# Physical Explosion
Source: https://tekrisk.com/en/docs/risk-models/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.

<Callout type="info" title="In validation · available soon">
Our team is validating this model. It cannot be created from the app yet; this page documents its method and it will be available soon.
</Callout>

## 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).

<Callout type="info" title="Scope of this model">
TekRisk models a vessel **filled with gas or vapour**. The blast of a **BLEVE** (flashing of the superheated liquid) is not included: it needs the internal energy of the liquid and the vapour (Yellow Book, step 4).
</Callout>

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## 2. Calculation sequence

<Mermaid>
{`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"]`}
</Mermaid>

---

## 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:

1. Find the scaled distance $z_s$ at which the TNT curve gives the surface overpressure.
2. Compute $R_s = z_s\,W^{1/3}$.
3. Subtract the radius of the idealized vessel, $r_v = 0.782\,V^{1/3}$, to get the **virtual distance** $R_s - r_v$.
4. 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".

<Callout type="warn" title="Validation note (CCPS Ex. 2.23)">
The CCPS spreadsheet matches the **absolute** surface pressure (10.21 bar) against the **over**pressure curve. TekRisk uses the overpressure $(P_s - 1)\,p_a$, which is the consistent form; at 18.28 m the result is within 2 % of the book's 18.38 kPa.
</Callout>

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## 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.

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## 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.

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## 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.

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## 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.