Vapor Cloud Explosion

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#

1.3 Scope of This Model#

This model calculates, using the TNT equivalent method:

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

2. Calculation Sequence#

flowchart TD
  A["Input Data<br/>(mass, Hc, fe, method)"] --> B["TNT Equivalent<br/>m_TNT = fe*Hc*M / DH_TNT"]
  B --> C["For each distance x:<br/>z, P(z), I(z), td(z), ta(z)"]
  C --> D["Forward: P(x)"]
  C --> E["Inverse: x(P_target)<br/>Incremental Iteration"]
  D --> F["Probit Analysis"]
  E --> F
  F --> G["Lethality<br/>(people)"]
  F --> H["Structural<br/>Damage"]
  F --> I["Domino<br/>Effect"]
  G --> J["Fatality Calculation<br/>(Concentric Rings)"]
  H --> J
  I --> J

  style A fill:#e1f5fe
  style J fill:#c8e6c9
  style C fill:#fff3e0

The VCE calculation follows these stages:

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

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

3. 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.

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

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

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

3. Principal Equations#

3.1 TNT Equivalent#

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

3.2 Scaled Distance (Hopkinson-Cranz)#

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

3.3 Overpressure Calculation#

3.4 Impulse Calculation#

3.5 Duration and Arrival Time#

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:

Unit conversion for target overpressure:

3.7 Probit Analysis — Effects on People#

3.8 Probit Analysis — Structural Effects#

3.9 Domino Effect#

3.10 Probit to Probability Conversion#

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

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:

1. Dynamic initial radius: $r_{min} = 0.0674 \cdot m_{TNT}^{1/3}$, effective $r_{initial} = \max(1, \lceil r_{min} \rceil)$ m
2. For each ring $i$ at distance $r_i$ (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)
3. Ring area: $A_{ring} = \pi (r_{outer}^{2} - r_{inner}^{2})$
4. Fatalities per ring: $F_i = A_{ring} \times \rho_{pop} \times P_i / 100$
5. Total fatalities: $F_{total} = \sum F_i$

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:

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

4. Justification of Selected Methods#

5. Model Limitations#

6. Input/Output Summary#

6.1 Required Inputs#

6.2 Outputs#

6.3 Receiver Effect Categories#

Population/Structural Effects:

Domino Effects on Equipment:

7. References#