Fire Ball

Technical documentation for the BLEVE-FireBall consequence model — thermal radiation, probit analysis, domino effects, and fatality estimation

1. Introduction and Physical Phenomenon

1.1 BLEVE and FireBall

A BLEVE (Boiling Liquid Expanding Vapor Explosion) occurs when a pressurized vessel containing a liquid at a temperature above its atmospheric boiling point fails catastrophically. The sudden depressurization causes instantaneous flash vaporization of a significant fraction of the liquid, generating a rapid two-phase release. If the substance is flammable and an ignition source is present, the resulting combustion produces a characteristic fireball.

The fireball is a luminous, approximately spherical mass of burning vapor/aerosol that rises due to buoyancy. It produces intense thermal radiation over a short duration (typically seconds to tens of seconds), capable of causing:

Burns (first and second degree) to exposed persons
Fatalities from lethal thermal dose
Domino effects through failure of nearby equipment and vessels

1.2 Industrial Context

High-severity scenario

BLEVE/fireball events are among the most severe accident outcomes in industrial QRA. Historical incidents such as San Juan Ixhuatepec (Mexico, 1984) and the Feyzin refinery explosion (France, 1966) demonstrate their catastrophic potential.

LPG Storage & Transport

Propane and butane terminals, tank farms

Petroleum Refineries

Pressurized hydrocarbon vessels and process units

Chemical Plants

Flammable liquid storage under pressure

Rail & Road Transport

Tank cars and road tankers carrying pressurized flammable liquids

1.3 Scope of This Model

This model calculates:

Fireball geometry (diameter, height, duration)
Thermal radiation intensity at any distance
Distance to a specified radiation threshold (inverse problem)
Thermal dose and probit-based probability of burns/fatalities
Domino effect probability for nearby equipment (Cozzani method)
Population fatalities using concentric ring analysis

2. Calculation Sequence

Cargando diagrama...

The BLEVE-FireBall calculation follows these stages:

Geometry Calculations — Compute maximum diameter ( $D_{max}$ ), fireball duration ( $t_{fb}$ ), and center height ( $H_{fb}$ ) from the released fuel mass.

Combustion Parameters — Calculate mass burning rate ( $\dot{m}''$ ) and Surface Emissive Power ( $SEP_{max}$ ) from fuel properties.

Atmospheric & View Factor — For each distance, compute surface distance ( $X_{surface}$ ), atmospheric transmissivity ( $\tau_{atm}$ ), and geometric view factor ( $F_{view}$ ).

Thermal Radiation — Calculate incident radiation using three available models (solid plume, point source, empirical).

Inverse Distance — Solve for the distance at which radiation equals a target threshold using Newton-Raphson iteration.

Thermal Dose & Probit — Compute thermal dose and convert to burn/fatality probabilities via probit functions.

Domino Effect — Estimate time-to-failure for nearby vessels using Cozzani correlations.

Fatality Estimation — Integrate fatality probability over concentric rings to estimate total casualties.

3. Principal Equations

3.1 Geometry: Diameter, Duration, Height

Maximum Fireball Diameter

D_{max} = 5.8 \cdot M^{1/3} \tag{2.1}

Symbol	Description	Unit
$D_{max}$	Maximum fireball diameter	m
$M$	Mass of flammable fuel released	kg

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., Eq. 2.2.32, p. 207

Alternative correlation

The TNO Yellow Book (CPR 14E) provides: $D_{max} = 2 \times 3.24 \cdot M^{0.325}$ (Eq. 6.119), but is not used as the primary method.

The initial diameter at ground level accounts for the expansion phase before lift-off:

D_{ground} = 1.3 \cdot D_{max} \tag{2.2}

Fireball Duration

The duration depends on whether the fireball is momentum-dominated or buoyancy-dominated:

t_{fb} = \begin{cases} 0.45 \cdot M^{1/3} & \text{if } M < 30{,}000 \text{ kg} \\ 2.6 \cdot M^{1/6} & \text{if } M \geq 30{,}000 \text{ kg} \end{cases} \tag{2.3/2.4}

Symbol	Description	Unit
$t_{fb}$	Fireball duration	s
$M$	Mass of flammable fuel	kg

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., pp. 207-208

Fireball Center Height

H_{fb} = 0.75 \cdot D_{max} \tag{2.5}

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., p. 211

Height factor

Kakosimos proposes $H_{fb} = 1.0 \cdot D_{max}$ , placing the fireball higher. The CCPS factor of 0.75 is used by default — a more conservative (closer to ground) assumption that yields higher radiation at ground-level receptors.

3.2 Combustion: Burning Rate and SEP

Burning Rate

$\dot{m}'' = \frac{M}{\pi \cdot D_{max}^{2} \cdot t_{fb}}$

Symbol	Description	Unit
$\dot{m}''$	Burning rate	kg/(m $^2$ s)
$M$	Fuel mass	kg
$D_{max}$	Maximum diameter	m
$t_{fb}$	Fireball duration	s

Reference: Kakosimos, Safety in Chemical Engineering, p. 102

Surface Emissive Power (SEP)

$SEP_{max} = f_s \cdot \dot{m}'' \cdot \Delta H_c$

Symbol	Description	Unit
$SEP_{max}$	Maximum surface emissive power	kW/m $^2$
$f_s$	Radiation fraction	dimensionless (0.2–0.4)
$\dot{m}''$	Burning rate	kg/(m $^2$ s)
$\Delta H_c$	Heat of combustion	kJ/kg

Reference: TNO Yellow Book (CPR 14E) and Kakosimos p. 102

3.3 Atmospheric Transmissivity

Surface distance from the fireball to the ground-level receptor at horizontal distance $x$ :

$X_{surface} = \sqrt{H_{fb}^{2} + x^{2}} - \frac{D_{max}}{2}$

Partial vapor pressure of water:

p_a = 1013.25 \cdot RH \cdot \exp\left(14.4114 - \frac{5328}{T_a}\right) \tag{2.8}

Symbol	Description	Unit
$p_a$	Partial pressure of water vapor	Pa
$RH$	Relative humidity	fraction (0–1)
$T_a$	Ambient temperature	K

Atmospheric transmissivity:

\tau_{atm} = 2.02 \cdot (p_a \cdot X_{surface})^{-0.09} \tag{2.7}

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., Eqs. 2.2.42–2.2.43, p. 209

Zero humidity guard

If relative humidity is zero, it is replaced by 0.001 to avoid division by zero in the transmissivity calculation.

3.4 View Factors (4 Methods)

The view factor $F_{view}$ represents the geometric fraction of the fireball's radiation that reaches the receptor.

$F_{view} = \frac{x \cdot (D_{max}/2)^{2}}{\left(x^{2} + H_{fb}^{2}\right)^{3/2}}$

This is the view factor used in the solid plume radiation model (qTerm0).

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., Eq. 2.2.47, p. 209

3.5 Thermal Radiation Models (3 Equations)

Each model calculates the incident heat flux $q$ (kW/m $^2$ ) at a ground-level receptor located at horizontal distance $x$ from the fireball center projection.

q_1(x) = \frac{2.2 \cdot \tau_{atm}(x) \cdot f_s \cdot \Delta H_c \cdot M^{2/3}}{4\pi \cdot X_{center}^{2}} \tag{2.6}

where $X_{center} = \sqrt{H_{fb}^{2} + x^{2}}$ .

Symbol	Description	Unit
$q_1$	Thermal radiation at receptor	kW/m $^2$
$\tau_{atm}$	Atmospheric transmissivity	dimensionless
$f_s$	Radiation fraction	dimensionless
$\Delta H_c$	Heat of combustion	kJ/kg
$M$	Fuel mass	kg
$X_{center}$	Center-to-receptor distance	m

The constant 2.2 is an empirical correction factor derived from experimental data. This is the primary radiation model used for all downstream calculations (dose, probit, fatalities, inverse distance).

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., Eq. 2.2.41, p. 208

3.6 Inverse Calculation (Newton-Raphson)

To find the distance $x$ at which thermal radiation equals a target value $q_{target}$ , the model solves:

$f(x) = q_1(x) - q_{target} = 0$

using the Newton-Raphson iterative method.

Parameter	Value
Initial guess	$x_0 = D_{max}/2$ (fireball radius)
Solver	`newton-raphson-method` npm package
Unit conversion	$1 \text{ BTU/(s ft}^2) = 11.3565 \text{ kW/m}^2$

3.7 Thermal Dose

$D = t_{fb} \cdot (q_1(x) \times 1000)^{4/3}$

Symbol	Description	Unit
$D$	Thermal dose	W $^{4/3}$ s m $^{-8/3}$
$t_{fb}$	Fireball duration	s
$q_1$	Thermal radiation (converted from kW to W)	W/m $^2$

The exponent 4/3 accounts for the non-linear relationship between radiation intensity and skin damage.

Reference: TNO Green Book (CPR 16E), Methods for the Determination of Possible Damage, Chapter 3

3.8 Probit Analysis (Burns and Fatalities)

Probit functions transform a physical exposure parameter into a normally-distributed probability. The general probit equation is $Y = a + b \cdot \ln(V)$ .

Probit equations:

Effect	Equation	Reference
First degree burn	$Y = -39.83 + 3.0186 \cdot \ln(D)$	TNO Green Book, Eq. 3.4, p. 20
Second degree burn	$Y = -43.14 + 3.0186 \cdot \ln(D)$	TNO Green Book, Eq. 3.7, p. 20
Fatality (CCPS)	$Y = -14.9 + 2.56 \cdot \ln(D / 10{,}000)$	CCPS, p. 269
Fatality (TNO)	$Y = -36.38 + 2.56 \cdot \ln(D)$	TNO Green Book, Eq. 3.5, p. 20

FireBall uses CCPS methodology for probit deaths by default. Both CCPS and TNO equations are mathematically equivalent.

Probit to probability conversion:

$P(\%) = f_k \cdot 50 \cdot \left[1 + \text{sgn}(Y - 5) \cdot \text{erf}\left(\frac{|Y - 5|}{\sqrt{2}}\right)\right]$

Symbol	Description	Value
$f_k$	Protection factor	1.0 (no protection)
$\text{erf}$	Error function (Taylor series, 50 terms)	—

Bounds: If $Y < 0$ → $P = 0\%$ . If $Y > 8.09$ → $P = 100\%$ .

3.9 Domino Effect (TTF — Cozzani)

The domino effect analysis estimates the probability that nearby equipment will fail under thermal radiation exposure.

Time to Failure (TTF)

TTF correlations by vessel type:

Vessel Type	Equation	Reference
Atmospheric	$TTF = \exp(-1.13 \ln(q) - 2.667 \times 10^{-5} V + 9.877)$	Cozzani et al.
Pressurized	$TTF = \exp(-0.95 \ln(q) + 8.845 V^{0.032})$	Cozzani et al.
Full engulfment	$TTF = \exp(-1.29 \ln(q) + 10.97 V^{0.026})$	Cozzani et al.

Symbol	Description	Unit
$TTF$	Time to failure	s
$q$	Incident thermal radiation	kW/m $^2$
$V$	Vessel volume	m $^3$

Full engulfment criterion

Equipment is considered fully engulfed when its distance from the fireball center is less than $1.1 \times D_{max}/2$ (10% safety margin accounting for thermal radiation gradients at the fireball boundary).

Domino Probit

$Y_{domino} = 9.25 - 1.847 \cdot \ln\left(\frac{TTF}{60}\right)$

TTF is divided by 60 to convert from seconds to minutes.

Reference: Cozzani, V. et al., Journal of Hazardous Materials, p. 300

Equipment Type Mapping

Database Type	Cozzani Category
`atmospheric_tanks`, `storage_tanks`	Atmospheric
`pressurized_vessels`, `lpg_tanks`, `gas_cylinders`	Pressurized
`reactors`, `heat_exchangers`, `columns`	Pressurized

3.10 Fatality Calculation (Concentric Rings)

Population fatalities are estimated by dividing the affected area into concentric rings centered on the fireball ground projection.

Algorithm:

For each ring $i$ $i$ at distance $r_i$ $r_{i}$ (increment = 5 m, max = 10 km):
- Calculate thermal radiation: $q = q_1(r_i)$
- Calculate thermal dose: $D = t_{fb} \times (q \times 1000)^{4/3}$
- Calculate CCPS probit: $Y = -14.9 + 2.56 \times \ln(D / 10000)$
- Convert probit to percentage: $P = \text{probitToPercent}(Y)$
- If $P < 0.1\%$ : STOP (negligible fatalities beyond this distance)
Ring area: $A_{ring} = \pi (r_{outer}^{2} - r_{inner}^{2})$
Fatalities per ring: $F_i = A_{ring} \times \rho_{pop} \times P_i / 100$
Total fatalities: $F_{total} = \sum F_i$

Parameter	Default Value
Ring increment	5 m
Maximum radius	10 km
Minimum probability threshold	0.1%
Rounding rule	If $F > 0.6$ → $\lceil F \rceil$ ; otherwise 0

Population density conversion:

Input Unit	Conversion Factor to p/m $^2$
p/m $^2$	1
p/ha	$\div$ 10,000
p/km $^2$	$\div$ 1,000,000
p/mi $^2$	$\div$ 2,589,988

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., p. 273; TNO Purple Book (CPR 18E)

3.11 Polygon Receiver Exclusion

When polygon-type receivers (e.g., residential zones, industrial areas) are defined, the model avoids double-counting population:

Polygon receivers overlapping with analysis rings are identified using geographic intersection
For each ring, the polygon area is subtracted: $A_{effective} = A_{ring} - A_{excluded}$
Fatalities from polygon areas are calculated separately using distributed grid analysis with their own population counts

Parameter	Description	Unit
`mass`	Flammable fuel mass	kg, lb, g, ton
`hckjkg`	Heat of combustion	kJ/kg
`radiationFraction`	Fraction of energy radiated ( $f_s$ )	0.2–0.4
`tempAmb`	Ambient temperature	C, F, K
`humidityRel`	Relative humidity	% (0–100)
`populationDensity`	Population density	p/m $^2$ , p/ha, p/km $^2$ , p/mi $^2$
`thermalZones`	Risk zones with radiation thresholds	kW/m $^2$

6.2 Outputs

Output	Description	Unit
`diameterMax`	Maximum fireball diameter	m
`durationFireBallCombustion`	Fireball duration	s
`heigthFireBall`	Fireball center height	m
`burningRate`	Mass burning rate	kg/(m $^2$ s)
`SEPmax`	Maximum surface emissive power	kW/m $^2$
`zones`	Array of risk zones with distances	m
`zones[i].dose`	Thermal dose at zone boundary	W $^{4/3}$ s m $^{-8/3}$
`fatalidades`	Fatality calculation results	Object or 0
`receiverEffects`	Effects on each receiver	Array

6.3 Receiver Effect Categories

Category	Effect	Probit Source
Thermal	1st degree burn	TNO Eq. 3.4
Thermal	2nd degree burn	TNO Eq. 3.7
Thermal	Fatality	CCPS p. 269
Domino	Equipment failure	Cozzani p. 300

Fire Ball

1. Introduction and Physical Phenomenon

1.1 BLEVE and FireBall

1.2 Industrial Context

LPG Storage & Transport

Petroleum Refineries

Chemical Plants

Rail & Road Transport

1.3 Scope of This Model

2. Calculation Sequence

3. Principal Equations

3.1 Geometry: Diameter, Duration, Height

Maximum Fireball Diameter

Fireball Duration

Fireball Center Height

3.2 Combustion: Burning Rate and SEP

Burning Rate

Surface Emissive Power (SEP)

3.3 Atmospheric Transmissivity

3.4 View Factors (4 Methods)

3.5 Thermal Radiation Models (3 Equations)

3.6 Inverse Calculation (Newton-Raphson)

3.7 Thermal Dose

3.8 Probit Analysis (Burns and Fatalities)

3.9 Domino Effect (TTF — Cozzani)

Time to Failure (TTF)

Domino Probit

Equipment Type Mapping

3.10 Fatality Calculation (Concentric Rings)

3.11 Polygon Receiver Exclusion

4. Justification of Selected Methods

5. Model Limitations

6. Input/Output Summary

6.1 Required Inputs

6.2 Outputs

6.3 Receiver Effect Categories

7. References

On this page

Fire Ball

LPG Storage & Transport

Petroleum Refineries

Chemical Plants

Rail & Road Transport

Why qTerm1 (Point Source) as Primary Model

Why CCPS for Probit Fatalities

Why Cozzani for Domino Effect

Why Concentric Ring Analysis for Fatalities

Geometric Limitations

Atmospheric Limitations

Combustion Limitations

Population / Receptor Limitations

Domino Effect Limitations

Numerical Limitations

CCPS — Guidelines for Chemical Process QRA, 2nd Edition

TNO Green Book (CPR 16E, 3rd Edition)

TNO Purple Book (CPR 18E)

TNO Yellow Book (CPR 14E, 3rd Edition)

Cozzani, V. et al.

Casal, J. (2008)

EPA/NOAA — ALOHA Technical Documentation

Kakosimos, K.E.

Lees, F.P. (2004)

On this page