Fire Ball

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#

1.3 Scope of This Model#

This model calculates:

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

2. Calculation Sequence#

flowchart TD
  A["Input Data<br/>(mass, Hc, fs, weather)"] --> B["Geometry Calculations<br/>Dmax, t_fb, H_fb"]
  B --> C["Combustion Parameters<br/>m'', SEP_max"]
  C --> D["For each distance x:<br/>X_surface, tau_atm, F_view, q"]
  D --> E["Forward: q(x)"]
  D --> F["Inverse: x(q_target)<br/>Newton-Raphson"]
  E --> G["Thermal Dose<br/>D = t * q^(4/3)"]
  F --> G
  G --> H["Probit Analysis<br/>Burns / Deaths"]
  G --> I["Domino Effect<br/>TTF - Cozzani"]
  H --> J["Fatality Calculation<br/>(Concentric Rings)"]
  I --> J

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

The BLEVE-FireBall calculation follows these stages:

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

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

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

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

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

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

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

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

3. Principal Equations#

3.1 Geometry: Diameter, Duration, Height#

Maximum Fireball Diameter#

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

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

Fireball Duration#

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

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

Fireball Center Height#

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

3.2 Combustion: Burning Rate and SEP#

Burning Rate#

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

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

Surface Emissive Power (SEP)#

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

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:

Atmospheric transmissivity:

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

3.4 View Factors (4 Methods)#

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

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.

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.

3.7 Thermal Dose#

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

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:

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 $Y > 8.09$ → $P = 100\%$.

3.9 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:

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#

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:

1. For each ring $i$ at distance $r_i$ (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)
2. Ring area: $A_{ring} = \pi (r_{outer}^{2} - r_{inner}^{2})$
3. Fatalities per ring: $F_i = A_{ring} \times \rho_{pop} \times P_i / 100$
4. Total fatalities: $F_{total} = \sum F_i$

Population density conversion:

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:

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#

7. References#