

component is modeled. ALOHA uses one of two methods for finding the evaporation rate 

depending upon whether or not the puddle is sufficiently close to its boiling point. Brighton's 

formulation (Brighton 1985) is used when the average puddle temperature is sufficiently below its 

boiling point, and an energy balance method is used when the puddle approaches its boiling point.

The volatilization rate of non-boiling puddles is governed primarily by the wind speed, area of the 

puddle, and vapor pressure of the chemical,2 or partial pressure in cases involving mixtures. 

ALOHA accounts for changes in temperature and composition, in cases involving mixtures. 

Temperature is held at the boiling point for boiling puddles, and the volatilization rate is defined by 

the energy balance. Turbulent diffusion is expected to be the primary mechanism transporting 

vaporized material away from the puddle. Temperature and composition are assumed to be 

spatially uniform throughout the puddle. As evaporation removes material, the puddle is assumed 

to grow thinner, but its radius is assumed to be constantALOHA accommodates spills of volatile 

chemicals on a variety of solid surfaces and on bodies of water. ALOHA assumes that the spilled 

liquid does not penetrate into the soil, sink, or dissolve in water.

The methods used to compute evaporation rates from puddles of fixed area are also used with the 

Tank Option to estimate evaporation rates from expanding puddles from liquid leaking from a 

Tank. The modifications to accommodate spreading puddles are described in section 3.4.

3.3.1 EVAPORATION FROM NON-BOILING PUDDLES

When the average puddle temperature is below its ambient boiling point,3 ALOHA uses a model by 

Brighton (Brighton 1985) to predict the evaporation rate. The model assumes that the layer of air 

2 The vapor pressure is strongly dependent upon the puddle skin temperature, which can be significantly 

lower than the temperature of the underlying liquid. Kawamura and Mackay (Kawamura et al. 1985) 

measured skin temperatures that were 5 to 6°C lower than bulk temperatures in the same puddles. ALOHA 

approximates the temperature of the puddle as spatially uniform set to the spatially averaged puddle 

temperature; this can cause ALOHA to overestimate evaporation rates. 3 ALOHA defines the upper temperature limit for Brighton’s model when the correction factor for highly 

volatile liquids reaches 4.

12 Source Strength Models

 
in contact with the surface of the puddle holds vapors in equilibrium with the liquid in the puddle. 

As a turbulent airstream passes across the pool surface, vapor molecules diffuse passively from the 

top of this vapor layer and are advected downwind.

Brighton formulated an expression for the evaporative mass flux, E xt ( ) , , in terms of the friction 

velocity of the air, U* , the chemical’s vapor-phase saturation concentration, Cs , and a 

dimensionless mass transfer coefficient, j x( ) : 

( ) * , () E x t CU j x = s . 

The mass transfer coefficient is spatially dependent on the position along the axis oriented parallel 

to the wind direction. By integrating the mass transfer coefficient over the puddle dimension,4 a 

spatially averaged mass transfer coefficient, j , is found using

( ) 0

1 DP

P

j j x dx D

⌠

⌡ = . 

As the vapor pressure of the puddle, Pv , approaches ambient air pressure, Pa , the evaporating 

vapors perturb the boundary layer and affect the turbulent diffusion above the puddle. A 

correction for highly volatile liquids is applied to the mass transfer coefficient to account for this 

behavior:

ln 1 a v

c

v a

P P j j P P

  =− −    

. 

The total evaporative flux is based on the corrected, spatially-averaged mass transfer coefficient, 

c j : 

( ) E t CU j = s c * . 

ALOHA estimates the friction velocity, U* , using a formulation from Deacon (Deacon 1973) for 

neutral conditions over the ocean described as

*

10 0.03

n

U U

z

  =     , 

where U is the user-specified wind speed at height, z . 

4 The diameter of a uniform circular puddle, DP , is used as the puddle dimension along the along-wind axis.

Source Strength Models 13

 
ALOHA uses a power-law wind speed profile to approximate the wind speed profile above the pool 

surface. The values of the power-law exponent, n , are shown in Table 7, for the six Pasquill stability 

classes (Havens and Spicer 1985). 

Table 7. Values for the power-law exponent for six Pasquill stability classes.

Stability 

Class

n

A 0.108

B 0.112

C 0.120

D 0.142

E 0.203

F 0.253

The average mass transfer coefficient is expressed in terms of dimensionless variables as

( )

( )

( )

( ) ( )

( ( ) )

2 2

2 2

2 2 2

1 1 ln 1

arctan

2 ln

1 1 1 1 ln

6

ln

1 e

1

T e 1

1

e X

e X

k j n Sc e X

e X

γ

π π π

γ π

π

Λ

Λ

Λ

Λ

    −   − +     +  

≈ +     +− +   +

+  

. 

The dimensionless distance is evaluated at the downwind edge of the puddle as

2

1 1

0

P

n T

nk D X

Sc z e

= , 

where 

k is the VonKarmon constant = 0.4 and

0 z is the surface roughness length.

Brutsaert (Brutsaert 1982) describes Λ as a measure of the ratio of the scalar roughness length of 

the puddle and the momentum roughness length of the terrain as

14 Source Strength Models
( ) ( ) 1 1 2ln 1 2 1 ( ), e

T

k

n n f Sc

n Sc

Λ= + + + − + + γ

where 

e γ is Euler’s constant (taken to be 0.577) and

T Sc is the turbulent Schmidt number set equal to 0.85 in accordance with measurements by 

Fackrell and Robins (Fackrell and Robins 1982). 

The form of f Sc ( ) depends on the roughness Reynolds number and uses 

( )

2 1

3

0

1

4

0 0

3.85 1.3 ln for 0.13: smooth, or 0.13

7.3 5 for 2 : rough,

Sc

Sc T Sc Re

k f Sc

Re Sc Sc Re T

         

 



 − + <  = 

 − >



where the roughness Reynolds number, Re0 , is defined by the ratio of the wind friction velocity, 

U* , the surface roughness length, 0 z , and the kinematic viscosity of air, ν : 

* 0

0

U z Re

v = . 

When 0 0.13 2 ≤ ≤ Re , ALOHA estimates f Sc ( ) by a straight-line interpolation between the 

maximum and minimum values of the roughness Reynolds number.

The laminar Schmidt number, Sc , is the ratio of the molecular kinematic viscosity of air to the 

molecular diffusivity of the contaminant gas in air:

c Sc κ

ν = . 

Molecular diffusivities of only a few chemicals have been measured. Unless a value has been 

entered into the chemical library by the user, ALOHA uses Graham’s Law to estimate the molecular 

diffusivity (Thibodeaux 1979)as shown by

w c w c

M

M κ κ = , 

where 

Mw is the molecular weight of water,

Mc is the molecular weight of the chemical, and

w κ is the molecular diffusivity of water vapor in air set equal to 2.39 x 10-5 m2 s-1. 

Source Strength Models 15
The treatment of evaporation from solutions follows this same method, but the solute and solvent 

are treated independently using the physical properties of each accordingly. One simplification is 

used in calculating the dimensionless mass transfer coefficient; the laminar Schmidt number is 

based on a weighted average molecular weight and a single mass transfer coefficient is used for 

both solute and solvent.

3.3.2 EVAPORATION FROM BOILING PUDDLES

ALOHA limits the upper bound of the puddle temperature to its ambient boiling point. Non-boiling 

puddles can absorb enough thermal energy from their surroundings to boil, but it would be 

extremely rare for a puddle to exceed its boiling point. Users may also set the initial puddle 

temperature at, but not above, its ambient boiling point. In cases where the calculated puddle 

temperature approaches boiling, or more precisely, exceeds the upper limit for Brighton’s nonboiling puddle model, ALOHA transitions to a boiling puddle model. The model is based on an 

assumption of a steady-state temperature fixed at the boiling point. The evaporation rate and 

associated evaporative cooling of the boiling puddle is set equal to a value which balances the 

thermal energy fluxes, thereby maintaining a constant puddle temperature at its boiling point.

ALOHA allows puddles to transition from boiling to non-boiling, or the reverse. ALOHA constantly 

compares the evaporation rate calculated with the boiling puddle model with the evaporation rate 

calculated with Brighton’s model at its temperature limit. ALOHA chooses the method that yields 

the larger evaporation rate.

3.3.3 PUDDLE ENERGY BALANCE 

The magnitude of the evaporative mass flux depends on the temperature of the puddle, which is 

assumed to be spatially uniform. Puddle temperature can increase or decrease with time 

depending on the magnitude and sign of the six energy sources considered in ALOHA:

•

•

•

•

•

•

the net shortwave solar flux into the puddle, FS ; 

the longwave radiation flux down from the atmosphere, F↓ ; 

the longwave radiation flux upward into the atmosphere, F↑ ; 

the heat exchanged with the substrate by thermal conduction, FG ; 

the sensible heat flux from the atmosphere, FH ; and

the heat lost from the puddle by evaporative cooling, FE . 

16 Source Strength Models
Energy fluxes into and out of the puddle can either increase or decrease puddle temperature, T

p , as

described by

( ) ( ) ( ) ( ) d 1

d

p

G H E

pl l p

T (t) F F Ft Ft F t Ft s t c d (t) ρ ↓ ↑ = ++ + + +    , 

where 

ρl is the liquid density,

p d is the depth of the puddle, and

pl c is the heat capacity of the liquid (J kg-1 K-1). 

3.3.3.1 Solar Radiation, FS

The solar irradiance, the flux of solar shortwave radiation at the top of the atmosphere, averages 

about 1,368 W m-2 (Lide 1993). The net transmissivity of the atmosphere is the fraction of solar 

irradiance that reaches the surface of the earth. Clouds and atmospheric particles both reflect and 

absorb part of the solar irradiance before it reaches the earth’s surface. Frouin et al. (Frouin et al. 

1989) reported that the maximum solar flux reaching the surface at continental and maritime 

locations varied from 1,090 to 1,130 W m-2, indicating that atmospheric transmissivity at these 

locations ranged from about 80 to 83 percent.

ALOHA estimates the solar radiation flux that reaches the ground at a location, given degree of 

cloudiness, time, and date (Raphael 1962) as

( ) [ ] ( ) 2 1111 1 0.0071 sin .1 0

0 otherwise,

: sin( ) > 0.1, I s s

C s F  − φ − φ = 



where 

CI is the cloudiness index (fraction of sky covered by clouds on a scale of 0 to 10) and

φS is the solar altitude (the angle of the sun above the local horizon).

This formula includes a correction for the fraction of solar radiation that is reflected back into the 

atmosphere based on an average value for the earth’s albedo. ALOHA computes solar altitude as a 

function of latitude, longitude, the time of day, and the Julian day of the year. ALOHA approximates 

the solar radiation as constant over the time of the release.

Source Strength Models 17
3.3.3.2 Longwave Radiation, F↑ and F↓

Any temperature difference that exists between the puddle and the atmosphere can produce a net 

loss or gain of energy through radiative energy transfer. Radiative (long wavelength) loss is based 

on the Stefan-Boltzman radiation law, 

4 F TP ↑ = −εσ , 

where

ε is the puddle emissivity (set equal to that of water = 0.97) and

σ is the Stefan-Boltzman constant (= 5.67 x 10-8 W m-2K-4).

ALOHA uses the following equation developed by Thibodeaux (ref. 1979) to estimate longwave 

radiation downward from the atmosphere into the puddle:

( ) 4 1 F rBT ↓ = − σ a , 

where 

r is the surface reflectivity to longwave radiation (set to that of water = 0.03), 

Ta is the air temperature, and

B is the atmospheric radiation factor, which is a function of partial pressure of water vapor, we

(Pa), and cloud cover: B a be = + w . 

Table 8. Radiation factor coefficients as a function of the cloudiness index CI . 

CI a 6 b ×10

0 0.740 44.3

1 0.750 44.3

2 0.760 44.3

3 0.770 44.2

4 0.783 40.7

5 0.793 40.5

6 0.800 39.9

7 0.810 38.4

8 0.820 35.4

9 0.840 31.0

10 0.870 26.6

18 Source Strength Models
The partial pressure of water in the atmosphere, we (Pa), can be derived from the relative humidity,

RH , and air temperature, Ta , as

5431.3 99.89 exp 21.66 100 w

a

RH

e

T

  =   −

 

. 

3.3.3.3 Heat Exchange with the Substrate, FG

The substrate, whether ground or water, can be a large source of thermal energy for a puddle. To 

estimate FG , ALOHA requires an initial value for ground or water temperature, GT . This should be 

a value for the temperature far enough below the surface that temperature is relatively unaffected 

by daily changes in solar radiation, rather than the surface temperature. When a general range for 

substrate temperature is known or suspected, choosing a relatively high value from within this 

range represents a conservative choice because it can result in a higher evaporative flux estimate. 

In most cases, inaccuracies of a few degrees in either direction in an estimate of GT are expected to 

have little effect on predicted evaporation rates.

3.3.3.3.1 SPILLS ON WATER

ALOHA can model insoluble floating liquids spilled on water that is warmer than 5°C. In cases 

where the puddle temperature is greater than the underlying water, heat flow to the water is 

neglected. This simplification acts to produce larger evaporation rates, but the effect tends to be 

short-lived because evaporative cooling often quickly lowers the temperature of the puddle to that 

of the water. 

In cases where the puddle temperature is less than the underlying water, thermal energy flows 

from the water to the puddle. Convective heat exchange can generate a large sustained heat flux. 

Modeling the convective process in the water column is beyond the scope of ALOHA, so a simple 

empirical model is used to estimate the heat transfer based on the temperature difference between 

the water and the puddle, ∆T . The model was proposed by Webber (Webber 1991), and matches 

measured results for butane on water:

2 G( ) 500 W F t T

m K

= ∆ . 

3.3.3.3.2 SPILLS ON LAND

The heat exchange between a puddle and a solid substrate is limited by the conductive heat flow 

within the substrate and is described by Fourier’s Law, 

0

( ) G G

z

T F t

z

∂

α

∂ =

= − , 

where α G is the thermal conductivity of the ground.

Source Strength Models 19
The temperature gradient is a function of time and is found by solving time dependent heat 

equation for the ground temperature, 

2

2

1

G

T T

t z

∂ ∂

κ∂ ∂ = , 

where κ G is the thermal diffusivity of the ground. The ground is represented by a slab of finite 

thickness, infinite in the horizontal directions. The thickness of the slab is based on an estimate of 

the depth over which the temperature is affected by the puddle. We assume that the ground is 

initially at a uniform temperature, the interface with the puddle is at the temperature of the puddle, 

and the other interface is at the initial ground temperature. The heat equation has a simple 

analytical error-function solution when the puddle temperature is constant; however, when the 

puddle temperature varies with time, the solution becomes more complicated, so the heat equation 

and heat flux expressions are solved numerically using a forward-stepping finite difference method.

The distance between nodes is adjusted to assure numerical stability, minimize numerical error, 

and manage the computational time. To insure numerical stability the following condition should 

be met:

dz dt ≥ 3κ G . 

The depth over which the ground temperature is affected by the puddle is approximated by finding 

the depth at which the ground temperature changes by one degree after one hour, assuming a 

constant puddle temperature. If the initial temperature differential between the puddle and ground 

is small, a minimum of 20°C is used to estimate the affected depth.

Comparison between this heat transfer model and experimental data suggests that a correction 

factor is needed for the effective thermal conductivity of soils (Briscoe and Shaw 1980). The

effective thermal conductivity appears to be about 9 times that of the measured values for soils.

Table 9. Thermal properties of solid substrates with correction factor applied (Briscoe and Shaw 

1980). 

 

Thermal

Conductivity

(Wm-1 K-1) 

Thermal

Diffusivity

(m2 s-1) 

Dry Sandy Soil 2.34 1.74 x 10-6

Concrete 8.28 3.74 x 10-6

Default Soil 8.64 4.13 x 10-6

Moist Sandy Soil 5.31 3.02 x 10-6

20 Source Strength Models
3.3.3.4 Evaporative Heat Flux

The evaporative heat flux, FE (J m-2), represents the effect of evaporative puddle cooling. Once the 

average evaporative flux, E has been estimated, ALOHA estimates FE as F LE E C = , where C L (J 

kg-1) is the specific latent heat of vaporization for the chemical at the puddle temperature.

3.3.3.5 Sensible Heat Flux

The sensible heat flux, FH , is the thermal conduction to or from the air as a result of temperature 

differences between the puddle and the air above it (Brighton 1985, 1990), and is shown as

( ) F c CU T T H a H ap = ρ p a, * − , 

where 

a ρ is the density of air (kg m-3), 

p a, c is the heat capacity of air (J kg-1 K-1), approximated as a constant 1,004 J kg-1 K-1, and

CH is the sensible heat transfer coefficient = ( )

2

3

c

thermal diffusivity j molecular diffusivity . 

3.4 CHEMICALS RELEASED FROM A TANK

ALOHA can estimate the amount of substance released into the air as a result of a rupture of a tank. 

ALOHA treats tanks containing pressurized gases, liquids at ambient pressure, gases liquefied by 

refrigeration, and gases liquefied under pressure. ALOHA deals only with tanks containing a single 

chemical. It does not deal with tanks that are pressurized with a second substance such as air or an 

inert gas. As material is released from the tank, ALOHA reevaluates the conditions within the tanks 

and can change the release rate calculation as needed.

The rate at which a chemical enters the atmosphere is limited, but not entirely determined, by the 

rate at which the chemical is released from the tank. The release rate from the tank depends upon 

the phase released (liquid, gas, or mixed phase), the driving pressure, and the nature of the rupture.

Two types of ruptures can be modeled: a hole in the tank wall, or a release through a short pipe or 

valve. Models for a release through a hole are used for both types of ruptures when a gas or nonboiling liquid are involved; they provide reasonable overestimates for short pipes. However, for 

superheated liquids and 2-phase fluids, the release rate through a hole and short pipe can differ 

significantly, so the release rate model differentiates these two rupture types.

In tanks containing liquids, the release point can occur above the level of the liquid, below the level 

of the liquid, or at the level of the liquid. The location of the rupture largely determines the 

pressure driving the release and the phase of the effluent. Top vents are assumed to be absent. 

Source Strength Models 21
Material released from a hole or short pipe can be a gas, a non-boiling liquid, a superheated liquid, 

or a mixture of liquid and gas. ALOHA predicts that a released gas enters the atmosphere 

immediately and disperses downwind, or ignites to form a jet fire. ALOHA predicts that a liquid 

released at a temperature below its boiling point forms a puddle which then evaporates to generate 

an airborne cloud. So, for non-boiling liquids, the tank release rate is linked to a puddle 

evaporation model. ALOHA does not allow the user to model a pool fire that starts with a tank 

release. ALOHA assumes that a superheated liquid (liquids released at a temperature above their 

ambient boiling point) enters the atmosphere immediately and disperses downwind, or ignites to 

form a jet fire. We expect that in many cases a superheated liquid would partially rain out to form a 

puddle; however, ALOHA employs this assumption to ensure that it can provide a reasonable 

overestimate of the actual source strength.

3.4.1 INTERNAL TEMPERATURE OF THE TANK 

The temperature at the onset of the release is defined by the user.

To account for temperature changes during the release, ALOHA considers thermal conduction 

through the tank walls that are in contact with liquid contents, and evaporative cooling due to 

liquid vaporizing within that tank to fill the void created by the escaping fluid. (Cooling due to 

adiabatic expansion of a gas within a tank is neglected.) This is shown as

d

d

T e c Hw tw

l pl l

T QL F A

t cV ρ

+ = , 

where 

FHw is the thermal energy flux across the walls, 

Atw is the area of the tank walls in contact with liquid, and

, , l pl l ρ c V are the density, heat capacity, and volume of the liquid within the tank. 

The mass evaporation rate into the head space of the tank, Qe , is simply related to the total mass 

loss from the tank, QT , and the density of the effluent fluid, ρ X , and is determined using

( ) T l g

e

Xl g

Q Q t ρ ρ

ρ ρρ

  =   −  

, 

where ρ g is the density of the gas.

The thermal energy flux across the walls of the tank that are in contact with the liquid contents 

assumes that the tank walls are constructed of 2 cm of steel and 10 cm of insulating foam.

22 Source Strength Models
3.4.2 INTERNAL PRESSURE OF THE TANK

Tanks are assumed to be unvented and initially contain a single component that is in 

thermodynamic equilibrium with its own vapors in the headspace (ALOHA does not allow for tanks 

containing other gases in the head space). For tanks containing liquid, the internal pressure is 

defined by the tank temperature and set equal to the vapor pressure of the liquid. Tanks containing 

only gas have an additional degree of freedom, so the user must define the initial temperature and 

pressure of the tank.

3.4.3 PHASE OF THE EFFLUENT FROM THE TANK

For tanks containing pressurized gases, or liquids stored below their boiling points, the phase of the

effluent is easily determined from the location of the rupture. However, for tanks containing 

superheated liquids, the determination of the phase of the effluent can be complicated when the 

rupture occurs in the head space. When a rupture is in or intersects the headspace of a tank 

containing a liquid stored above its normal boiling point, pressure is quickly released causing the 

liquid to vaporize quickly. This process can result in the formation of bubbles throughout the liquid 

(homogeneous nucleation). If gas bubbles are generated faster than they can rise and pass out of 

the liquid, the volume of the fluid can swell and fill the tank. The rupture in the head space can then 

release a 2-phase mixture of gas and liquid resulting in a large release. In most cases, ALOHA 

assumes that any rupture in the head space of a tank containing a liquid stored above its boiling 

point will swell to fill the tank with a uniform bubbly 2-phase mixture. However, there are two 

chemicals in ALOHA for which a more refined model is used to predict swelling: ammonia and 

chlorine. 

Ammonia and chlorine are treated differently than other gases liquefied under pressure. Instead of 

assuming that a rupture in the head space always results the vessel filling with a uniform 2-phase 

mixture, ALOHA uses a vapor disengagement calculation. The DIERS method (Fisher et al. 1992) 

compares the superficial vapor velocity to the drift flux to determine whether the fluid swells to fill 

the vessel. The superficial vapor velocity is defined as the volume release rate divided by the cross 

sectional area of the vessel and can be interpreted as the speed at which a slug of vapor must travel 

upward through the cylinder to match the release rate of the exit port. The drift flux of the boiling 

2-phase fluid can be interpreted as the volume flux of gas bubbles in the boiling fluid relative to the 

average flow velocity of the fluid (in ALOHA the fluid is approximated as stationary). At the surface 

of the 2-phase fluid, the drift flux can be viewed as the gas flux through the surface.

Source Strength Models 23
The drift flux is the product of two terms, the bubble rise velocity, Urise , and the reference 

dimensionless superficial vapor velocity, Ψ . These two terms are found from empirical studies 

and take different forms depending on the flow regime. According to Leung (Leung 1994) most 

single component liquids exhibit churn-turbulent (CT) flow. The measurements of the flow 

behavior of ammonia made by Fauske and Associates and transmitted to us by Karlins (Karlins 

1994) suggests that ammonia behaves as a CT fluid. Our assumption that chlorine behaves as a CT 

fluid is based solely on Leung's expectation and is not based on empirical data. For CT flow, the 

bubble rise velocity is defined as 

( ) 1

4

1

2

1.53 l g

rise

l

g

U

σρρ

ρ

  −   = , 

where 

σ is the surface tension,

ρl is the density of the liquid phase,

ρ g is the density of the gas phase, and

g is the acceleration of gravity.

The reference dimensionless superficial vapor velocity depends on the average void fraction of the 

2-phase fluid and is defined as 

0

2

1 C

α

α

Ψ = − , 

where 

α is the average void fraction and

C0 is the correlating parameter (= 1.0 for conservative estimate).

If the superficial vapor velocity is greater than the drift flux (UΨ), then the 2-phase fluid is 

predicted to swell. Since the drift flux depends on the average void fraction, it varies as the fluid 

swells and changes volume. If, at the point where the fluid fills the vessel, the drift flux is still less 

than the superficial vapor velocity, then 2-phase venting is predicted. As material is lost, the 

average void fraction of the fluid changes, and when the superficial vapor velocity becomes less 

than the drift flux, 2-phase venting is predicted to cease. Generally, small leaks lead to gas release 

and larger leaks lead to 2-phase venting.

The DIERS approach described above was developed for vertically oriented cylinders of constant 

cross section with a release point at the top of the cylinder. Grolmes and Fisher (Grolmes and 

Fisher 1994) recommend, as an acceptable approximation, modeling spherical and horizontalcylindrical vessels as vertical cylinders of equivalent volume and height. ALOHA accommodates 

vessels with ruptures below the top of the vessel by modeling these systems as vertical cylinders 

with heights equal to the height of the actual exit port and a volume equal to the volume of the 

actual vessel below the level of the exit port.

24 Source Strength Models
3.4.4 RELEASE OF A LIQUID STORED BELOW ITS AMBIENT BOILING POINT

The following method is applied to holes and short pipes below or intersecting the level of the 

liquid in a tank containing a liquid stored below its ambient boiling point. ALOHA calculates the 

release of the liquid from the tank, the puddle formation, and the evaporation rate from the puddle. 

When the liquid level drops below the bottom of the hole or pipe, ALOHA neglects any further 

release of gas from head space above the liquid. The same calculation methods are used for a hole 

in the tank wall and a short pipe.

3.4.4.1 Mass flow rate

ALOHA uses Bernoulli’s equation to compute the mass flow of liquid from the hole, 

( ) ( ) ( )

1

2 2 , Qt CA P P T = dis f h a l − ρ

where Cdis is the discharge coefficient (0.61), 

Af is the area of the hole times the fraction of the hole lying below the level of the liquid,

Pa is the ambient atmospheric pressure, and

Ph is the internal pressure in the tank at the height of the hole.

ALOHA assumes that the tank is not vented. When the liquid level is above the top of the hole, the 

internal pressure driving the flow, Ph , is the combination of the vapor pressure of the liquid and the 

hydrostatic pressure of the liquid above the hole. However, as the liquid level drops, the internal 

pressure calculated in this manner can approach the ambient air pressure causing the flow to stop. 

Commonly, air can be ingested back into the tank allowing flow to continue as a series of gushes. To 

approximate this in ALOHA, when the computed value for Ph reaches 1.01 Pa , it is held constant at 

that value (Belore and Buist 1986, Dodge, Bowels, and White 1980). 

When the liquid level intersects the hole, the internal pressure at the hole is the ambient air 

pressure plus the hydrostatic pressure of the liquid above the bottom of the hole.

Source Strength Models 25
3.4.4.2 Puddle Growth

The liquid release creates a spreading puddle that evaporates or burns. Evaporation or burning 

acts to reduce the mass of a puddle that has been formed. The mass loss is approximately 

proportional to the area of the puddle. When the tank release rate is greater than the computed 

evaporation rate, ALOHA predicts that the puddle will spread. As the puddle evaporates, ALOHA 

does not permit the puddle’s area to shrink; it allows the puddle depth to decrease. For spreading 

puddles, ALOHA uses the method proposed by Briscoe and Shaw (Briscoe and Shaw 1980) to 

compute the radius of the puddle, p

r , which is described as

2 , p

p

dr

gd dt =

where p d is the depth of the puddle.

ALOHA approximates the puddle depth as uniform and terminates the spreading when the depth 

drops to 5 mm, an arbitrarily chosen value.

3.4.4.3 Puddle Temperature

The temperature of the puddle has a large impact on the evaporation rate. In addition to the 

sources and sinks of thermal energy described in the section on the Puddle Option, liquid added to 

the puddle from the tank can either raise or lower the average temperature of the puddle. The 

change in the puddle temperature is defined by the sum of the thermal flux terms: 

( ) ( ) ( ) ( ) ( ) ( ) d 1

d

p

G H E dM

pl l p

T (t) F Ft Ft Ft F t Ft F t s t c d (t) ρ ↓ ↑ = ++++ ++    . 

The thermal energy flux from the addition of liquid from the tank, FdM , is proportional to the 

temperature difference between the liquid in the tank and that in the puddle, and is described as

( ) ( )

2

T pl T P

dM

p

Qc T T F t

π r

− = , 

where QT is mass flow rate into the puddle. 

The thermal energy flux from the ground for a growing puddle is modeled in a way that is 

conceptually consistent with the stand-alone puddle. The puddle is modeled as discrete concentric 

rings of liquid in contact with the ground; as the puddle expands rings are added. The entire puddle 

is assumed to be a uniform temperature, but the underlying ground temperature varies depending 

upon its duration of contact with the puddle. Conduction through the ground is treated separately 

for each ring; the thermal energy flux is the sum of the individual heat flux terms from each ring.

As with the puddle scenario described above, ALOHA uses one of two methods for finding the 

evaporation rate depending upon whether the puddle is sufficiently close to its boiling point. 

26 Source Strength Models
Brighton's formulation is used when the average puddle temperature is below its boiling point, and 

an energy balance method is used when the puddle approaches its boiling point.

3.4.5 RELEASE OF A 2-PHASE MIXTURE OR A LIQUID ABOVE ITS AMBIENT 

BOILING POINT

ALOHA uses the Homogeneous Nonequilibrium Model for release scenarios involving an effluent 

that is a superheated liquid, or a 2-phase mixture resulting from the flash boiling of a superheated 

liquid (Fauske 1985, Fauske and Epstein 1987, Henry and Fauske 1971). The mass flux, G , 

through a short pipe or hole is described by 

( ) ( ) 1 c 2 P

g l

L G Nc T

v v

−

= − , 

where,

c L is the specific latent heat of vaporization (J/kg), 

g ν is the specific volume of the vapor (m3/kg), 

ν l is the specific volume of the liquid (m3/kg), 

P c is the heat capacity of the fluid (J/kgK), 

T is the temperature (K), 

( )

2

2 2 2

c l

e dis g l P

L v l N

PC v v Tc l = +∆ −

, 

∆P is the difference between inside and outside pressure (Pa), 

l is the length of the pipe (m), and

el = 0.1 m.

The Homogeneous Nonequilibrium Model is limited to a 2-phase flow that is predominately liquid. 

The maximum value of Quality, X , for which the Homogeneous Nonequilibrium Model is valid 

(Quality is defined as the mass of the vapor divided by the mass of the mixture.) is

( ) (max) P gl

c

Pc T v v

X

L

− = , 

where P is the pressure of the fluid within the tank.

When the Quality of the effluent exceeds this limit, ALOHA computes the release rate based on a 

fixed quality set to X ( ) max . This provides a reasonable overestimate of the actual release.

The Homogeneous Nonequilibrium Model was developed to treat release scenarios involving pipes 

that are shorter than 10 cm, including a pipe length set to zero and a hole in the tank wall. ALOHA 

allows the user to specify whether the release is through a hole or short pipe, the latter corresponds 

Source Strength Models 27
to a pipe length of 10 cm, the upper limit of the model. At the lower limit of pipe length, the 

Homogeneous Nonequilibrium Model reduces to the Bernoulli equation. At the upper limit of pipe 

length, 10cm, the Homogeneous Nonequilibrium Model approximately reduces to the homogeneous 

equilibrium model, which has been shown to be applicable to typical valve geometries (Huff 1985). 

For holes in a tank wall, the Homogeneous Nonequilibrium Model predicts that the release rate is 

proportional to the density and pressure difference at the exit port. In contrast for 10 cm pipes, the 

Homogeneous Nonequilibrium Model is insensitive to both density and pressure differential. 

Flashing is assumed to occur in the pipe resulting in choked flow. As a result, release rates of 

superheated liquids through short pipes are generally much smaller than through holes.

Upon release from the tank, ALOHA assumes that the liquid undergoes an adiabatic vaporization. 

Evaporation cools the liquid as it vaporizes. If the liquid has not vaporized completely before its 

temperature drops to its ambient boiling point, ALOHA assumes that the cold 2-phase mixture is in 

the form of a fine aerosol and treats it as a dense gas. ALOHA does not allow the liquid to rain out 

and form a puddle.

The Homogeneous Nonequilibrium Model is limited by the diameter of the pipe. For large diameter 

pipes, ALOHA uses an approximation to insure that the predicted release rate does not 

underestimate the actual rate. For pipes greater than one meter in diameter, ALOHA reverts to the 

Bernoulli equation. For pipe diameters between 0.1 and 1 meter, ALOHA interpolates between the 

Homogeneous Nonequilibrium Model and the Bernoulli equation.

3.4.6 GAS RELEASE THROUGH A HOLE OR SHORT PIPE

ALOHA predicts a gas release under two conditions: the tank contains a pressurized gas, or there is 

a small leak in the head space containing chlorine or ammonia liquefied under pressure. ALOHA 

uses a single model which produces identical source strength estimates for holes and pipes 

regardless of which rupture type its user chooses. ALOHA considers the release rate from ruptures 

in the head space of tanks containing liquids stored below their boiling points as insignificant.

ALOHA uses algorithms incorporated in LEAKR, a computer model developed for Environment 

Canada (Belore and Buist 1986), to predict the release rate of gases from holes and short pipes. 

From the ratio of tank to atmospheric pressure, the ratio of hole width to tank length, and the 

critical pressure ratio for sonic flow (a threshold pressure ratio value), ALOHA first determines 

whether gas flow will be supersonic (choked) or subsonic (unchoked). If the pressure difference is 

great enough, ALOHA models flow as supersonic until the pressure drops to the point at which flow 

is subsonic. From then on, ALOHA computes a subsonic release rate until it predicts that tank 

pressure has dropped to atmospheric pressure. The estimated rate of gas release drops over time 

because the tank pressure is expected to drop as gas exits the tank, and because adiabatic 

expansion is expected to cool the tank contents, further reducing pressure. In gas release cases, 

ALOHA does not account for the effect of heat flux across the tank wall or for frictional differences 

between tank holes and short pipes/valves.

28 Source Strength Models
ALOHA predicts whether gas outflow will be either subsonic or supersonic by computing three 

ratios. The first is the ratio of atmospheric pressure to tank pressure:

/ R PP P aT = . 

The second important ratio is the ratio of hole diameter to tank diameter, 

/ c hT β = l l , 

where

T

l is the diameter of a spherical or cylindrical tank, and

hl is the either the diameter of a round hole or the square root of the area of a rectangular hole.

The third important ratio is the critical pressure ratio, 

2 1

0.2 (big hole) 1

Rc c

γ

γ

β

γ

−   =   >

  + , and

1 2

1 1 4 0.2 (small hole) 2 2

R R c c c c

γ

γ γ γ γ β β

−

− +

+= ≤ , 

where γ is the ratio of gas heat capacity at constant pressure and gas heat capacity at constant 

volume. 

3.4.6.1 Choked Flow

When RP is less than or equal to Rc , mass flows from the tank at the speed of sound and is usually 

choked, so that reducing downstream pressure further does not appreciably change the flow rate. 

The mass flow rate is given by

( )

1

2 1

() () 1 Qt C A tP t dis h g T

γ

γ

ρ γ

γ

+

−   =     + .

Source Strength Models 29
3.4.6.2 Unchoked Flow

When RP is greater than Rc , the flow is subsonic and the mass flow is given by

( ) 2 1/ )

() 2 1

a a

dis h g T

T T

P P Qt C A P P P

γ γ γ

γ ρ

γ

+

    =     − −    

. 

Outside the tank, the gas expands until its pressure equilibrates with the atmosphere. This process 

is modeled as an adiabatic expansion with an associated reduction in temperature. The 

temperature of the gas after the gas has fully expanded is given by

1

T

g T

a

P T T

P

γ

γ

−

  =    

. 

Detailed discussions on the relationship between choked and unchoked flow is given by Shapiro 

(Shapiro 1953) and Blevins (Blevins 1984). How this relationship can be applied to hazard analysis 

is discussed by Hanna and Strimaitis (American Institute of Chemical Engineers. Center for 

Chemical Process Safety. 1989). 

3.5 GAS PIPELINE

3.5.1 GENERAL COMMENTS

Only pure gas releases from pipe ruptures are considered by ALOHA. ALOHA’s gas pipeline

algorithm is based on modifications made by Wilson (Wilson, Alberta. Pollution Control Division., 

and University of Alberta. Dept. of Mechanical Engineering. 1981, Wilson et al. 1979) to a model 

developed by Bell (Bell 1978). Measurements show that heat transfer to the moving gas through 

the pipe walls maintains an almost isothermal condition throughout the length of the pipe except 

for the last 200 hole diameters. Within 200 diameters of the hole, the flow is assumed to be 

adiabatic because of the large acceleration near the end of the pipe.

30 Source Strength Models
3.5.2 THEORY

Wilson showed that an exponential is a solution for isothermal quasi-steady state pipe flow, and 

that the release of gases from a finite length of pipe can be approximated by a double exponential of 

the form

( ) ( ) 2 0 ( ) 1

Q t t Qt e e αβ β α

α

− − = +

+ , 

where

Q is the rate of mass discharge per unit time (kg s-1), 

Q0 is the initial mass flow at the time of the rupture (kg s-1), 

α is a non-dimensional mass conservation factor, and 

β is the release rate time constant (s).

The pressure in most pipelines is much greater than ambient pressure. Therefore, Q0 is calculated 

assuming a choked flow condition, where, for a pipe, the discharge coefficient is one and the 

internal pressure and temperature at the time of rupture are known, using

1

0 0 h

g

MW Q PA RT

γ − = Γ , 

where

P0 is the initial pipe pressure (Pa),

Ah is the area of the hole (m2), 

MW is the molecular weight of the gas (kg/mole), 

γ is the ratio of the heat capacity at constant pressure to heat capacity at constant volume, 

and 

1

1 1

2

γ

γ γ +

−   + Γ =     . 

Source Strength Models 31
The time constant, β , is computed by the equation (Wilson 1989), 

3 2 2 2 32 3 1 1

3

H

pF H

F

K K K

K β τ

       

   

− =Γ + −

Γ , 

( )

,

,

,

p

p

Fp p

H hp

L

c

K d L and

K AA

τ

γμ

=

=

=

where

L

p is the length of the pipe, 

c is the speed of sound in the pipe,

p d is the pipe diameter, 

μ is the Darcy-Weisbach friction factor, and

A

p is the cross-section area of the pipe. 

The speed of sound for an isothermal pipe is

1

RT 2

c

MW

  γ =     . 

The friction factor is computed according to Blevins (Blevins 1984) as

( ( ))

2

0.25

0.57 log p d

μ

ε

=

−

, 

where the roughness coefficient, ε , is set to 0.0001 m for normal conditions and is increased to 

0.002 m for rough pipe conditions. For typical pipe conditions, 0.01 0.02 < < μ . 

The time constant can be simplified for small- and large-hole conditions. When ( ) 2 K K H F Γ is 

small, the pipe can be treated like a long isothermal storage tank: 

1 β τ p H K− ≈ Γ (small hole).

A hole is considered to be large if ( ) 2 30 K K H F Γ > and

2

3 β τ ≈ p F K (large hole).

32 Source Strength Models
The total mass in the pipeline is based on the ideal gas law, 

0 p p

T

P A L MW

M

RT = , 

and the mass conservation factor is

0

MT

Q

α

β = . 

For each time step, the exit temperature must be re-evaluated by the methods described below. 

Adiabatic decompression of the gas within 200 diameters of the hole is assumed. Beyond 200 

diameters of the hole, the flow is approximately isothermal with frictional heating and adiabatic 

cooling in near balance. The exit temperature of the current time step is given by

( ) 1 /

a

f g

ia

P T T

P

γ γ −

  =    

, 

where 

T

g is the exit temperature at the previous step, 

Pa is the ambient air pressure, and 

Pia is the pressure at the isothermal-adiabatic interface, ia

p

Qc P A = .

At the isothermal-adiabatic interface, for steady-state conditions, the model assumes an infinite 

reservoir attached to a finite length of pipe with the hole the same size as the pipe. For this case, 

the flow rate is calculated as

QA v = p ia ia ρ , 

where ia v is the velocity of the gas at the isothermal-adiabatic interface, and ρia is the density at 

the interface. Choked flow is assumed and leads to the relationships

ia v c = γ and

ρ ρ ia r ia r = P P , 

Source Strength Models 33
where the subscript r refers to the reservoir. A non-linear equation for the velocity of the gas out of 

the reservoir must be solved to find a value for Pia to be used in the above equation. If Ma is the 

Mach number (ratio of discharge speed to the speed of sound), the equation for isothermal pipe 

flow is (Blevins 1984) 

2

2

2 0 4

1 1 d d p L

p

Ma L Ma

d Ma Ma

μ γ γ

γ

  − =     ∫ ∫ , 

where the approximation has been made that the length of the pipe where isothermal flow 

predominates is much larger than the part where adiabatic flow occurs, so that the upper limit of 

the left-hand integral is set to be the whole length of the pipe. Integration of this expression leads 

to an expression for the pipe friction parameter, 

1 2 22 2

2 2

v 21n ir ia ir

F

ir ia ia

c vc v K

vv v

γ γ

γ γ

−

  − −  

= −+        

, 

where ir v is the gas velocity at the reservoir-pipe interface.

The pipe pressure at the adiabatic-isothermal interface, Pia , is estimated from the relationship

1 2

2

ir

ia r r ir

ia

v P Pv

v

ρ

   = −       . 

3.5.3 COMPUTATIONAL NOTES

The gas pipeline release algorithm computes an array of gas release rates at different times. As in 

other source routines, the length of the time interval varies such that equal amounts of mass are 

released in each time step. If n is the total number of time steps, then each new time step is found 

from the previous one by using a Newton-Raphson iteration scheme to find the roots of the 

equation, 

( ) 1

1 ( )d T i i

i

M t F t Q t t

n t

+ + = − ∫ , 

where i t is the previous time and i 1 t + is the new time. Minimum time interval is 60 seconds and the 

release is terminated if it extends beyond one hour.

34 Source Strength Models 
4 AIR DISPERSION MODELS 

4.1 INTRODUCTION

The accidental release of a volatile chemical can present a threat to life and health far from the 

point of release. Some chemicals are toxic by inhalation, others can pose a fire hazard. Air 

dispersion models are central to predicting hazard zones associated with toxic or flammable gas 

clouds. These models are used to predict how the concentration of a pollutant, once released into 

the air, varies with time and position. ALOHA provides information about the concentration and 

duration of exposure, but does not resolve the probability that an exposed individual will be 

injured. 

Vapor clouds containing a flammable chemical mixed with air can burn when an ignition source is 

present. When a large quantity of a flammable chemical is released into the air and disperses to the 

point where a flammable fuel/air mixture forms before ignition, the fire can travel from the ignition 

source to the edges of the flammable cloud. The rate at which the combustion propagates through 

the cloud determines whether a damaging shock wave is generated (a vapor cloud explosion). The 

hazards associated vapor cloud explosions are described in Chapter 5. Hazards associated with 

flash fires, which do not generate damaging shock waves, are described in Chapter 6. The 

dispersion models described in this chapter are used to estimate the flammable area of the cloud. 

The movement of the ambient air is responsible for the advection and diffusion of a neutrallybuoyant pollutant in all but the very near field. The movement of pollutants with densities 

significantly different than the ambient air is also affected by gravity and can loft or hug the ground. 

But even for dense gases and dense aerosols, the movement of the ambient air usually dominates 

their dispersion.

A chemical vapor cloud is composed of a pollutant chemical and air in a ratio that changes with time 

and location. Clouds containing chemicals with high molecular weight, or aerosols, can be 

sufficiently dense that gravity has a significant effect on their movements. These dense gas clouds 

are modeled differently in ALOHA than clouds that are not directly affected by gravity. ALOHA 

incorporates two semi-empirical air dispersion models: the Gaussian model is appropriate for 

pollutants clouds that are not directly affected by gravity; the Heavy Gas model is appropriate for 

pollutant clouds with densities greater than the ambient air and affected in a significant way by 

gravity. These air dispersion models used for neutral and dense pollutant plumes in ALOHA can 

account for vertical gradients in wind speed and atmospheric turbulence, but do not account for 

topographic steering or winds that vary with time.

The ambient wind usually dominates the advection and diffusion processes for both dense and 

neutrally buoyant gas clouds. The movement of air is characterized by its velocity field and 

turbulence. The roughness elements of the ground provide a drag on the moving air, producing a 

velocity gradient where the speed at the surface itself is zero and increases logarithmically with 

height. The surface roughness elements not only affect the vertical wind profile, but also generate 

turbulence within the wind field. Topography and large structures also affect the velocity field and 

turbulence as the wind moves around and over these features. Turbulence is also generated by 

thermal effects; the sun warms the ground which, in turn, warms surface air causing it to rise.

Air Dispersion Models 35
4.2 METHODS FOR CHARACTERIZING WIND FIELDS AND TURBULENCE

Atmospheric turbulence has a major impact on the rate of dispersion of a pollutant cloud. Stability 

is a concept used to characterize the property of the low-lying atmosphere that governs the vertical 

movement of air. Specifically, stability refers to the tendency of the atmosphere to resist or 

enhance vertical motion and thus turbulence. A stable atmosphere inhibits the vertical mixing; a 

neutral atmosphere neither enhances nor inhibits vertical mixing; and an unstable atmosphere 

enhances vertical mixing and turbulence. Solar radiation plays a large role in atmospheric stability. 

With strong solar radiation, the ground warms and warms the low-lying air; the warm air rises 

generating eddies and a high level of turbulence. In contrast, when the air temperature increases 

with height, buoyancy forces act to inhibit vertical mixing creating a highly stable atmosphere. 

Stability classification is a central factor in ALOHA’s air dispersion models.

Pasquill (Pasquill 1961) defined six atmospheric stability classes (now usually called the 

Pasquill-Gifford-Turner stability classes). Classes A to F each represent a different degree of 

atmospheric turbulence. Stability class D represents a neutral stability condition. Unstable 

conditions are associated with atmospheric stability classes A, B, and C, where A is extremely 

unstable, B is moderately unstable, and C is slightly unstable. Stability classes E and F represent 

increasingly stable atmospheric conditions.

ALOHA can use either of two estimation methods to determine atmospheric stability class, 

depending on whether the user enters weather information manually or a portable weather station 

transmits readings to it. In addition, ALOHA allows the user to override these methods and directly 

specify the stability class.

4.2.1 MANUAL DATA ENTRY OF ATMOSPHERIC CONDITIONS

When the user manually enters atmospheric data, ALOHA uses a stability class method that 

incorporates the Pasquill-Gifford-Turner stability typing scheme. The method was developed using 

air dispersion measurements collected during experimental releases conducted during the 1950s 

and 1960s. During these experiments, most concentration measurements were made at distances 

within 1 kilometer from the release point. Because only in a few instances were concentrations 

measured at distances as far as 10 kilometers from the release location, it is not certain whether the 

method is reliable at greater distances.

Solar insolation (the solar energy flux) and wind speed are the two factors affecting the choice of 

stability class in ALOHA. Table 10 forms the basis for the determination of stability class for plumes 

over land in ALOHA. This table is identical to one presented by Turner (Turner 1994) with the 

exception that Turner did not assign stability classes for nighttime cases when wind speed is less 

than 2 m s-1. According to Turner, strong insolation corresponds to a solar altitude greater than 60° 

on a clear day, moderate insolation corresponds to a solar altitude between 60° and 35° on a clear 

day, and slight insolation corresponds to a solar altitude between 15° and 35° on a clear day. 

Nighttime is considered the time between one hour before sunset to one hour after sunrise.

ALOHA includes solar altitude and cloud cover in its determination of stability class. User-entered 

information on date, time, location, and cloud cover are used to determine the solar insolation. If 

36 Air Dispersion Models
more than one stability class fits the indicated conditions, ALOHA selects the most stable of these 

classes. ALOHA provides the user the option to directly select the stability class.

Table 10. Solar Insolation and stability class table.

Wind Speed Day Night

at 10 meters Solar Insolation Cloud Cover

(m s-1) Strong Moderate Slight >50% <50%

<2 A A – B B E F

2 – 3 A – B B C E F

3 – 5 B B – C C D E

5 – 6 C C – D D D D

>6 C D D D D

Stability is D for completely overcast conditions during day or night.

For plumes over water, ALOHA assumes stable conditions regardless of solar insolation or wind 

speed.

ALOHA classifies the solar insolation based on the net energy flux to the ground. Solar insolation 

greater than 851 W/m2 is considered strong insolation; computed insolation between 851 and 526 

W/m2 is considered moderate insolation; and computed insolation between 526 and 176 W/m2 is 

considered slight insolation. ALOHA estimates solar insolation absorbed by the ground from the 

location, time, and degree of cloudiness (Raphael 1962): 

( ) 2 1111 1 0.0071 sin 0.1

0 otherwise

sin( ) > 0.1 s s

C s I F

φ φ      

 − −  = 



.

CI is the cloudiness index (on a scale of 0 to 10) and 

S φ is the solar altitude (the angle of the sun above the local horizon) in degrees.

ALOHA computes solar altitude, S φ , as a function of

the latitude in radians, θ , 

the longitude in radians, λ , 

the hour of the day in Greenwich Mean Time (GMT) , Z (as 1 to 24 hours), 

the Julian day (the day number in the year) in days , J (as 1 to 365 days), 

sin( ) = sin sin cos cos cos φ θδ θ δ S ( ) ( ) ( ) ( ) ( ) s s + hs , 

( ) 2 2 23.45 sin 0.986 80 radians

360 360 S J π π δ     =   −       , and

Air Dispersion Models 37
2 15 12- radians

360 15 s h Z    π λ     =      −         

. 

4.2.2 AUTOMATIC WIND DATA ENTRY USING A SAM

ALOHA can be connected to a Station for Atmospheric Measurements (SAM) for direct wind 

measurements. Stability class can be derived from the wind speed and measured standard 

deviation in the wind direction, σθ ,m , in combination with user-entered surface roughness and 

solar radiation. The method used to estimate stability class was described by the U.S. 

Environmental Protection Agency (United States. Environmental Protection Agency 1987). 

The method is based on the wind speed at 10m, and a Surface Roughness Length, 0 z , equal to 15 

cm. If the roughness length is different than 15 cm, the standard deviation in wind angle is adjusted

to

0.2

0

, 0.15 m

z

σ σ θ θ

  =     . 

An initial estimate of stability class is based on the standard deviation in wind direction measured 

at 10 meters. 

Table 11. Initial estimate of stability class and standard deviation in wind direction.

Initial estimate of 

stability class

Corrected σθ

in degrees

A 22.5 ≤ σθ

B 17.5 ≤ σθ< 22.5

C 12.5 ≤ σθ< 17.5

D 7.5 ≤ σθ< 12.5

E 3.8 ≤ σθ< 7.5

F σθ < 3.8

If the measurement height is other than 10 meters, the table is adjusted. The lower bound to σθ on 

the table is multiplied by ( ) 10 P

Z m −

. The exponent is P = (0.06, 0.15, 0.17, 0.23, 0.38) for stability 

classes A through E, respectively.

38 Air Dispersion Models
The stability class is finally derived from the initial estimate and the wind speed measured at 10m. 

Nighttime is the period from 1 hour before sunset to 1 hour after sunrise.:

Table 12. Stability class for daytime and nighttime.

Daytime

Initial Class U10 (m s-1) Final Class

A U10 < 3 A

A 3 < U10 < 4 B

A 4 ≤ U10 < 6 C

A 6 ≤ U10 D

B U10 < 4 B

B 4 ≤ U10 < 6 C

B 6 ≤ U10 D

C U10 < 6 C

C 6 ≤ U10 D

D, E, F any speed D

Nighttime

Initial Class U10 (m s-1) Final Class

A U10 < 2.9 F

A 2.9 ≤ U10 < 3.6 E

A 3.6 ≤ U10 D

B U10 < 2.4 F

B 2.4 ≤ U10 < 3.0 E

B 3.0 ≤ U10 D

C U10 < 2.4 E

C 2.4 ≤ U10 D

D any speed D

E U10 < 5.0 E

E 5.0 ≤ U10 D

F U10 < 3.0 F

F 3.0 ≤ U10 < 5.0 E

F 5.0 ≤ U10 D

4.2.3 WIND PROFILE

The air dispersion models used in ALOHA require the vertical wind-speed profile. A steady-state 

wind varies significantly with height near the surface. Dispersion in the alongwind direction is 

generated mostly by this vertical wind shear. ALOHA uses the entered wind speed and reference 

height along with atmospheric stability and ground roughness data to compute a wind profile. 

Wind profile formulations are based on the assumption that wind speed and direction, as well as all 

other atmospheric conditions, remain horizontally homogeneous and do not change during the 

duration of a release.

The wind profile is affected by the irregularities of the solid surface over which the air moves. 

Obstacles such as trees, bushes, even blades of grass have an impact on the vertical profile of the 

wind. The ALOHA user chooses ground roughness conditions by entering a numerical value for 

Air Dispersion Models 39
ground roughness length, 0 z (m), or selecting one of three terrain conditions: Choosing Open 

Country sets the surface roughness to 0.03 m, Urban or Forest sets the surface roughness to 1 m, 

Open Water sets the surface roughness to a value based on the wind speed at 10 meters, U10 (m s1):

2.5

0 10 z U = 0.0000026 ( )

The dispersions models in ALOHA use surface roughness lengths in a limited fashion. The Gaussian 

Model only distinguishes between large and small ground roughness, and the Heavy Gas model uses 

a roughness length of 0.10 m for all user-specified roughness lengths greater than 0.10 m.

In parts of the model, ALOHA approximates the wind profile within the surface sublayer using

( ) ( ) * 0

0

ln U z z U z

k z ψ ζ   + =   +

 

, 

where 

U* is the friction wind velocity, 

U z( ) is the mean wind at reference height z, 

k is the von Kármán constant, taken to be 0.4, 

0 z is the ground roughness length, and

z

L ζ = is the non-dimensional height.

ALOHA uses the following empirical equations to relate the Obukhov length, L, to the surface 

roughness length under each Pasquill stability class:

0.10

0

0.17

0

0.30

0

0.30

0

0.17

0

11.4 Class A,

-26.0 Class B,

123 Class C,

Class D,

123 Class E,

26.0 Class F.

z

z

z L

z

z

−





 − = 

 ∞







40 Air Dispersion Models
The function ψ ζ( ) depends on the sign of the dimensionless height. For unstable, neutral, and 

stable surface sublayers (Businger 1973), ALOHA uses empirical equations, 

2 2ln (1+ ) ln (1+ ) + 2arctan 0, 2 2

0 0,

4.7 0,

a a

a

a

π ζ

ψ ζ ζ

ζ ζ



 − +<

 ( )=  =

− > 



where 

( ) 1

4 a = −1 15ζ . 

In most cases, ALOHA uses a power function (Brutsaert 1982), rather than a logarithmic function, 

to describe the wind profile within the surface sublayer: 

2

1

2 1

n

z U U

z

  =     . 

The exponent n greatly affects heavy gas dispersion computations. DEGADIS matches the 

logarithmic profile for the given ambient conditions to estimate the exponent, n . For 0 z cm =15

and Pasquill stability classes A to F, typical values of n are 0.109, 0.112, 0.120, 0.142, 0.203, 0.253.

4.3 GAUSSIAN MODEL FOR NEUTRALLY BUOYANT GASES

ALOHA’s Gaussian dispersion model is intended to be used with vapor clouds that do not 

significantly affect the ambient air flow, and are not affected by gravity; they are considered passive 

pollutants. 

The Gaussian model predicts that the concentration distribution of a steady-state release of 

neutrally buoyant gas will approach a Gaussian distribution with increasing down-wind distance. 

The parameters characterizing the distribution are based on empirical measurements. Increasing 

averaging times of the measurements also tend to drive the distribution to a Gaussian shape, as well 

as widen the spatial distribution. The actual concentration distribution from a release can vary 

significantly from Gaussian at any single instant in time.

The method used in ALOHA accommodates instantaneous, continuous, and time-dependent 

releases of finite duration up to one hour. An instantaneous release is modeled as a single oneminute steady-state release which gives rise to a single cloud. A continuous release is modeled as a 

single one-hour steady-state release which gives rise to a single cloud.

Air Dispersion Models 41
A time-dependent release is modeled as series of five finite-duration steady-state releases; each 

giving rise to a cloud that does not interact with the other clouds. The concentration at a point in 

space and time is found by summing the contributions from each cloud.

The model used to describe the dispersion of each cloud is based on a Gaussian dispersion model 

developed by Palazzi (Palazzi et al. 1982) that describes the behavior of steady state releases of 

short duration. This model is described as

( )

( ) ( )

( )

erf erf 

2 2 2 ,

,,, , -

 erf erf 

2 2 2

x x Ut

t t

r

x x C xyzt

t t xU t t x Ut r r

x x

χ

σ σ

χ

σ σ

         

    

                                 



 −  − ≤  = 

 − < <∞ −  − 



where

σ x , σ y , and σ z are the dispersion parameters,

rt is the duration of the release.

The term χ represents a well-known Gaussian distribution from a continuous steady-state point 

source (Hanna et al. 1982), 

( ) ( ) ,,, ( ) ( ) , , y z

Q t xyzt g xy g xz U χ

  =    

, 

where 

( ) ( ) ( )

2

1 1 , exp 2 2 y

y y

y g xy

πσ x σ x

    =   −          

, 

and, when no inversion is present,

( ) ( ) ( ) ( )

2 2

11 1 , exp + exp , 2 2 2

s s

z

z z z

z h z h

g xz

πσ x σ σ x x

         − + =− −                

Where s h is the height of the release.

The presence of an inversion can have a significant effect on the vertical diffusion of the cloud. 

When a layer of unstable air at the ground lies beneath a layer of stable air, the interface acts to 

contain the pollutant cloud. The effect of an inversion can be quite complex and simplifications 

42 Air Dispersion Models
were required in ALOHA. As a conservative approximation, ALOHA models the interface between 

the stable and unstable air as impervious to the transport of the pollutant. The ground and the 

inversion act as a pair of reflecting surfaces for the pollutant cloud. Note that ALOHA only 

accommodates situations where the inversion height, i h , is above the height of the release, s h . The 

form of z ( ) ,zxg represents the repeated reflection, up to J=5, of the pollutant cloud, and is 

described as

( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

2 2

1

2

2 2

2 2

2 2

2

1

2

1 1 1 , 2 exp exp 2 2

1 1 2 2

exp - exp - 2 2

2 1

 exp - 2

s s

z

z z z

i s i s

z z J

n

z

z h z h

g zx

x x

z nh h z nh h

x x

z nh h i s

x

π

σσ σ

σ σ

σ

−

=

           − + =      − +− +                  

    − −   + −        +            

 − +

+

∑ ( )

( )

2

2

2 1

exp - 2 z

z nh h i s

σ x

 

 

          + +                +              

, 

where i h is the height of the inversion.

Far downwind, the repeated reflections of the cloud tend to produce a uniform concentration 

beneath the inversion. ALOHA approximates the vertical distribution as homogenous at distances 

downwind such that ( ) z i σ ≥ 2hx . 

4.3.1 DETERMINATION OF DISPERSION PARAMETERS

The concentration distribution described above is a Gaussian distribution parameterized by three 

empirical dispersion parameters. These are the standard deviations of the Gaussian distribution: 

.,, yx z and σσσ The dispersion parameters depend upon the stability class, and in some cases the 

surface roughness. They also depend upon the averaging times associated with their measurement. 

Briggs developed formulas for estimating σ y and σ z for each of the Pasquill-Gifford-Turner 

stability classes, and for Urban (large surface roughness) and Rural (small surface roughness) 

conditions (Briggs 1973, Hanna et al. 1982): 

( ) ( ) ( ) 1 3

1 2

2

and 1 .

1

y z y z z z

y

s x s

x x s x s x

s x

σ σ = = +

+

The values of the parameters are presented as a function of atmospheric stability and roughness in 

Table 13. 

Air Dispersion Models 43
Briggs analysis employed only two surface roughness classifications: Urban corresponds to large 

ground roughness, or Rural which corresponds to small surface roughness. If the user-specified 

ground roughness is less than 20 cm, Rural is selected, otherwise Urban is selected.

Averaging times varied for the experiments used to determine dispersion parameters; however, 

Briggs did not adjust the coefficients used to estimate σ y to account for the different averaging 

times associated with the data sets from which they were derived (Blewitt 1990). The rural 

coefficients were derived from data averaged over 3 minutes, and the urban coefficients were 

derived from data averaged over 1 hour. When the coefficients are corrected for this difference, the 

formulas produce similar curves (Blewitt 1990). For this reason, ALOHA uses only the rural 

coefficients based on 3-minute averaging times to estimate σ y under all surface roughness 

conditions.

To estimate σ z , Briggs derived a set of analytical solutions from two different experiments. Data 

used for values of σ z greater than 100 m relfect a 1-hour averaging time, while data used for 

smaller values of the σ z reflect averaging times of 10 to 15 minutes (Briggs 1991). There is no 

definitive averaging time associated with this parameter, and we do not expect σ z to be strongly 

dependence on averaging time.

To estimate σ x , the standard deviation function in the alongwind direction, ALOHA uses the 

following formula, which is based on Beals (Beals 1971) and is also used in the heavy gas model 

DEGADIS (Havens and Spicer 1985). Beals made no reference to the associated averaging times for 

σ x , and we do not expect σ x to be strongly dependent on averaging time. The formula is shown as

( ) 2

0 10 . x s

x x σ x sx =

44 Air Dispersion Models
Table 13. Values for the coefficients used to estimate the dispersion parameters.

Stability Class

Roughness Coefficient A B C D E F 

Both

sx1 0.02 0.02 0.02 0.04 0.17 0.17

sx2 1.22 1.22 1.22 1.14 0.97 0.97

sy1 0.22 0.16 0.11 0.08 0.06 0.04

sy2 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

Small 

Surface 

Roughness 

(Rural)

sz1 0.2 0.12 0.08 0.06 0.03 0.016

sz2 0 0 0.0002 0.0015 0.0003 0.0003

sz3 0 0 -0.5 -0.5 -1 -1

Large 

Surface 

Roughness 

(Urban)

sz1 0.24 0.24 0.2 0.14 0.08 0.08

sz2 0.001 0.001 0 0.0003 0.0015 0.0015

sz3 0.5 0.5 0 -0.5 -0.5 -0.5

Note: An incorrect value of 0.00015 for sz2 was presented in Briggs (1973) and many later 

references. ALOHA uses the correct value, 0.0015.

4.4 HEAVY GAS MODEL FOR DENSE GASES

The Heavy Gas dispersion calculations used in ALOHA are based on the DEGADIS model (Havens 

and Spicer 1985, Spicer and Havens 1989). DEGADIS, in turn, is an adaptation of the Shell HEGADIS 

model described by Colenbrander (Colenbrander 1980, Colenbrander and Puttock 1983); DEGADIS 

also incorporates some techniques used by van Ulden (van Ulden 1974, 1983). DEGADIS was 

selected as the basis for ALOHA’s dense gas computations because of its general acceptance and the 

extensive testing that was carried out by its authors.

Some simplifications were introduced into ALOHA-DEGADIS, making it different from the DEGADIS 

model. ALOHA-DEGADIS is limited to releases at ground level, and does not account for the initial 

momentum from a jet release. Some of the numerical methods used in ALOHA-DEGADIS have been 

simplified compared to those used in DEGADIS. The ambient air pressure used in ALOHA is 

independent of position and time. The pollutant and air are assumed to behave as non-interacting 

ideal gases.

The method for handling releases that vary with time is unique to ALOHA. DEGADIS is designed to 

model steady-state releases, while ALOHA is designed to model time-dependent releases. The 

ALOHA-DEGADIS model is essentially the DEGADIS steady-state plume model that has been 

adapted to handle time-varying sources. For any non-instantaneous sources in ALOHA, the model 

generates a five-step release; each step is a finite-duration steady-state release. The Heavy Gas 

model generates infinite-duration steady-state plumes for each of these sources. A composite timevarying cloud is created by adding together pieces of the five steady-state plumes. The pieces start 

Air Dispersion Models 45
as steady-state plumes ten meters in length. Each piece is converted into a Gaussian in the 

downwind direction, conserving mass, but maintaining its vertical and horizontal dimensions. The 

pieces are summed to create the time-varying cloud. The Guassian dispersion parameter used, σ x , 

is the same as that used in the Gaussian dispersion model (Beals 1971). 

ALOHA-DEGADIS has beenverified by comparisons to field experiments, the original DEGADIS 

model, and other reference models. As part of the verification process, the authors of both models 

reviewed each model's code and computations line by line (Havens 1990). 

Estimates made by ALOHA-DEGADIS and DEGADIS for a series of release scenarios were also 

compared. Using a fractional factorial experimental design (Cochran and Cox 1957), a set of 24 test 

scenarios was prepared. Eight DEGADIS input variables—stability class, wind speed, spill size, 

release duration, gas density, ground roughness, chemical-ground temperature difference, the 

coefficients of the equation of state, and level of concern—were varied in the 24 scenarios in order 

to span the expected variability within model parameters. We believe that these factors are the 

most important influences on concentration estimates made by DEGADIS.

For the 24 cases, the correlation between the estimates of concentration and downwind distance 

made by the two models was 0.997. For steady-state, continuous release cases, ALOHA-DEGADIS 

tended to predict a longer distance than was predicted by the full DEGADIS model. ALOHADEGADIS threat zones averaged 10% longer than those predicted by DEGADIS. For short-term 

releases, the arrival time of the peak concentration at a given location downwind and the maximum 

value of the peak were also examined. Arrival times predicted by ALOHA-DEGADIS were slightly 

later than those predicted by DEGADIS, although the correlation between estimates made by each 

model was 1.0. ALOHA-DEGADIS predicted downwind movement of gas clouds to average about 

9% slower than the travel rate predicted by DEGADIS. ALOHA-DEGADIS predicted maximum 

concentrations that exceeded DEGADIS estimates by an average of 8%. The correlation between 

maximum concentration estimates made by the two models was 0.994%. 

These results demonstrate that the ALOHA-DEGADIS model is consistently slightly more 

conservative than DEGADIS. It can be expected to produce peak concentration estimates for a given 

point that are, on average, about 10% greater than estimates made by the full DEGADIS model.

4.4.1 CRITERIA FOR USE OF THE HEAVY GAS MODEL

The ALOHA Heavy Gas model is designed to account for the gravitational effects on pollutant clouds 

with densities significantly different than air. Gravitational effects can be insignificant for dense 

pollutants if the dispersion is dominated by the wind or thermal convection in the atmosphere, or 

the evaporative flux from a pool is small. The Heavy Gas model can still be used with these clouds, 

but the Gaussian Model is preferred. The Gaussian model is also recommended when the pollutant 

cloud is less dense than air.

In its default mode, ALOHA incorporates a decision algorithm to choose between the Heavy Gas and 

Gaussian models. ALOHA uses the critical Richardson's Number (also called the friction 

Richardson’s Number), Ric, as the criterion for distinguishing between passive or non-passive 

dispersion. When Ric is less than 1, ALOHA considers the gas to be passive and, unless the user 

46 Air Dispersion Models
overrides this decision, ALOHA computes dispersion using the Gaussian dispersion model (Spicer 

and Havens 1989). 

The critical Richardson number depends upon the density of the pollutant, the wind speed, and the 

release rate, 

2

*

ˆ c

H g Ri

U

∗ = , 

where the reduced gravity is

( ) ˆ a

a

g g ρ ρ

ρ

− = , 

ρ is the density of the chemical, 

ρa is the density of the ambient air, 

g is the acceleration of gravity, 

U* is the friction velocity of the wind, and

H is the characteristic dimension of the source.

For instantaneous releases, H is the characteristic height of the source, 

0

0

V H A = , 

where V0 is the volume of the release and A0 is the area of the source.

For puddle sources, the characteristic height of the source is related to the release rate and 

diameter of the puddle and is found using

10

E H

ρU D = , 

where 

E is the mass evaporation rate (kg s-1),

D is the diameter of the puddle, and

U10 is the wind speed at 10 meters.

For continuous or semi-continuous sources other than the puddle source, the characteristic 

dimension is

10 4

E H

U

π

ρ = . 

Air Dispersion Models 47
4.4.2 CONCEPTUAL MODEL

Havens and Spicer (Havens and Spicer 1985) described three stages in a heavy gas release. This is 

the conceptual model used in DEGADIS and ALOHA-DEGADIS.

1. Buoyancy-Dominated Dispersion. Near the point of a release, buoyancy forces, inertial 

forces, and ambient air motions all affect cloud development. Immediately after it is released, a 

heavy cloud “slumps” over the ground. The rapidly spreading cloud forces itself under the ambient 

atmospheric boundary layer. As it does so, shear at the ground and at the top of the cloud creates a 

vortex ring at the forward edge of the cloud (Britter 1989). During this initial spreading, the cloud 

entrains approximately 10 times its original mass of ambient air. That is, the density ratio is 

reduced by a factor of 10. By the end of this stage, a thin blanket of gas has formed. The blanket 

constitutes the secondary source.

The buoyancy-dominated stage evolves through three overlapping sub-stages: 

(i) During the slumping phase, which lasts only a few seconds, strong, buoyancy-dependent 

spreading forces are balanced by counterflow of the ambient fluid. 

(ii) During the buoyancy-inertial phase, buoyancy spreading forces are balanced by inertial 

forces. Concentrations within the cloud are greatly diluted as the spreading “head” of the 

cloud entrains ambient fluid.

(iii) During the viscous phase, buoyancy forces are balanced by viscous forces. Turbulent 

mixing is suppressed, and the dilution rate slows.

2. Stably Stratified Shear Flow. In this region, buoyancy forces and ambient air flow determine 

the cloud development, acting to produce a stably stratified cloud embedded in the mean wind flow. 

The cloud remains thin and spreads laterally as it is advected downwind. The stable density profile 

in the cloud suppresses turbulence, and turbulent fluctuations in the cloud are characteristically 

below-ambient levels. Air is entrained into the spreading cloud primarily through its top as well as 

along its edges.

3. Passive Turbulent Dispersion. The cloud continues to be diluted until its density only 

slightly exceeds the density of the air. The small remaining density difference only negligibly 

affects cloud dispersion, and natural levels of turbulence return. The gas becomes a passive 

contaminant.

4.4.3 SECONDARY SOURCE IN ALOHA

ALOHA does not explicitly model the buoyancy-dominated dispersion that leads to the formation of 

a heavy gas blanket, also called the Secondary Source. Instead, it uses a simplified method to 

estimate the blanket dimensions.

Four source types are described in Chapter 2: direct release, tank release, pipeline release, and 

evaporating puddle. For heavy gas modeling, these are referred to as the Primary Sources. The 

Primary Source is generally time dependent and approximated as a series of five finite-duration 

steady-state sources.

48 Air Dispersion Models
The heavy gas model is fundamentally a steady-state plume model, while the Primary Sources are 

generally time-dependent. To approximate the time-dependence of the plume, ALOHA generates 

steady-state plumes based on the five time steps of the Primary Source, treating each as a timeindependent steady-state source. Time dependence is later introduced by combining the timeindependent plumes.

The buoyancy-dominated stage of a heavy gas release generates a “blanket” of dense gas which is 

centered on the location of the Primary Source and does not move with the ambient air flow. The 

Primary Source feeds this blanket at a constant rate, while the uptake and transport of gas from the 

top of the blanket removes gas from the blanket. The atmospheric uptake rate is proportional to 

the area of the blanket, so the blanket reaches its steady-state size when the atmospheric uptake 

rate equals the Primary Source. The blanket is assumed to be circular.

Most of the Primary Sources in ALOHA are point sources; however, the evaporating puddle has a 

defined area. If the area of the evaporating puddle is greater than the steady-state area of the heavy 

gas blanket, the area of the blanket is set equal to the puddle area.

The total mass of the developing blanket is

() () () () M t At H t t b b = ρ , 

where 

A t( ) is the area of the blanket,

( ) H t b is the blanket height, and

ρ ( )t is the overall mean density of the blanket.

( ), ( ), ( ), ( ) M t At H t t b b ρ are unknowns.

The rate of change in the blanket mass depends on the entrainment of air and contaminant into the 

cloud, 

*max

,

( ) ( ) () () d () ( ) a b

c p c

dM E t Q t Q t At

t w t w t

 

= +−    

, 

where 

E t( ) is the contaminant evolution rate from the Primary Source (kg s-1), 

( ) w t c p, is the contaminant mass fraction in the primary source, 

( ) Q t a is the entrainment rate into the air above the blanket (kg s-1), 

Q*max is the maximum mass flux into the air from the blanket (kg s-1m-2), and

( ) w t c is the mass fraction of contaminant in the developing cloud.

( ) Q t a , ( ) Q t *max , and ( ) w t c are unknowns.

Air Dispersion Models 49
The terms E t( ) and ( ) w t c p. depend on the primary source and are provided as external 

parameters.

In the final phases of the developing cloud, (humid) air entrainment is primarily through the top, 

and its rate is given by

( )

( ) 2

2 ( ) ( ) ( )( ( ) ( ) ˆ( ) ( )/ ( )

b ba f

a

b f

R tH t t u t

Q t

gtH t u t

π ρε = , 

where 

gˆ is the reduced gravity defined in a previous section,

( ) R t b is the radius of the blanket,

ε is the entrainment rate coefficient at the front of the cloud, and

uf is the rate of advance of the cloud front, ( ) ( )/ f b u t dR t dt = . 

The total mass of contaminant in the source blanket is

2 ( ) ( )2 ( ) ( ) ( ) ( ) ( ) M t w t R t H t t w tM t c c b b cb = π ρ = , 

and its rate of change is

max *

d () () () () d

M t c Et AtQ t

t = − . 

The amount of air in the source blanket is

( ) (1 ( ))2 ( ) ( ) ( ) (1 ( )) ( ) M t w t At H t t w t M t a c b = − = ρ − c b , 

and its rate of change is

( )

, *max

,

1 d () ( ) ( ) ( ) ( ) () () d t 1 1 ( )

c p a a

a

aa c c p

w M t E t t Q t Q t w t At

t w q q w t

  −

=   + −   + +  

, 

where qa is the absolute humidity.

The total energy in the cloud is

2 () () () () () U t h tR t H t t b bb b = π ρ , 

where 

Ub is the total energy in the cloud and

b h is the mean enthalpy (J kg-1) for the blanket.

50 Air Dispersion Models
The rate of change is

2 2 max *

,

d () () () ( ) () () () ( ) ( ) d ( ) ( )

b s

a a b b b HS

c p c

U t h tEt Q t hQ t h t R t R t F t

t w t w t

π π

 

= +−   +

 

, 

where 

s h is the enthalpy of the primary source,

a h is the enthalpy of air, and

( ) F t HS is the surface heat flux.

These relationships are solved simultaneously to obtain the steady state limit of the blanket 

variables.

4.4.4 STABLY STRATIFIED SHEAR FLOW AND PASSIVE DIFFUSION STAGES

The heavy gas dispersion model in DEGADIS is almost identical to the similarity model proposed by 

Colenbrander (Colenbrander 1980). 

Air flow over the blanket creates a plume of heavy gas. To simplify computation, the cylindrical 

blanket is treated as a square prism of equal volume and height. The plume is assumed to be 

composed of (i) a horizontally homogeneous core of width 2b that has vertical dispersion, and (ii) 

Gaussian-shaped edges. As the plume transitions from stably stratified shear flow to passive 

turbulent diffusion, the width of the homogeneous core goes to zero. The following equation for the 

concentration of the pollutant is applicable to both regions:

2 1

1

( ) exp ( )

(, ,)

( ) exp 

n

c y

n

c z

y bx z c x y b x

S x Sz

cxyz

z

c x y b x

S

+

+

    − ( )    − −  > ( ),          = 

       −  ≤ ( ).        





Four variables are functions of x and must be computed for each downwind step: 

( ) c c x is the centerline ground-level concentration (ppm), 

( ) y S x is the lateral dispersion parameter (m), 

( ) z S x is the vertical dispersion parameter (m), and

b is the half-width of the homogeneous core section (m).

The dispersion parameters are based on measurements. We do not expect the vertical dispersion 

parameter to be strongly dependent on the averaging times used in its measurement. However, the 

horizontal dispersion does have a dependence on averaging time; it is thought to scale with the 

averaging time raised to the 0.2 power. ALOHA-DEGADIS adjusts the horizontal dispersion 

parameter to correspond to an averaging time of 5 minutes for toxic clouds, and 10 seconds for 

flammable clouds.

Air Dispersion Models 51
In the discussion to follow, a coupled set of empirical parametric equations will be presented. This 

set of equations must be solved simultaneously at each step in the x direction.

Figure 2. The plume model proposed by (Colenbrander 1980). 

4.4.4.1 The Effective Cloud Width, Height, and Velocity

Empirical formulations for the plume/cloud dispersal require a cloud height parameter. The 

effective cloud height, Heff , is based on the centerline concentration of contaminant in the form

0

1 ( ) ( ,0, ) ( ) eff

c

H x c x z dz

c x

∞

= ∫ . 

The effective cloud height is

( ) 1 ( ) 1 1

z

eff

S x H x

n n

  = Γ   + +  , 

where Γ (1/(1 + n)) is the gamma function. This term will appear several times in the development 

below and will be written simply as Γ. 

52 Air Dispersion Models
We define an effective cloud width by the sum of the constant concentration central core, b(x) 

and the tail-off region on each side as

() () () 2B x bx S x eff y

π

= + . 

The lateral spread of the cloud is modeled by a Froude number representation as

1/2

2

d ˆ ( )

d ()

n

eff R eff

E

z R

B Z gH x C

x Sx U

   = Γ      

, 

where CE is a constant.

At some x downwind, b x() 0 → and the cloud width is defined by ( ) y S x . The cloud initially 

spreads under the influence of the reduced gravity, g ˆ , then as the cloud is diluted and ( ) z S x grows, 

the core area diminishes to zero, at which point the heavy gas effects cease.

The effective cloud velocity in the plume is the concentration-weighted mean of the wind speed, 

0 ( ,0, ) d ( ) ( )

( ,0, )d

n

z R

eff

R

cx zU z Sx U U x

cx z z Z

∞

  = =     Γ

∫

∫ , 

where UR is the wind speed at reference height ZR . 

4.4.4.2 Bulk Richardson Number

The Bulk Richardson Number is 

* 2

*

ˆ() () Ri ( ) eff gxH x

x

U = . 

Experiments show that in continuous releases the plume has well-defined sharp edges when Ri*>1 

and lateral concentration profile approaches a Gaussian shape when Ri*<1.

Air Dispersion Models 53
4.4.4.3 Corrections to Ri* for Heat Flux

When the surface temperature under the plume, TS , is greater than the mean temperature of the 

cloud, ( ) T x c , a positive heat flux reduces the cloud stability. In this case, Ri* is corrected in the 

form

( )2

Ri ( ) Ri ( ) * * * w x xU σ ′ = , 

where σ w is the root-mean-square (rms) vertical velocity at the top of the gas cloud. The ratio 

( ) σ w U* to the top of the cloud is estimated by

( )1 2 2 3

T

*

1 Ri w

U

σ

= + , 

where the bulk temperature Richardson Number, RiT ,is given by

*

Ri

n

eff s c R

T

c R eff

T T H Z

g T UU H

  −   =          

. 

This equation introduces a new unknown, the cloud temperature, ( ) T x c . A cloud energy balance 

must be added to the equation set in order to estimate this term.

4.4.4.4 Heavy Gas Dispersion Coefficients

Two-dimensional diffusion equations represent a balance between the downwind gradient in 

concentration and either the vertical or the lateral turbulent diffusion. By separation of variables 

(, ,) z

cxyz c U K

x zz

∂ ∂∂

∂ ∂∂

  =     and

(, ,)

y

cxyz c U K

x yy

∂ ∂∂

∂ ∂∂

  =    . 

The equations are solved using the power-law approximation for the wind profile described earlier 

in this chapter, and the empirical approximations of

( ) *

*

(,) Ri ( ) z

kU z K xz φ x = ′ and

1

0 () () K x K UB x y eff

γ = , 

54 Air Dispersion Models
where γ1 is a constant, K0 , is a dimensionless constant ( ) ( ) 1 1

n −γ = and φ ( ) Ri*

′ is a stability factor 

given by

( )* ** ( ) ( ) 1.04 25 5.7 φ Ri 0.88 0.099 Ri 1.4 10 Ri Ri 0 − ′ ′ ′′ = + +∗ ≥ . 

The previous equations are solved to yield the similarity form of the dispersion equation in the x-z 

plane:

( )

( )

*

*

1

d ( ) 1

d 1 Ri

n

RR z

R

UZ S x kU n

x nZ φ

+       +     =     + ′      

. 

4.4.4.5 Mass and Energy Balance

The mean density of the cloud gas mixture depends on the ( ) c c x and temperature, Tc . These are 

approximated from a combined mass and energy balance for a differential slice with thickness dx , 

width of Beff , and height Heff . 

The mass balance at x is given by (ρUHB eff eff eff ) and

( )

( ) ( ) ( ) '

*

d 1

d Ri

a w

eff eff eff eff

k n

UHB B x

x x

ρ σ

ρ φ

+   =   . 

When the surface temperature is greater than the cloud temperature, then Ri* is corrected and the 

cloud temperature is a new dependent variable, Tc(x). The temperature of the cloud will change as 

heat is added to the cloud by flux of sensible heat from the ground, FH . The added energy density 

of the cloud, Dh , in (J kg-1) is computed from

( ) ( ) ( ) (0) ( ) Dx c x Tx T h pg c c = − . 

The energy budget for a transverse slice of the cloud is

d

d

D U H B FB h eff eff eff H eff L x

  ρ δ =   , 

where FH is the surface heat flux under the cloud, and δ L is an empirical constant.

When the core width b = 0, the energy balance becomes

d /

d

DUH F h eff eff H L x

  ρ δ =   . 

Air Dispersion Models 55
The steady-plumes generated by the Heavy Gas model are combined to create a time-dependent 

cloud, as described in the General Comments section above.

4.5 THREAT ZONE

ALOHA’s threat zone represents the area within which the ground-level gas concentration exceeds 

the level of concern at any time.

4.5.1 ESTIMATING MAXIMUM CONCENTRATION

The concentration of the pollutant is a function of both location and time. In order to generate a 

threat zone, ALOHA calculates the peak concentration as a function of time for all points in space. It 

is important to note that this is not a true peak in the concentration since the concentrations 

predicted by ALOHA are time-averaged over different time intervals, depending upon the 

dispersion model employed. Implicit averaging times within the dispersion models vary.

The maximum concentration function M(x,y) for any point (x,y) is defined to be the maximum over 

all time of concentration at that point, and is given by

( ) ( ) , max , , t

M xy C xyt −∞≤ ≤∞

=    . 

ALOHA implements this function as follows. First, it identifies the source step associated with the 

highest release rate, and finds tm (s), the time corresponding to the midpoint of that source step. It 

then finds t0 (s), the time at which the cloud associated with that source step is centered at the point 

(x,0) as

0 m

x

t t

u

= + . 

ALOHA initially guesses the maximum concentration at (x,y) to be C(x,y,t0). To test this, it compares 

the value of C(x,y,t0) to C(x,y,t0±30). If one of these values is larger, it compares concentrations in 

that direction using 30-second intervals for as long as concentration continues to increase.

4.5.2 DRAWING THE THREAT ZONE

The threat zone contour runs through points where the maximum concentration equals the LOC. At 

points within the threat zone, the maximum concentration function exceeds the LOC.

56 Air Dispersion Models 
ALOHA constructs the threat zone contour in two steps. First, it performs a binary search of the 

maximum concentration function to find xl, the downwind distance to the LOC (m). For a puddle 

source, ALOHA performs a second binary search to find the off-axis distances to the LOC from nine 

points equally spaced along the x-axis at distances 0, xl/8, 2xl/8, 3xl/8...xl. In the cases of point 

sources, ALOHA finds these off-axis distances using 

( )

1

2 ,0 2log , y

M x

y LOC

σ

    =        

where x is the distance of each of the nine points along the x-axis. The threat zone contour runs

through each of the (x,y) points found by this method. ALOHA uses a Bezier fit to connect these 

points by smooth curves.

4.5.3 CONFIDENCE CONTOURS

Uncertainties in the average wind direction are represented on the threat zone graph as confidence 

lines. The axis of the threat zone is rotated through an angle on either side of the nominal wind 

direction and the contour of the rotated threat zone is drawn. The angle, α , is based on the width 

of the Guassian plume at a distance 5 minutes downwind of the source, 

3

2

arctan

300

y

U

σ

α

  =    

, 

where U3 is the wind speed at 3 meters (m s-1).

4.6 INDOOR AIR CONCENTRATION

It is generally recognized that a well-insulated house or building provides excellent protection 

against a toxic cloud of finite time duration (Wilson 1987). Assuming the infiltration rate is small 

and that the air inside is always well-mixed, the building acts like a low-pass filter and an electrical 

R-C filtering circuit offers an excellent analog.

The infiltration time constant is the time required after a step-increase in the outside concentration 

of a gas for the concentration inside to reach 63% of the step difference. This number can vary 

from 1 to 0.1 hours depending on the building tightness, the wind, and the inside-outside 

temperature difference. The wind blowing against a building creates pressure differences and 

these drive infiltration and exfiltration depending on the direction of the pressure gradient. 

Temperature differences create a pressure gradient and thus enhance the infiltration process.

Because of the extreme variability in weather and building types, ALOHA must make some broad 

generalizations. Sherman (Sherman 1980, Sherman, Wilson, and Keil 1984) studied 196 houses and 

found that an effective leakage area is 0.00059 times the area of the house. ALOHA uses this figure 

and assigns an average house floor area of 160 m2 (1722 ft2). An average building ceiling height of 

Air Dispersion Models 57
2.5 m is taken for a single-story house and 5m is used for a two-story building. A reasonable 

approximation of inside temperature is 20°C. It is assumed that all leakage occurs evenly over the 

structure and that there is no difference between floor and ceiling leakage.

We assume that the outside and inside concentrations are uniform and that environmental 

conditions are stationary in time. The balance of inflow and outflow of the chemical is 

( ) ( ) () () / i

oi E

dc t

c t ct

dt = − τ , 

where 

( ) C t i is the inside concentration, 

( ) C t o is the outside concentration, and 

E τ is the infiltration time constant.

The infiltration time constant can be specified by the user during setup. If not specified, it is 

approximated using a method based on the work of Sherman (Sherman 1980). The infiltration time 

constant is assumed to be proportional to the square root of the pressure differential between 

inside and outside, which in turn are dependent upon wind loading and temperature differences 

(stack effect). The infiltration time constant is 

1

2 2

2 2

S

E

S W

V

Q Q

τ

  =     + , 

where Vs is the total volume of the building.

The temperature-induced volume inflow rate depends on the absolute value of the temperature 

difference, ∆T (Grimsrud, Sherman, and Sonderegger 1983), and is described as

Q Af T s es = ∆ , 

where

AE is the effective leakage area = 0.00059x (floor area of the building). 

The simplified stack parameter is

1 /2

3

S

s

i

R gH f T+ = , 

where 

R is the ratio of vertical to total leakage set equal to 0.5, 

HS is the structure height, and 

Ti is the uniform internal temperature.

58 Air Dispersion Models
The infiltration from wind loading is dependent upon the wind speed at the building height, UH , 

and the sheltering of the building, and is given by

1

3 (1 ) Q AC R U w e sh = − H , 

where 

{ 0.24 sheltered

0.32 unsheltered Csh = .

Air Dispersion Models 59
5 MODELS FOR CALCULATING BLAST EFFECTS FROM VAPOR CLOUD 

EXPLOSIONS

5.1 BACKGROUND

As an airborne chemical travels downwind, it mixes with air. A cloud containing a flammable 

chemical within its flammability limits can ignite if it encounters a spark, flame, or other ignition 

source. The combustion reaction can propagate away from the source by one of two mechanisms: 

Deflagration reactions propagate by means of diffusion of reactive species through the cloud. 

Detonation reactions propagate through a reactive fuel-air mixture by means of a pressure wave 

that travels at the local speed of sound. Deflagrations propagate more slowly than detonations; 

however, in either case, the reaction can cause temperatures and pressures within the cloud to 

increase dramatically. Both detonations and deflagrations can generate pressure waves with sharp 

onsets and significant overpressures; a pressure wave capable of causing damage to individuals or 

structures is called a blast wave in ALOHA.

Most vapor cloud combustions are deflagrations that propagate slowly and do not produce blast 

waves; these are usually referred to as flash fires. For some highly reactive chemicals, the flame 

speed (the propagation speed) within part of the cloud is accelerated by turbulence caused by 

obstacles or confinement resulting in a fast deflagration or transition to detonation; either is 

referred to as an explosion. These events can generate blast waves; usually only a small part of the 

flammable cloud is involved, so the blast effects are limited. In rare events, a high-power triggering 

event such as condensed-phase explosive or confined vapor cloud explosion can set off the 

detonation of the entire flammable cloud. The American Institute for Chemical Engineers 

(American Institute of Chemical Engineers 1994) estimates that direct initiation of detonation 

requires approximately one million Joules. The blast wave from the detonation of a large 

flammable cloud can have far-reaching effects; they can extend well beyond the area affected by the 

thermal radiation.

Damage is associated with both the shape and magnitude of the blast wave; both change as the 

wave travels outward from the reacting cloud. Peak overpressure and impulse are commonly used 

to characterize the blast wave. Wiekema reports a correlation between lung damage and the 

combination of overpressure and impulse (Wiekema 1984). Most studies of condensed phase 

explosions correlate injury with overpressure only. Clancey explained that the relationship 

between overpressure and impulse for condensed phase explosions is unique, so a single parameter 

is sufficient to describe the blast wave (Institution of Chemical Engineers (Great Britain). North 

Western Branch. 1982). Injuries and damage to structures from vapor cloud explosions depend on 

both overpressure and impulse; however, as with condensed phase explosions, most correlations 

use overpressure only. This simplification seems to be driven by the lack of data rather than any 

unique relationship between overpressure and impulse for vapor cloud explosions.

ALOHA uses only peak pressure to characterize the damaging effects associated with a blast wave. 

Harm can be due to direct effects or indirect effects of the pressure wave. Direct effects include 

damage to pressure-sensitive organs like ear and lung. Indirect effects can result from glass 

fragments from broken windows, collapse of buildings, or debris that is accelerated by the blast 

60 Models for Calculating Blast Effects from Vapor Cloud Explosions
wave. ALOHA includes three LOCs that quantify indirect and direct effects. Glass windows can 

shatter at 1 psi over ambient; at 3.5 psi there is a significant risk of eardrum rupture and injuries of 

serious nature from flying debris; at 8 psi there is significant risk of ear and lung damage and 

indirect effects from the collapse of unreinforced buildings (Baker 1983). 

5.2 METHOD

ALOHA only models combustion reactions. ALOHA estimates the blast wave from unconfined vapor 

cloud explosions (fast deflagrations and detonations). Unconfined means that the cloud is not 

entirely or partially bounded by solid walls or ceilings. Confined vapor cloud explosions generally 

produce more damaging blast waves than unconfined or partially confined explosions.

The Baker-Strehlow-Tang (BST) model is the basis for the ALOHA overpressure calculation 

(Pierorazio et al. 2005); it uses non-dimensional, empirically derived blast curves to predict 

overpressure. The overpressure is based on the propagation speed of the flame front and the mass 

of fuel involved in the reaction. The basic principle of this method is that within the vapor cloud 

there are regions where physical structures can cause an acceleration of the flame front. These 

areas are characterized by the structure density using a parameter termed congestion. Flame 

speed is related to the chemical properties of the fuel, the level of congestion, and the nature of the 

ignition source.

Table 14. Baker-Strehlow-Tang flame speeds (Mach number). Mach 5.2 is used for deflagration to 

detonation transition (DDT).

 Low

congestion

medium

congestion

High 

congestion

High Reactivity 0.36 DDT DDT

Medium Reactivity 0.11 0.44 0.5

low Reactivity 0.026 0.23 0.34

A reactivity term is used to characterize the chemical properties of the fuel. Reactivity ratings used 

by Zeeuwen and Wiekema (Zeeuwen and Wiekema 1978) classify reactivity based upon chemical 

laminar burning velocity in the BST model (Woodward 1998). Low-reactivity chemicals have 

velocities less than 45 cm/sec. High reactivity applies to those chemicals with burn velocities 

greater than 75 cm/sec and anything in between is labeled medium reactivity. Some chemicals in 

the ALOHA database were classified using this criteria; however, most of the flammable chemicals 

in ALOHA have not been classified. In these cases, ALOHA uses medium reactivity, since few 

chemicals have a laminar burning speeds exceeding 75 cm/sec.

A congestion parameter is used to quantify the way small structures within the vapor cloud affect 

the flame speed. Congestion refers to the density of obstacles that generate turbulence. Obstacles 

of this nature are generally small, like a shrub, and do not impede the flame front. Larger objects, 

like a building, can impede the flame front, so they should not be considered obstacles for the 

Models for Calculating Blast Effects from Vapor Cloud Explosions 61
purposes of congestion. Greater turbulence allows the flame front to accelerate, thereby generating 

a more powerful blast wave. The experiments that form the basis for the BST model found the 

flame speed could be related to area blockage ratio (cross sectional area of the structures divided 

by area of the cloud) and pitch (distance between rows of structures). The experiments used small 

structures arranged in regular patterns. Three levels of congestion could be distinguished: low 

congestion with an area blockage ratios less than 10%; medium congestion with area blockage ratio 

between 10% and 40%; and high congestion with higher blockage ratios. Extrapolating the 

laboratory parameters to what is found in an accidental explosion is highly imprecise so ALOHA 

greatly simplifies the use of congestion. Guidance in ALOHA requires the user to determine 

whether significant congestion is present. A congested zone is defined as one with so many closely 

spaced obstacles that it is difficult or impossible to walk through it; for example, pipe racks in 

industrial facilities and some forested areas where the trees and branches are closely spaced may 

be characterized as congested zones. To err on the side of caution, congestion defined this way is 

correlated with the high-congestion flame speed in the BST model.

The source of ignition also affects the flame speed. In ALOHA, the user inputs the ignition source. If 

the source is designated as a detonation, the model assumes that the triggering event is of sufficient 

power to cause the entire reactive cloud to detonate.

ALOHA’s method for finding the normalized overpressure as a function of distance from the center 

of the explosive cloud is based on a set of empirically determined graphs (Pierorazio et al. 2005). 

BST reported normalized overpressure versus normalized distance from the center of the 

congested region with a different graph for different flame speeds. To implement these in ALOHA, 

the graphical data reported by BST were fit to functions of the form

0

1/

 ,

 ,

atm

x C

atm

P D if x x P

P else A B x

P

∆

= <

∆ = ⋅

where ∆P is the maximum overpressure and A, B, C, D, and x0 are constants. Table 15 gives values 

of these constants for various flame Mach numbers. 

Table 15. Curve fit constants for various Mach numbers for use in the BST method. 

 Mach 

0.2

Mach 

0.35

Mach 

0.7

Mach 

5.2

A 0.0335 0.1041 0.3764 0.2932

B 0.8359 0.8642 0.7439 1.399

C -1.1192 -1.0568 -1.2728 -1.1591

D 0.065 0.22 0.65 20

x0 0.35 0.32 0.3 0.16

62 Models for Calculating Blast Effects from Vapor Cloud Explosions
The normalized distance, x is defined as 

1/3 Patm x r

E

  =     , 

where

Patm is the atmospheric pressure and

r is the distance from the center of the exploding cloud.

The energy contributing to the blast wave is 

E ref H Mass = ⋅⋅ c , 

where

ref is a ground reflection factor, 

Hc is the heat of combustion of the fuel, and

Mass is the mass of fuel involved in the explosion. 

The ground reflection factor is set to 2 in ALOHA, consistent with the treatment by BST. It accounts 

for the reflection of the blast wave off the ground. ALOHA’s treatment is based on a simplified 

conceptual model of the explosive cloud. The cloud is treated as a hemisphere at ground level with 

a uniform concentration. Elevated clouds would have a smaller reflection factor.

The most significant difference between the method in ALOHA and the Baker-Strehlow-Tang model 

is the method for determining the mass of fuel in the explosion. In the BST method, areas of 

congestion within a flammable cloud define the mass of fuel contributing to an explosion. The 

flame front propagates from the point of ignition, accelerates within regions of congestion, and 

decelerates outside the region of congestion. Only the mass of the fuel within a congested area 

contributes to the explosion. A flammable cloud emanating from a single release may give rise to as 

many explosions as there are distinct congested areas. The flame speeds are defined by the levels 

of congestion within those regions. Outside the congested areas the flame speed is assumed to be 

so slow that no significant overpressure is generated.

ALOHA uses a different approach for determining the mass of fuel involved in the explosion based 

on the recommendations of the American Institute for Chemical Engineers (American Institute of 

Chemical Engineers 1994). AIChE suggests that the BST blast curves can be coupled with an air 

dispersion model for determining the mass of the explosive cloud, using all the fuel within 

flammable limits times an efficiency factor. An efficiency factor of 5% to 20% is recommended; 

ALOHA uses 20%. ALOHA deviates slightly from the AIChE recommendation by using the fuel 

within a concentration range between the upper explosive limit and 90% of the lower explosive 

limit. This minor variation was introduced in ALOHA to create another conservative bias in the 

hazard zone calculation; it was not based on standard practices, theory, or measurement. Gas 

concentrations above the upper limit are presumed to be too rich, and those below the lower limit 

too lean, to participate in the explosion. The explosion of the flammable cloud is approximated as a 

single uniform event; the flame speed is based on the average level of congestion within the cloud. 

If the explosion is triggered by a high-power source, termed a hard ignition in ALOHA, or the 

Models for Calculating Blast Effects from Vapor Cloud Explosions 63
average level of congestion indicates a transition to detonation, 100% of the mass of the flammable 

cloud is used and Mach 5.2 is used for the flame speed.

The center of the explosive cloud is equated with the center of mass of flammable cloud in ALOHA. 

For non-steady-state releases the location of the center and mass of the flammable cloud changes 

with time. Users may choose the time of ignition; ALOHA then finds the flammable mass and center 

of the cloud, and generates overpressures as a function of distance. ALOHA can also compute all 

possible explosions from a single release and show the composite overpressure threat zone if the 

ignition time is not specified.

64 Models for Calculating Blast Effects from Vapor Cloud Explosions
6 THERMAL RADIATION AND FLAMMABLE AREA MODELS

6.1 BACKGROUND

ALOHA models flame hazards associated with some common scenarios involving the combustion of 

liquids and gases. Fire scenarios involving chemicals with flashpoints exceeding 300°F cannot be 

modeled using ALOHA. Combustion scenarios are divided into two types in ALOHA: those that 

involve a fuel mixed with air to form a cloud with concentrations within the flammability range, and 

those involving an overly-rich core of fuel that burns at its outer edges (this includes pool fires 

where the rich core is the pool itself). ALOHA uses different methods for estimating the threat 

zones for these two types of scenarios. 

Fires that burn on the surface of a rich core can burn for a long time. The hazard from thermal 

(infrared) radiation can extend well beyond the boundaries of the fire itself. Three models in 

ALOHA deal with these types of fires: Fireballs occur when a tank containing a flammable liquid 

explodes due to overpressurization and immediately ignites, commonly referred to as a BLEVE (the 

fire occurs at the surface of the fireball); Jet Fires occur when a flammable gas escapes from a pipe 

or tank (the fire occurs at the exit and burns at the edges of the rich core); and Pool Fires occur 

when a flammable liquid in a pool ignites and burns directly over the pool. There are two common 

empirical modeling approaches for these types of fires: point source models in which some fraction 

of the combustion energy is converted to thermal radiation and radiates from the center of the fire, 

and solid flame models in which the size and shape of the flame surface is found and radiation 

emitted from its surface are computed.

ALOHA employs solid flame models to compute thermal radiation hazards from fireballs, jet fires, 

and pool fires. In these three scenarios, the flux of thermal radiation emitted from the surface of the 

flame is computed, and the radiation impinging upon a distant target is found using

q EF =⋅⋅τ , 

where 

q is the thermal radiation flux incident on a vertical surface (W m-2), 

E is the thermal radiation energy flux at the surface of the fireball (W m-2), 

F is the geometric view factor, and 

τ is the transmissivity of the atmosphere to thermal radiation.

Fuels that have been mixed with air before ignition burn quickly; the duration of the fire and the 

duration of the generated thermal radiation is generally quite short. Since the effects of thermal 

radiation scale with duration, the damaging effects usually do not extend very far beyond the 

boundaries of the flame itself. ALOHA address the hazard associated with fires that occur when a 

vapor cloud disperses downwind and forms a flammable mixture with air, ignites, and burns 

through the cloud by modeling the flammable area. ALOHA does not explicitly model the thermal 

radiation associated with flash fires.

Thermal Radiation and Flammable Area Models 65
6.2 LEVELS OF CONCERN FOR THERMAL RADIATION

The effects of thermal radiation are a function of the energy flux and the duration of exposure. For 

fires with expected durations exceeding about 30 seconds, the LOC is based upon the thermal

radiation generated by the flame. Eisenberg analyzed data from nuclear weapons tests to 

determine the relationship between the intensity of thermal radiation and the probability that an 

exposed individual would be injured or killed (Eisenberg, Lynch, and Breeding 1975). He reported 

that the probability of a fatality is generally proportional to the duration of exposure (td) times the 

incident radiation flux (I) raised to the 4/3 power. For non-lethal effects, a power law of 1.15 fits 

the data better. The threshold for first degree burns was found when the exposure (in the form I1.15 

td) is 550,000 (W/m2)1.15 sec. Mudan reviewed Eisenberg's work as well as other studies that dealt 

with radiation sources that are less intense, but have longer duration (National Fire Protection 

Association 1995). The exposure laws described by Eisenberg obviously fail for very low, longlived radiation sources; Mudan reports that an incident flux of 1.7 kW/m2 will not even cause pain, 

regardless of exposure time. The apparent breakdown of the exposure law reported by Eisenberg 

is also reflected in the Safety Standards for Liquid Natural Gas Facilities in Title 49 of the Code of 

Federal Regulations. In calculating the radiation effects of pool fires, long-lived events, the Title 49 

states that the radiation flux in an area of public assembly should be less than 5kW/m2, regardless 

of exposure time (United States Office of the Federal Register). 

Burning pools, jets, and fireballs associated with a BLEVE can burn with sufficient longevity to 

create a persistent thermal radiation source that can cause harm far from the flame front itself. The 

impact on people depends on the both the duration of exposure and the intensity of the radiation. 

ALOHA computes the energy flux associated with the radiation as a function of distance and the 

duration of the fire and reports them separately. Even though dose values are associated with 

damage, energy flux values are used as LOCs in ALOHA since exposure time is a function of both the 

duration of the fire and the movement of individuals seeking some protection. ALOHA’s LOCs are 

consistent with the recommendations of Mudan and Croce (National Fire Protection Association 

1995): 10kW/m2 is a fatality threshold; 5kW/m2 can cause second degree burns on unprotected 

skin; and 2kW/m2 can cause pain. ALOHA refers to exposure durations of 60 seconds or less for 

these effects. The duration of the fire is included in the text summary screen of the ALOHA output 

allowing the user to adjust the flux levels to conform to the fire duration time.

6.3 BLEVE-FIREBALL

The BLEVE-Fireball model in ALOHA is a solid flame model based on studies of fireballs resulting 

from BLEVEs (boiling liquid expanding vapor explosions) involving flammable gases liquefied 

under pressure and stored at ambient temperatures, such as LPG (American Institute of Chemical 

Engineers 1994). In a common course of events, a fire impinges upon the tank, thermally stressing 

the tank and causing the internal pressure to rise. Pressure relief valves fail to adequately relieve 

the pressure and the tank explodes from the combination of heat and pressure. The contents are 

almost instantaneously released and quickly flash boil as they experience a drop in pressure. Much 

of the fuel, both liquid droplets and gas, is thrown into the air and ignites. The bulk of the burning 

mass is too rich to burn, but the fire burns at the surface where sufficient air can mix with the fuel. 

The resulting fireball burns for tens of seconds and often lifts into the air.

66 Thermal Radiation and Flammable Area Models
For a BLEVE scenario ALOHA assumes a near-instantaneous rupture and defaults to an assumption 

that the entire contents of a tank contribute to the fireball. However, in cases where the initial tank 

pressure is relatively low the user has an option of specifying the fraction of the contents that 

become part of the fireball. Based on the mass of fireball, the maximum diameter of the fireball is 

computed, approximating the fireball as a spherical object with its surface just touching the ground. 

(The fireball’s diameter is a function of time; the maximum diameter is used in ALOHA to insure 

that hazard zones are not underestimated.) The flux of thermal radiation emitted from the burning 

surface is computed, and the radiation impinging upon a distant target is found. When the user 

specifies that less than 100% of the contents of the tank are in the fireball ALOHA assumes the 

remaining fuel forms a pool fire, and the pool fire model is used to find an alternate threat zone for 

thermal radiation. A limit of 5000 metric tons is employed as the maximum amount of chemical 

that can be modeled. This limit is of the order of the largest historical single BLEVE incident. 

Though the probability of explosive rupture and fireball production is much lower for nonpressurized tanks, ALOHA extends the BLEVE-Fireball model to gases liquefied by refrigeration and 

volatile liquids with flash points below 300°F. Explosions of tanks containing pressurized gases can 

also generate fireballs of comparable size and duration (Zalosh and Weyandt 2005). However, 

ALOHA does not allow users to model the thermal radiation from fireballs associated with 

pressurized gases.

6.3.1 EMISSIVITY

AIChE states that 350 kW/m2 is a reasonable emissive energy flux for large fireballs involving 

hydrocarbon fuels (American Institute of Chemical Engineers 1994). ALOHA applies this to a 

broader range of chemicals and sizes, but adjusts this value by multiplying by the ratio of the heat 

of combustion of the chemical divided by the heat of combustion of propane, and described as

,

350,000 c

c propane

h E

h

  ∆ =   ∆ 

, 

where 

∆hc is the heat of combustion of the chemical (J kg-1), and

propanec h∆ , is the heat of combustion of propane (J kg-1).

Thermal Radiation and Flammable Area Models 67
6.3.2 VIEW FACTOR

The geometric view factor is a function of the size of the fireball and distance between a receptor 

and the fireball. The size depends upon the amount of the fuel contributing to the fireball. The user 

may choose to let ALOHA use the entire contents of the tank as the mass of the fireball, or have 

ALOHA calculate the fraction of the tank contents that contribute to the fireball based on the 

pressure or temperature of the tank at the time of rupture. If tank pressure is specified, the 

temperature at the time of rupture, T , is found using the Clausius-Clayperon equation, based on 

the assumption that two phases are present at equilibrium in the vessel, and described as

1

0

1 ln c

B V

R P T

T HP

−

        = −             ∆    

, 

where 

TB is the normal boiling point, 

Rc is the gas constant (8.3144 J K-1 mole-1),

∆HV is the heat of vaporization (J mole-1), 

0 P standard pressure (101,000 Pa), and

P is the rupture pressure.

From the temperature, the fraction of liquid that flashes upon rupture is calculated. For an 

isenthalpic process the fraction that flashes is given by

p b ( )

v

CTT

f H

− = ∆ , 

where 

T is the chemical temperature at tank failure (K), 

Tb is the ambient boiling point (K), 

Cp is the specific heat capacity at constant pressure (J mole-1 K-1), and 

∆Hv is the heat of vaporization (J/mole).

This somewhat overestimates the fraction that flashes in an adiabatic (isentropic) process, but 

provides a reasonable approximation. 

An empirical correlation indicates that mass of chemical participating in the fireball is about three 

times the mass of fuel that flash boils upon rupture, limited to the total mass of the contents of the 

tank (Hasegawa and Sato 1977). 

ALOHA adopted the recommendation of Roberts for computing the diameter of a fireball (Roberts 

1982): 

( ) ( ) 1/3

max D meters mass kg = 5.8 . 

68 Thermal Radiation and Flammable Area Models
The geometric view factor for a vertical surface, F , was given by AIChE (American Institute of 

Chemical Engineers 1994) as

2

max

max 3/2 2

2 max

2

2

2

D

x

D F for x

D

x

        = >           +      

, 

where x is the distance between a receptor and a point at ground level directly below the center of 

the fireball.

6.3.3 TRANSMISSIVITY

The transmissivity of the atmosphere to thermal radiation, τ , depends upon the distance between 

the emitting and receiving surfaces, x . It is sensitive to fog, rain, smoke, carbon dioxide, and water 

vapor in the atmosphere. ALOHA only accounts for water vapor. By neglecting the other factors, 

ALOHA overestimates the transmitted radiation in cases where the other attenuating phenomena 

occur. Transmissivity of the atmosphere to thermal radiation, τ , is well studied. ALOHA uses the 

formula reported by Cook et al. (Cook, Bahrami, and Whitehouse 1990), 

10 1.389 0.135log ( ) P xw τ = − . 

ALOHA uses the formula reported by Thibodeaux for the partial pressure of water in the 

atmosphere (Thibodeaux 1979), 

5431.3 99.89 exp 21.66 100

H

w

a

R P

T

  =   −

 

, 

where 

Ta is the ambient air temperature and 

RH is the relative humidity.

6.3.4 DURATION

The duration of the fire, burn t , is not used in the computation of the hazard zone, but is displayed. 

ALOHA uses the formula recommended by the Dutch Committee for Prevention of Disasters (Duiser 

1992), which was based upon averages obtained from reported literature values, 

0.26 (sec) 0.852 ( ) burn t = mass kg . 

Thermal Radiation and Flammable Area Models 69
6.4 JET FIRES

The jet fire analysis in ALOHA is designed to address the thermal radiation hazards associated with 

gases and aerosols released from pressurized tanks and pipes which ignite before the vapors 

disperse downwind. Jet fires differ from flash fires in that they completely burn immediately upon 

release at the surface of a fuel-rich core. Jet fires differ from fireballs in that jet fires are associated 

with sustained releases, while fireballs are associated with an explosive tank rupture due to 

overpressurization.

The Jet Fire model in ALOHA can be applied to an upward vertical jet release: a pipe oriented 

vertically or a hole at the top of a tank. The method in ALOHA is based on an empirical solid flame 

model developed by Shell Research (Chamberlain 1987). Fuel released from the pipe or tank 

expands, mixes with air, and burns on its surface emitting intense thermal radiation that

propagates outward. The thermal energy incident upon a distant target is a product of the 

emissivity of the flame surface, the geometric view factor, and the transmissivity of the atmosphere 

to thermal radiation.

70 Thermal Radiation and Flammable Area Models
Figure 3. Source release for jet fire flowchart.

Thermal Radiation and Flammable Area Models 71
6.4.1 EMISSIVITY

From Cook et al (Cook, Bahrami, and Whitehouse 1990), the emissive power of the flame, E, is 

calculated as

rad c f Qh E

A

∆ = , 

where 

Q is the mass discharge rate (kg s-1),

∆hc is the heat of combustion (J kg-1), and

A is the surface area of the flame (m2). 

From Chamberlain (Chamberlain 1987), the fraction of heat radiated from the flame surface is 

0.21 exp( 0.00323 ) 0.11 rad MW j fC u = −+ . 

Where CMW is a correction factor introduced by Cook et al (Cook, Bahrami, and Whitehouse 1990) 

equal to unity if the molecular weight is less than 21 g mole-1, the square root of the molecular 

weight divided by 21 if the molecular weight is between 21 g mole-1 and 60 g mole-1, and equal to 

1.69 if the molecular weight is greater than 60 g mole-1. 

The gas velocity in the expanding jet is 

g cj

j j

gk

R T

u M

W

γ =

and

( ) 2

2

2 1

s

j

g j

T T

γ M = + − , 

where Ts is temperature inside vessel or at exit of pipe.

For unchoked flow, the Mach number of the expanded jet, M j , is calculated by 

( )

1

1 2

2 2 1 2( 1) 1

( 1)

g

j

g

F

M γ

γ

 

  +− − = −

 

and

5

2 3.6233 10 s

o g gk

Q T F

d Wγ

− = ⋅ , 

72 Thermal Radiation and Flammable Area Models
where 

o d is the diameter of the orifice,

Q is the mass release rate,

g γ is the ratio of specific heats,

Wgk is the molecular weight (kg mole-1), and

Rc is the gas constant (8.3144 J K-1 mole-1).

6.4.2 VIEW FACTOR

The flame surface is approximated as a frustum of a cone tilted by the wind.

The effective source diameter is the throat diameter of an imagined nozzle from which air at normal 

ambient density issues at the gas mass flow rate and exit velocity. The effective source diameter for 

a gas is

j

s o

air

D d ρ

ρ = , 

where 

o d is the diameter of the exit orifice, and

j ρ is the density of the gas.

For choked flow the Mach number is

( )

( )

( )

1

1 2

1

g

g c

g

o

j

g

P

P

M

γ

γ

γ

γ

−

 

+ −     = + , 

where 

Po is atmospheric pressure,

Pc is static pressure at the exit orifice, and

2 3.6713 c

c

o g gk

Q T P

d Wγ = , 

2

1

s

c

g

T T

γ = +

where Ts is the temperature inside the vessel or at the exit of the pipe.

Thermal Radiation and Flammable Area Models 73
The jet expands to atmospheric pressure at a plane downstream of the exit hole with the plane 

acting as a virtual source of diameter, d j . Then, 

j

s j

air

D d ρ

ρ = and

4 4 5 3.6233 10 c j j

j

j j o j g gk j g gk

Q Q R T Q T

d

πρ π γ u PM W M Wγ

− = = = ⋅ , 

where j u is the velocity of the gas in the expanded jet.

These formulas were modified slightly for the pipeline jet fire and two-phase release scenarios. 

ALOHA assumes that the gas expands adiabatically in the last 200 pipe diameters in the pipeline 

release. It exits at atmospheric pressure and therefore the effective source diameter, Ds for the 

choked option reduces to that for the unchocked option given earlier. For two-phase release, 

ALOHA uses a modification of the formula in Cook et al. (Cook, Bahrami, and Whitehouse 1990), 

1

4

2

j v

s j

air

D d ρ ρ

ρ

  =    

, 

where ρv is the pure vapor density. 

The modification of the Cook formula was necessary to insure that it would reduce to the proper 

algorithm when the two-phase case reduced to the pure gas scenario.

For a tilted jet, Kalghatgi (Kalghatgi 1983) showed in laboratory experiments that the flame length 

reduces as the jet is tilted into the wind. Chamberlain uses Kalghatgi’s empirical fit equation to 

determine the flame length, LB (Kalghatgi 1983) as

( ) 3 105.4 1 6.07 10 90 L D Bs j θ − = −⋅ −    . 

The flame length in still air is

( ) ( ) 3 0.51exp 0.4 0.49 1 6.07 10 90

B

Bo

j

L L

v θ − =

  − + −⋅ −     

. 

The angle, α, between the orifice axis and the flame depends on the velocity ratio, is

j

V R

u = , 

where V is the wind speed.

74 Thermal Radiation and Flammable Area Models
If R ≤ 0.05, then

( )( )( ( ))

( )

8000 90 1 exp 25.6 Bo j

Bo

R L R

L

ξ θ

α

ξ

+ − −− = , 

and if R > 0.05, then

( )( )( ( ))

( )

1726 0.026 134 90 1 exp 25.6 Bo j

Bo

RL R

L

ξ θ

α

ξ

  − + − −−   = and

( )

1

3

Bo Bo 2 2

s j

g L L

D u ξ

  =      

. 

The flame-lift off, b, is the distance along the axis of the cone, and is calculated as

0.015 B b L = for aerosols, and sin

sin B

K b L α

α = for gases. 

K has been correlated with experimental data with a best fit of

20 0.185 0.015 R K e− = + . 

The frustum length is given by the geometrical relationship between RL , LB , α and b, as

( ) 2 22 sin cos R Lb b L B =+ − α α . 

There appears to be a difference in the formula for the width of the frustum base as it appears in 

Chamberlain’s paper when compared to Lees (Lees 2001) presentation of Chamberlain’s formula. 

Based on sample calculations, we determined to use the Chamberlain’s version (Chamberlain p. 

303) with the width of the frustum base., W1 , as

( ) ( ( ) )

1

2

1

1 13.5exp 6 1.5 1 1 exp 70

15

CR a

s s

j

WD R D ρ ξ

ρ

          = − + −−       −                

, 

where

C R = −+ 1000exp 100 0.8 ( )

Thermal Radiation and Flammable Area Models 75
and the Richardson number, ξ( ) Ds , based on Ds is

( )

1

3

s s 2 2

s j

g D D

D u ξ

 

=      

. 

The Chamberlain formula for the width at frustum tip, W2, is

W L 2 = −+ − − B (0.18exp 1.5 0.31 1 0.47 exp 25 ( ) R )( ( ) R ). 

The surface area of the flame, A, is calculated as

( ) ( )

1

2 2

2 2 2 2 1

1 2 12 42 2 L

W W A W W WW R π π     − = ++ + +        

. 

The view factor F is defined by Sparrow and Cess (Sparrow and Cess 1978) as

2

cos cos

j i j

i jj i

A dA A

j

dA dA

dF

A r

β β

π − = ∫ , 

where

Aj is the area of the radiating surface; 

i dA is the receiving element; 

β i is the angle between the normal to the receiving element and the line between the element and 

the radiating surface; 

β j is the angle between the normal to the radiating surface at a point and the line between that 

point and the receiving element; and

r is the distance between the point on the radiating surface and the receiving element.

For a radiating surface of area Ai and a receiving element of area dAj , we can let

q' = incident radiation intensity per unit area, and

E'= emissive power per unit area,

so that

' ' ' j j

ii i i

q EF EAF A

q E F

dA dA dA dA

τ τ

τ ⋅ ⋅ ⋅ ⋅⋅   = = = =⋅ ⋅ ⋅     . 

Thus, it is actually ∫Aj

jji

r

dA

2

coscos

π

ββ that we need to calculate.

76 Thermal Radiation and Flammable Area Models
We calculate this integral numerically by dividing the flame surface into 1800 "tiles" (40 radial 

divisions x 25 axial divisions of the conical surface, plus 400 tiles for each circular "cap"). The value 

of the integrand is calculated at the center of each tile, and those values are added to produce an 

estimate of the integral. This process is carried out for three orthogonal orientations of the 

receiving surface, producing view factors 1f , 2 f , and 3f . The maximum view factor (over all 

orientations of the receiving surface) is then calculated as

2

3

2

2

2

1 ++= ffff . 

(For a true view factor, the integral omits portions of the radiating surface where β j cos <0. But 

since 1f , 2 f , and 3f are used to calculate the maximum view factor, portions where β j cos <0 are 

not excluded.)

6.5 POOL FIRES

There are 3 release scenarios that can be coupled with the Pool Fire model: The user may choose to 

model a pool of constant area that is not associated with a tank release; the Pool Fire model can also

be coupled with a model that estimates the pool formation dynamics when a tank of chemical is 

leaking; and the Pool Fire model is automatically applied to any fuel that pools during a BLEVE

scenario. In all cases, the pool is assumed to be circular, uniformly thick, and on a level surface. The 

pool temperature is approximated as a constant and set to either the initial pool temperature or the 

initial tank temperature. A 200-meter diameter limit applies in all cases.

A solid flame model is used to calculate the thermal radiation from pool fires. The size of the pool 

can be set by the user, or ALOHA will calculate the dynamic area and volume from the release of a 

liquid from a tank. The flames rising from the pool form a tilted cylinder; the surface of the cylinder 

radiates thermal radiation. Burn rate, flame height, angle of tilt, and emission of radiation from the 

surface are based on empirical correlations.

The thermal energy incident upon distant target is the product of the thermal radiation energy flux 

at the surface of the flame, the geometric view factor, and the transmissivity of the atmosphere to 

thermal radiation.

6.5.1 EMISSIVITY

The average emissive power per unit area, E, of the cylinder surface is estimated using the approach 

of Moorhouse and Pritchard (Moorhouse and Pritchard 1982), 

1 4

rad c f hm E

h

d

∆ =     +  

 , 

Thermal Radiation and Flammable Area Models 77
where 

h is the flame length (m), 

d is the diameter of the pool (m), 

∆hc is the heat of combustion (J kg-1), 

m is the mass burn rate per unit area (kg m-2 s-1), and 

frad is the fraction of energy released as thermal radiation.

ALOHA approximates the fraction of energy radiated, frad , as a constant 30% as suggested by 

Roberts (Roberts 1982). 

The mass burn rate per unit area, m , is calculated from ratios of the heats of combustion and 

vaporization. Mudan states that the following correlation for the mass burning rate fits a wide 

range of fuels including liquefied gases (Mudan 1984): 

0.001 c

v

h

m

h

∆ = ⋅ ∆  . 

ALOHA uses a modified version of this with a correction for temperature, given as

( ) 0.001 c

v pb

h

m

h cT T

∆ = ⋅ ∆+ −  , 

where 

p c is the specific heat capacity (J kg-1 K-1),

Tb is the ambient boiling temperature (K),

∆hv is the heat of vaporization (J kg-1), and

T is the pool temperature (K).

6.5.2 VIEW FACTOR

The flame of the pool fire is assumed to be an optically dense tilted cylinder which intersects a 

plane parallel to the ground in a circle. The flame length, h, is estimated by a modification of the 

Thomas formula for flame length (Thomas 1963). Let u* be a non-dimensional wind velocity 

defined as

1/3

* a u u

gmd

  ρ = ⋅  ⋅ ⋅    . 

The flame length is given by

( )

0.67

0.21 * 55

a

m h d u

ρ g d

  −

=⋅ ⋅  ⋅ ⋅ ⋅  

 , 

78 Thermal Radiation and Flammable Area Models
where

ρa is the density of ambient air (kg m-3). 

The angle of tilt is based on formulas from the American Gas Association (American Gas Association 

1973), and is given by

*

1 *

*

0 1,

1 cos 1.

if u

if u

u

θ

θ −

= ≤

= >

The view factor F is defined by Sparrow and Cess (Sparrow and Cess 1978) as

2

cos cos

j i j

i jj i

A dA A

j

dA dA

dF

A r

β β

π − = ∫ , 

where

Aj is the area of the radiating surface; 

i dA is the receiving element; 

β i is the angle between the normal to the receiving element and the line between the element and 

the radiating surface; 

β j is the angle between the normal to the radiating surface at a point and the line between that 

point and the receiving element; and

r is the distance between the point on the radiating surface and the receiving element.

For a radiating surface of area Ai and a receiving element of area dAj , we can let

q' = incident radiation intensity per unit area, and

E'= emissive power per unit area,

so, 

' ' ' j j

ii i i

q EF EAF A

q E F

dA dA dA dA

τ τ

τ ⋅ ⋅ ⋅ ⋅⋅   = = = =⋅ ⋅ ⋅     . 

Thus, it is actually ∫Aj

jji

r

dA

2

coscos

π

ββ that we need to calculate.

We calculate this integral numerically by dividing the flame surface into 1000 "tiles" (40 radial 

divisions x 25 axial divisions). The value of the integrand is calculated at the center of each tile5, 

and those values are added to produce an estimate of the integral. This process is carried out for 

5 For tiles at the base of the flame surface, the point used is on the ground, not the tile center. This permits 

greater accuracy when calculating view factors for observers near the surface.

Thermal Radiation and Flammable Area Models 79

 
three orthogonal orientations of the receiving surface, producing view factors 1f , 2 f , and 3f . The 

maximum view factor (over all orientations of the receiving surface) is then calculated as

2

3

2

2

2

1 ++= ffff . (For a true view factor, the integral omits portions of the radiating surface 

where β j cos <0. But since 1f , 2 f , and 3f are used to calculate the maximum view factor, portions 

where β j cos <0 are not excluded.)

6.5.3 POOL DYNAMICS

The maximum diameter of the pool is used in all the above calculations. In cases where the pool 

size is dynamic because chemical is released during a BLEVE or from a leaking tank, ALOHA uses 

the same tank release and pool growth methods as those used for non-thermal hazards. 

For tank release rates, ALOHA uses Bernoulli’s equation to compute QT(t) (kg s-1), the mass flow of 

liquid from the hole, 

( ) ( ) 2 Qt CA P P T = dis f h a l − ρ , 

where 

Cdis is the discharge coefficient (0.61), 

Af is the flow area (m2), 

Ph is the pressure of the liquid in the tank at the height of the hole (Pa),

Pa is the ambient atmospheric pressure (Pa), and

ρl is the density of the liquid in the tank (kg m-3), which is assumed to be uniform throughout the 

tank.

ALOHA estimates the flow area, Af (m2), differently depending on whether or not the liquid surface 

intersects the hole. The flow area is

( ) ( )

surface above hole,

/ surface intersects hole, 

h

f

hlh

A

A t

A h ζ

 = 



where 

hl is the height of the liquid above the bottom of the hole (m), and

h ζ is either the hole height or diameter (m), depending on whether the hole is rectangular or 

circular.

80 Thermal Radiation and Flammable Area Models
When the hole is below the liquid surface, ALOHA estimates the pressure at the hole as the sum of 

the pressure within the gas void at the top of the tank and the pressure exerted by the column of 

liquid above the bottom of 

( )

surface above the hole,

surface intersects the hole,

cs l l

h

a ll

e hg

P t

p hg

ρ

ρ

 +  = 

 +

where cs e , the saturated vapor pressure of the chemical (Pa), is assumed to be the pressure within 

the void space.

Because ALOHA assumes that the sequence of pressure drop, evaporation of liquid into the void 

space, and temperature change caused by evaporative cooling occurs rapidly during each new time 

step, it computes a new temperature and saturated vapor pressure for the vapor in the void space. 

As the liquid height above the hole bottom decreases, the pressure at the hole decreases until it 

eventually equals atmospheric pressure. According to the physical model represented by ALOHA’s 

algorithms, tank outflow then stops until an air bubble is ingested back into the tank, equalizing 

tank and atmospheric pressure and allowing flow to continue as a series of gushes. To approximate 

the effect of this process mathematically, once ALOHA’s computed pressure at the hole reaches 1.01

Pa, it is held constant at that value (Belore and Buist 1986). 

If liquid spills into a puddle faster than the puddle burns, the puddle will grow deeper and will 

spread outward under the influence of gravity. The puddle is approximated with uniform depth 

and temperature. ALOHA estimates the change in puddle radius, p r , as

1 2

p p

p l

dr gm

dt r πρ = , 

where mp is the dynamic mass of the puddle, g is the acceleration of gravity, p r is the puddle 

radius, and ρl is the density of the liquid in the puddle. ALOHA uses a constant factor of 2 to 

account for the fact that the inertia of the spreading liquid is only a fraction of the inertia of the 

whole liquid-pool mass moving with the acceleration at its leading edge (Briscoe and Shaw 1980). 

When the puddle depth drops below 0.5 cm, ALOHA stops spreading; the area is held constant until 

the liquid has completely burned away.

6.6 FLASH FIRES AND FLAMMABLE AREA

The Flammable Area analysis in ALOHA is designed to address the hazards associated with a fire or 

explosion of a cloud composed of a mixture of flammable chemical vapors and air in proportions 

that will support the propagation of a flame. The flammable area represents the region where an 

ignition source can lead to a flash fire or vapor cloud explosion, and the area where the fire can 

occur. While a vapor cloud explosion can generate a damaging shock waves outside the flammable 

area, the fire hazard associated with a flash fire usually does not.

Thermal Radiation and Flammable Area Models 81
Typically with flash fires of premixed clouds, the thermal radiation generated is highly transient. 

ALOHA does not explicitly model the thermal radiation associated with a flash fire, rather uses the 

approach of the US EPA (United States. Environmental Protection Agency. Chemical Emergency 

Preparedness and Prevention Office 1999) and assumes that the threat zone from a flash fire is 

closely related to the lower flammability or explosive limit threat zone (ground level concentration 

contour) for the cloud. Based upon recommendations of the project external review team the 

choice was made to use 60% of the lower flammability limit as the level of concern in defining this 

threat zone. This value reflects the fact that the concentration calculation in ALOHA involves some 

time averaging. Therefore, there is the possibility that a location with an average concentration 

below the LEL may have a fluctuating concentration that sometimes exceeds the LEL.

Modeling the thermal hazard associated with flash fires employs source strength models and air 

dispersion models that are part of the toxic air plume analysis in ALOHA. Any location where the 

gas concentration exceeds 60% of the LEL at any time following the release is included in the 

hazard zone.

82 Thermal Radiation and Flammable Area Models
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U.S. DEPARTMENT OF COMMERCE 

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Kathryn D. Sullivan, Ph.D.

Acting Under Secretary of Commerce for Oceans and Atmosphere, and Acting 

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