Physical Explosion

Technical documentation of the physical explosion model (catastrophic rupture of a compressed-gas vessel) — Brode, Brown and Crowl expansion energy, TNT equivalence with Prugh's virtual distance and fragments per CCPS.

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1. Physical phenomenon#

A physical explosion is the catastrophic rupture of a vessel that holds gas under pressure. The energy stored in the compressed gas is released at once and produces:

a blast wave that travels outward from the vessel, and
fragments (projectiles) of the shell, accelerated by the escaping gas.

There is no combustion: the energy is mechanical. If the gas is flammable, a later ignition is a separate scenario (fireball, flash fire or VCE).

Typical causes (CCPS §2.2.3; Yellow Book ch. 7):

overpressure after a failure of pressure regulation or relief;
wall thinning by corrosion, erosion or chemical attack;
loss of strength by overheating, material defects or fatigue;
runaway reaction (the vessel is treated as a gas burst at the failure pressure).

2. Calculation sequence#

flowchart TD
  A["Vessel<br/>p1, V, T, γ, M"] --> B["Expansion energy<br/>Brode · Brown · Crowl"]
  B --> C["Blast fraction<br/>100 % · 80 % · 40 %"]
  C --> D["TNT equivalent<br/>W = E / 4.69 MJ/kg"]
  A --> E["Overpressure at the surface<br/>Liepmann-Roshko (YB 7.12)"]
  D --> F["Prugh virtual distance"]
  E --> F
  F --> G["Overpressure at each distance<br/>TNT curve (CCPS Table 2.17)"]
  G --> H["Zones, receivers, probit,<br/>fatalities, IR and F-N"]
  A --> I["Fragments<br/>Baker · Moore · u²/g"]

3. Equations#

3.1 Expansion energy#

Brode (constant volume; CCPS Eq. 2.2.11, Yellow Book Eq. 7.1):

E = \frac{(p_1 - p_a)\,V}{\gamma - 1}

Brown (ideal-gas isothermal expansion, in the CCPS form Eq. 2.2.12):

W_{TNT}\,[\text{lb}] = 1.39\times10^{-6}\; V[\text{ft}^3]\;\frac{p_1}{p_0}\;R\,T\;\ln\frac{p_1}{p_2}

Crowl (thermodynamic availability; CCPS Eq. 2.2.13):

E = p_1 V\left[\ln\frac{p_1}{p_a} - \left(1 - \frac{p_a}{p_1}\right)\right]

where $p_1$ is the absolute burst pressure, $p_a$ the ambient pressure (standard atmosphere at the scenario altitude), $V$ the gas volume and $\gamma = C_p/C_v$ .

Only part of the energy goes to the blast (Saville, quoted by CCPS): 80 % for a brittle failure and 40 % when a major section is ejected. 100 % is the conservative assumption. The TNT-equivalent mass is

W = \frac{f\,E}{4.69\ \text{MJ/kg}}

3.2 Overpressure at the vessel surface#

Right after the rupture, the shock in air is much weaker than $p_1$ . It follows from the shock-tube relation (Yellow Book Eq. 7.12; CCPS Eq. 2.2.14):

\frac{p_1}{p_a} = (P_{so}+1)\left[1-\frac{(\gamma_1-1)(a_a/a_1)\,P_{so}}{\sqrt{2\gamma_a\left(2\gamma_a+(\gamma_a+1)P_{so}\right)}}\right]^{-\frac{2\gamma_1}{\gamma_1-1}}

where $P_{so} = p_{so}/p_a - 1$ and $a = \sqrt{\gamma R T/M}$ is the speed of sound. TekRisk solves it by bisection.

3.3 Prugh virtual distance#

The TNT curve assumes a point source, and a vessel is not one. Prugh corrects the near field as follows:

Find the scaled distance $z_s$ at which the TNT curve gives the surface overpressure.
Compute $R_s = z_s\,W^{1/3}$ .
Subtract the radius of the idealized vessel, $r_v = 0.782\,V^{1/3}$ , to get the virtual distance $R_s - r_v$ .
Evaluate the curve at $z = (r + R_s - r_v)/W^{1/3}$ for each distance $r$ measured from the vessel centre.

The curve is the Lees surface-burst curve (CCPS Table 2.17), valid for $z \in [0.0674,\,40]$ m/kg $^{1/3}$ , the same one the VCE model uses. If $R_s < r_v$ , the virtual distance is set to 0. The overpressure never exceeds the surface value, $p_1 - p_a$ , or the curve value at the vessel wall ( $r = r_v$ ); with a zero virtual distance the latter is lower than the surface value, and a threshold above it is reported as "within the vessel".

4. Fragments#

Fragments are computed as information: they do not enter fatalities, individual risk or the F-N curve, because almost no quantitative risk analysis quantifies them (CCPS §2.2.3).

Baker (CCPS Table 2.25): dimensionless initial velocity as a function of the scaled pressure

\bar P = \frac{(p_1-p_a)\,V}{M_c\,a_0^2}, \qquad \ln\frac{v}{K a_0} = a\ln\bar P + b

with coefficients per shape (sphere or cylinder) and number of fragments (2, 10 or 100). For 2 fragments of unequal mass, $K = 1.306\,f + 0.308446$ .

Moore (CCPS Eq. 2.2.22–2.2.24) is always reported as an upper bound:

u = 1.092\sqrt{\frac{E\,G}{M_c}}

Maximum range with no drag or lift, at 45°:

r_{max} = \frac{u^2}{g}

It is a conservative upper bound. The map draws it as a dashed circle.

5. Limitations#

The blast is assumed symmetric; a real rupture is directional because the crack starts at one point.
TNT equivalence is inaccurate in the near field (closer than 10–20 vessel diameters, per the Yellow Book). Prugh's virtual distance only partly corrects it.
Ideal gas: $\gamma$ and $M$ are constant during the expansion.
It does not cover the BLEVE blast (liquid flashing), runaway reactions with chemical energy or the decomposition of energetic materials.
Baker's method with its own vessel-burst curves (Yellow Book Fig. 7.5) is not implemented yet.

6. Inputs and outputs#

Input	Tab	Note
Burst pressure	Vessel	Gauge or absolute. The helper applies the Yellow Book Table 7.1 factors (external fire 1.21 × valve set pressure, design × 2.5) or MAWP × 4 (CCPS).
Gas volume	Vessel	Only the gas-filled part.
Gas temperature	Vessel	Used by Brown, Crowl and the speed of sound.
Shape	Vessel	Sphere or cylinder: fragment coefficients.
Gas	Gas	Scenario substance (γ from the ideal-gas $C_p$ ), reference gas or manual values.
Energy method and blast fraction	Model	Brode, Brown or Crowl; 100, 80 or 40 %.
Fragments	Fragments	Vessel mass, number of fragments and method.
Ambient temperature and altitude	Weather	Set $p_a$ (standard atmosphere) and the speed of sound in air.
Population density and probit	Vulnerability	Overpressure lethality (Eisenberg or Hurst).
Overpressure zones	Zones	Thresholds with their unit.

Outputs: energy and TNT equivalent, surface overpressure, virtual distance, distance, impulse, duration and arrival time per zone, receiver effects (lethality, eardrum, structures, windows and domino effect on equipment), ring-based fatalities and, when requested, the velocity and range of each fragment.

7. References#

CCPS (2000). Guidelines for Chemical Process Quantitative Risk Analysis, 2nd ed., §2.2.3 "Physical Explosion". AIChE.
TNO (2005). Methods for the calculation of physical effects (Yellow Book, CPR 14E), ch. 7 "Rupture of vessels".
Baker, W.E. et al. (1983). Explosion Hazards and Evaluation. Elsevier.
Prugh, R.W. (1988). Quantitative evaluation of "BLEVE" hazards. Journal of Fire Protection Engineering.
Crowl, D.A. (1992). Calculating the energy of explosion using thermodynamic availability. Journal of Loss Prevention in the Process Industries.
Lees, F.P. (1996). Loss Prevention in the Process Industries, 2nd ed.