Vapor Cloud Explosion

Technical documentation for the Vapor Cloud Explosion consequence model — TNT equivalent method, Lees/Brasie overpressure, probit analysis, and fatality estimation

1. Introduction and Physical Phenomenon

1.1 Vapor Cloud Explosion (VCE)

A VCE (Vapor Cloud Explosion) occurs when a significant quantity of flammable gas or vapor is released into the atmosphere, mixes with air forming a flammable cloud, and ignites. If the cloud encounters sufficient confinement or obstruction (equipment, buildings, piping), the combustion can accelerate to generate a destructive overpressure wave.

Unlike a confined detonation, a VCE is a deflagration phenomenon where the flame speed is subsonic, but the overpressure generation can be significant depending on the degree of confinement and congestion in the surrounding environment.

The main effects of a VCE include:

Overpressure (blast wave) capable of causing structural collapse
Impulse (pressure-time integral) that determines dynamic damage
Lethality in people from direct overpressure effects
Domino effects through failure of nearby vessels and equipment

1.2 Industrial Context

High-severity scenario

VCE events are among the most severe accident outcomes in industrial QRA. Historical incidents such as Flixborough (UK, 1974), Buncefield (UK, 2005), and Texas City (USA, 2005) demonstrate their catastrophic potential.

Petroleum Refineries

Gaseous hydrocarbon leaks in congested process areas

Chemical Plants

Flammable vapor releases in process zones

LPG Storage Facilities

Rapid vaporization of pressurized liquids

Offshore Installations

Natural gas releases on platforms and FPSOs

1.3 Scope of This Model

This model calculates, using the TNT equivalent method:

TNT equivalent mass from gas mass, heat of combustion, and energy fraction
Overpressure, impulse, duration, and arrival time at any distance
Distance to a specified overpressure threshold (inverse problem)
Probit-based probability of lethality, eardrum rupture, and structural damage
Domino effect probability for nearby equipment (Cozzani and Mingguang)
Population fatalities using concentric ring analysis

2. Calculation Sequence

Cargando diagrama...

The VCE calculation follows these stages:

TNT Equivalent — Convert combustion energy of the vapor cloud into an equivalent mass of TNT.

Scaled Distance — Compute Hopkinson-Cranz scaled distance $z = x / m_{TNT}^{1/3}$ for each receptor.

Blast Wave Parameters — Calculate overpressure $P(z)$ , impulse $I(z)$ , positive phase duration $t_d(z)$ , and arrival time $t_a(z)$ using Lees or Brasie method.

Inverse Distance — Find distance at which overpressure equals a target threshold via incremental iteration.

Probit Analysis — Convert overpressure/impulse to probabilities of lethality, eardrum rupture, structural damage, and domino effects.

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

3. Principal Equations

3.1 TNT Equivalent

m_{TNT} = \frac{f_e \cdot \Delta H_c \cdot M}{\Delta H_{TNT}}

Symbol	Description	Unit	Value/Range
$m_{TNT}$	TNT equivalent mass	kg	calculated
$f_e$	Explosion energy fraction	dimensionless	0.01–0.10
$\Delta H_c$	Heat of combustion of gas	kJ/kg	user input
$M$	Mass of flammable gas released	kg	user input
$\Delta H_{TNT}$	Heat of combustion of TNT	kJ/kg	4,760

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., p. 165

Energy fraction (fe)

The parameter $f_e$ (explosion efficiency or yield) is typically between 1% and 10%. Higher values correspond to greater confinement/congestion. The user enters the value as a percentage (1–10) and it is divided by 100 internally.

3.2 Scaled Distance (Hopkinson-Cranz)

z = \frac{x}{m_{TNT}^{1/3}}

Symbol	Description	Unit
$z$	Scaled distance	m/kg $^{1/3}$
$x$	Actual distance from explosion center	m
$m_{TNT}$	TNT equivalent mass	kg

Valid range for Lees method: $0.0674 \leq z \leq 40$

3.3 Overpressure Calculation

Overpressure is calculated using a degree-11 polynomial in the transformed variable $u$ :

u = a_1 + b_1 \cdot \log_{10}(z)

\log_{10}(P) = c_0 + c_1 u + c_2 u^2 + c_3 u^3 + \ldots + c_{11} u^{11}

P = 10^{\log_{10}(P)} \quad \text{(kPa)}

Range: $0.0674 \leq z \leq 40$

Errata

Some constants differ between the original Lees Loss and CCPS. The constants implemented correspond to the corrected CCPS version (p. 162), which is considered the authoritative version.

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., pp. 161-162; Lees, Loss Prevention, 3rd ed., p. 17-127

3.4 Impulse Calculation

The positive impulse uses two polynomials depending on the range of $z$ :

Range 1 ( $0.0674 \leq z \leq 0.955$ ): Degree-4 polynomial

Range 2 ( $0.955 < z \leq 40$ ): Degree-7 polynomial

Both use: $u = a + b \cdot \log_{10}(z)$ , then $I = 10^{\text{polynomial}(u)}$ (kPa·ms = Pa·s)

Reference: Lees, Loss Prevention, 3rd ed., p. 17-127

3.5 Duration and Arrival Time

Three polynomials for different ranges of $z$ :

Range 1 ( $0.178 \leq z \leq 1.01$ ): Degree-5 polynomial
Range 2 ( $1.01 < z \leq 2.78$ ): Degree-8 polynomial
Range 3 ( $2.78 < z \leq 40$ ): Degree-5 polynomial

$t_d = 10^{\text{polynomial}(u)}$ (ms)

Reference: Lees, Loss Prevention, 3rd ed., p. 17-127

3.6 Inverse Calculation (Incremental Iteration)

To find the distance $x$ at which overpressure equals a target value $P_{target}$ , the model uses incremental iteration:

Parameter	Value
Step size	0.05 m
Max iterations	100,000
Initial reference pressure	2,068 kPa (EPA maximum)

Unit conversion for target overpressure:

Input Unit	Factor to kPa
kPa	$\times 1$
psi	$\times 6.89476$
bar	$\times 100$
atm	$\times 101.325$

Why not Newton-Raphson?

The high-degree Lees polynomials can have non-monotonic derivatives, making Newton-Raphson convergence difficult. Incremental iteration always converges if the solution exists within the valid range.

3.7 Probit Analysis — Effects on People

Y_{death} = 1.47 + 1.35 \cdot \ln(P_{psi})

where $P_{psi} = P_{kPa} \times 0.145038$ .

Reference: Hurst, Nussey & Pape, 1989

3.8 Probit Analysis — Structural Effects

Y_{structure} = -23.8 + 2.92 \cdot \ln(P_{Pa})

Reference: CCPS, Guidelines for Chemical Process QRA, 2nd ed., p. 275

3.9 Domino Effect

All use overpressure in Pascals:

Equipment Type	Probit Equation
Atmospheric vessels	$Y = -18.96 + 2.44 \cdot \ln(P_{Pa})$
Pressurized vessels	$Y = -42.44 + 4.33 \cdot \ln(P_{Pa})$
Elongated equipment	$Y = -28.07 + 3.16 \cdot \ln(P_{Pa})$
Small equipment	$Y = -17.79 + 2.18 \cdot \ln(P_{Pa})$

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

3.10 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 $Y > 8.09$ → $P = 100\%$ .

3.11 Fatality Calculation (Concentric Rings)

Population fatalities are estimated by dividing the affected area into concentric rings centered on the explosion point.

Algorithm:

Dynamic initial radius: $r_{min} = 0.0674 \cdot m_{TNT}^{1/3}$ , effective $r_{initial} = \max(1, \lceil r_{min} \rceil)$ m
For each ring $i$ $i$ at distance $r_i$ $r_{i}$ (increment = 5 m, max = 10 km):
- Calculate overpressure $P$ using selected method
- If $P = 9999$ or $P \leq 0$ : STOP (out of range)
- Calculate lethality probit (Hurst or Eisenberg)
- Convert probit to percentage $P\%$
- If $P\% < 0.1\%$ : STOP (negligible fatalities)
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

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

3.12 Polygon Receiver Exclusion

When polygon-type receivers 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
`massRelease`	Flammable gas mass released	kg, lb, g, ton
`hckjkg`	Heat of combustion	kJ/kg
`energyFraction`	Explosion energy fraction ( $f_e$ )	% (1–10)
`VCEMethod`	Calculation method	"less" or "brasie"
`populationDensity`	Population density	p/m $^2$ , p/ha, p/km $^2$ , p/mi $^2$
`probitMethod`	Lethality probit method	"hurst" or "eisenberg"
`overpressureZones`	Risk zones with overpressure thresholds	kPa

6.2 Outputs

Output	Description	Unit
`masaTNTEQ`	TNT equivalent mass	kg
`zones`	Array of risk zones with distances	m
`zones[i].overpressureZone`	Zone overpressure	kPa
`zones[i].distance`	Distance at that overpressure	m
`fatalidades`	Fatality calculation results	Object or 0
`receiverEffects`	Effects on each receiver	Array
`receiverEffects[i].overpressure`	Overpressure at receiver	kPa
`receiverEffects[i].impulse`	Impulse at receiver	Pa·s
`receiverEffects[i].duration`	Positive phase duration	ms
`receiverEffects[i].arrivalTime`	Arrival time	ms

6.3 Receiver Effect Categories

Population/Structural Effects:

Category	Effect	Probit Source
Overpressure	Fatality	Hurst 1989 / CCPS p. 275
Overpressure	Eardrum rupture	CCPS
Overpressure	Structural damage	CCPS p. 275
Overpressure	Window breakage	TNO Green Book

Domino Effects on Equipment:

Equipment Type	Probit Source
Atmospheric vessels	Cozzani — Atmospheric
Pressurized vessels	Cozzani — Pressurized
Elongated equipment	Cozzani — Elongated
Small equipment	Cozzani — Small

Vapor Cloud Explosion

1. Introduction and Physical Phenomenon

1.1 Vapor Cloud Explosion (VCE)

1.2 Industrial Context

Petroleum Refineries

Chemical Plants

LPG Storage Facilities

Offshore Installations

1.3 Scope of This Model

2. Calculation Sequence

3. Principal Equations

3.1 TNT Equivalent

3.2 Scaled Distance (Hopkinson-Cranz)

3.3 Overpressure Calculation

3.4 Impulse Calculation

3.5 Duration and Arrival Time

3.6 Inverse Calculation (Incremental Iteration)

3.7 Probit Analysis — Effects on People

3.8 Probit Analysis — Structural Effects

3.9 Domino Effect

3.10 Probit to Probability Conversion

3.11 Fatality Calculation (Concentric Rings)

3.12 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

Vapor Cloud Explosion

Petroleum Refineries

Chemical Plants

LPG Storage Facilities

Offshore Installations

Polynomial Constants (from LessConstants.js)

Range 1 Constants (ip1)

Range 2 Constants (ip2)

Duration Range 1 Constants (td1)

Duration Range 2 Constants (td2)

Duration Range 3 Constants (td3)

Why Lees (CCPS) as Primary Overpressure Method

Why Hurst as Default Lethality Probit

Why Cozzani for Overpressure Domino Effect

Why Incremental Iteration for Inverse Calculation

TNT Equivalent Method

Scaled Distance Range

Incremental Iteration

Probit Limitations

Population / Receptor

Numerical

CCPS — Guidelines for Chemical Process QRA, 2nd Edition

Lees, F.P. — Loss Prevention in the Process Industries, 3rd Edition

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

TNO Purple Book (CPR 18E)

Brasie, W.C. & Simpson, D.W. (1968)

Hurst, N.W., Nussey, C. & Pape, R.P. (1989)

Cozzani, V. et al.

Kakosimos, K.E.

JRC — Joint Research Centre

Mingguang, Z. et al.

On this page