Pool Fire

Technical documentation for the Pool Fire consequence model — burning rate, flame geometry, thermal radiation, probit analysis, domino effects, and fatality estimation

1. Introduction and Physical Phenomenon

1.1 Pool Fire

The Pool Fire model simulates the steady-state combustion of a flammable liquid that has spilled onto a flat surface and ignited. The model computes:

Maximum pool diameter based on spill type
Mass burning rate per unit area (burning rate)
Flame geometry (height, wind-induced tilt angle)
Thermal radiation intensity at any given distance
Human effects (1st/2nd degree burns, fatalities) via Probit functions
Domino effects on neighboring vessels using Cozzani correlations
Expected fatalities by integrating thermal radiation with population density

1.2 Industrial Context

Thermal radiation hazard

Pool fires produce sustained thermal radiation capable of causing severe burns and fatalities. The hazard zone depends on pool size, fuel properties, wind conditions, and atmospheric humidity.

Continuous Spill

Steady-state release; pool diameter grows until equilibrium between inflow and combustion rate

Massive (Instantaneous)

Entire volume released at once; maximum diameter depends on total volume released

Circular Dike

Fixed diameter defined by the user (inner dike diameter)

Rectangular Dike

Equivalent diameter computed from length × width of the containment dike

1.3 Scope of This Model

This model calculates:

Pool diameter and geometry based on source type (continuous, massive, circularDike, rectangularDike)
Burning rate using Burgess-Strasser, Mudan, or tabulated methods
Flame height via Thomas or Pritchard-Binding correlations
Surface Emissive Power (SEP) accounting for soot shielding in large hydrocarbon fires
Thermal radiation at any distance — Point Source or Solid Plume (tilted cylinder) model
Distance to a specified radiation threshold (inverse problem via Newton-Raphson)
Thermal dose and probit-based probability of 1st/2nd degree burns and fatalities
Domino effect time-to-failure for neighboring vessels (Cozzani correlations)
Population fatalities using concentric annular ring integration

2. Calculation Sequence

Cargando diagrama...

Chemical Properties — Retrieve fuel properties from YAWS database at $T_{amb}$ : heat of vaporization $\Delta H_{vap}$ , liquid heat capacity $c_P$ , and liquid density $\rho_L$ .

Burning Rate — Compute $\dot{m}''$ [kg/(m²·s)] using Burgess-Strasser, Mudan, or tabulated value (gasoline).

Pool Diameter — Determine maximum pool diameter $D$ based on source type: continuous spill, massive release, circular dike, or rectangular dike.

Flame Geometry — Compute dimensionless wind speed $u^*$ , flame height $H$ (Thomas or Pritchard-Binding), tilt angle $\theta$ , and Surface Emissive Power (SEP).

Thermal Radiation — Compute radiation $q(x)$ [kW/m²] at target distances using Point Source or Solid Plume (tilted cylinder) view factor model.

Probit Analysis — Convert thermal dose into probability of 1st/2nd degree burns, fatalities (TNO or CCPS), and domino effects (Cozzani).

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

3. Chemical Properties (YAWS Correlations)

All fuel properties are retrieved from the YAWS chemical database evaluated at ambient temperature $T_{amb}$ (Kelvin).

3.1 Heat of Vaporization at $T_{amb}$

\Delta H_{vap}(T_{amb}) = \frac{h_{va} \cdot \left(1 - \dfrac{T_{amb}}{T_c}\right)^{h_{vn}}}{M_w} \times 10^3 \quad [\text{kJ/kg}]

Symbol	Description	Source
$h_{va}$	Vaporization coefficient A	YAWS, p. 109
$h_{vn}$	Correlation exponent	YAWS, p. 109
$T_c$	Critical temperature (K)	YAWS
$M_w$	Molecular weight (g/mol)	YAWS

Code: PoolFire.js, lines 81–85.

4. Burning Rate — `burningRate()`

The burning rate $\dot{m}''$ [kg/(m²·s)] is the mass of fuel consumed per unit area per unit time. It governs fire intensity and pool diameter.

4.1 Special Case: Gasoline

For gasoline (GASOLINE-s), a tabulated experimental value is used directly:

\dot{m}'' = 0.055 \quad [\text{kg/(m}^2\text{·s)}]

Code: PoolFire.js, lines 291–293.

Design note

The denominator in both Burgess-Strasser and Mudan methods includes $c_P \cdot (T_b - T_{amb})$ to account for the sensible heat required to bring the liquid to its boiling point before evaporation.

5. Pool Diameter — `poolDiameter()`

5.1 Continuous Spill (CCPS p. 228)

The maximum diameter is reached when the horizontal spreading rate equals the combustion rate. The vertical burning rate is:

y_{max} = 1.27 \times 10^{-6} \cdot \frac{\Delta H_c}{\Delta H_{vap}(T_b)} \quad [\text{m/s}]

The equilibrium maximum diameter:

D = 2 \sqrt{\frac{Q_{spill}}{\pi \cdot y_{max}}} \quad [\text{m}]

where $Q_{spill}$ is the volumetric spill rate [m³/s].

Special cases

For GASOLINE: $y_{max} = 7 \times 10^{-5}$ m/s; for LP GAS: $y_{max} = 2 \times 10^{-4}$ m/s (tabulated values).

Reference: CCPS, p. 228. Code: PoolFire.js, lines 331–345.

7. Dimensionless Wind Speed — `ux()`

u^* = u_w \cdot \left(\frac{g \cdot \dot{m}'' \cdot D}{\rho_{air}}\right)^{-1/3}, \quad u^* \geq 1.0

The physical minimum $u^* = 1.0$ corresponds to the no-wind condition. Dry air density uses ISA 1976 altitude correction:

P = 101\,325 \cdot \left(1 - 2.5577 \times 10^{-5} \cdot H\right)^{5.25588} \quad [\text{Pa}], \quad \rho_{air} = \frac{P}{287.05 \cdot T_{amb}}

Reference: ISA 1976. Code: PoolFire.js, lines 414–430.

8. Flame Height — `alturaFlama()`

8.1 Thomas Method — No Wind ( $u_w = 0$ )

H = 42 \cdot D \cdot \left(\frac{\dot{m}''}{\rho_{air} \sqrt{g \cdot D}}\right)^{0.61}

Reference: Thomas, P.H., The size of flames from natural fires, 1963; Kakosimos p. 86. Code: PoolFire.js, lines 464–470.

Method comparison

Thomas slightly overestimates height for large pools (D > 50 m) with strong wind. Pritchard-Binding was developed specifically for hydrocarbons and yields somewhat lower heights; recommended for conservative analyses in industrial facilities at low $u^*$ values.

9. Surface Emissive Power (SEP) — `SEP()`

The SEP [kW/m²] is the radiant power emitted per unit area of the flame surface.

For large hydrocarbon pools (alkanes, gasoline, diesel, jet fuel), soot significantly reduces effective radiation. The bi-exponential correlation models the shielding effect:

SEP = 140 \cdot e^{-0.12 D} + 20 \cdot (1 - e^{-0.12 D}) \quad [\text{kW/m}^2]

The first term represents radiation from the luminous core; the second, background radiation from the smoke column.

Hydrocarbon identification

The code identifies hydrocarbons by the presence of suffixes ANE, GAS, GASOLINE, DIESEL, or TURBO in the substance name.

Reference: Mudan & Croce, SFPE Handbook, 1995; Kakosimos p. 88. Code: PoolFire.js, lines 505–508.

10. Flame Tilt Angle — `anguloFlama()`

Wind tilts the flame from the vertical. The angle $\theta$ [rad] is computed from the Froude and Reynolds numbers:

Fr = \frac{u_w^2}{g \cdot D}, \quad Re = \frac{u_w \cdot D}{\nu}, \quad c = 0.666 \cdot Fr^{0.333} \cdot Re^{0.117}

\theta = \arcsin\!\left(\frac{\sqrt{4c^2 + 1} - 1}{2c}\right) \quad [\text{rad}]

For $u_w \leq 0$ , $\theta = 0$ (vertical flame). Kinematic viscosity $\nu$ [m²/s] is obtained from an empirical polynomial in $T_{amb}$ [K].

Code: PoolFire.js, lines 523–542.

11. View Factor — `viewFactor(x)`

11.1 Point Source Model

Assumes all energy radiates from a geometric point at the flame center:

F = \frac{1}{4\pi x^2}

Near-field limitation

Valid for $x \gg D$ . Underestimates radiation in the near field.

Code: PoolFire.js, line 610.

12. Atmospheric Transmissivity — `ta(x)`

Atmospheric humidity attenuates thermal radiation. Transmissivity $\tau$ uses Wayne's correlation (cited in CCPS):

\tau = 2.02 \cdot (P_w \cdot x)^{-0.09}

The partial pressure of water vapor $P_w$ [Pa]:

P_w^0 = \frac{\exp\!\left(77.345 + 0.0057\,T - 7235/T\right)}{T^{8.2}}, \qquad P_w = \frac{HR}{100} \cdot P_w^0

where $HR$ is relative humidity [%] and $T = T_{amb}$ [K]. Code: PoolFire.js, lines 620–638.

13. Thermal Radiation — `qTermAtX(x)`

q(x) = \tau(x) \cdot F_\eta \cdot \dot{m}'' \cdot \Delta H_c \cdot F_{vista}(x) \cdot A_{pool} \quad [\text{kW/m}^2]

where:

$F_\eta$ — radiated energy fraction (0.15–0.35, user-configurable; CCPS p. 230–232, Table 2.27)
$A_{pool} = \pi D^2 / 4$ — pool area [m²]

Code: PoolFire.js, lines 652–663.

14. Distance to a Given Radiation Level — `xTerm(q_target)`

Given a target radiation level $q_{target}$ [kW/m²], the distance $x$ [m] is found via Newton-Raphson. For the Point Source model:

f(x) = \frac{Z \cdot P_w^{-0.09}}{4\pi} \cdot x^{-2.09} - q_{target} = 0, \qquad Z = 2.02 \cdot F_\eta \cdot \dot{m}'' \cdot \Delta H_c \cdot A_{pool}

f'(x) = -\frac{2.09 \cdot Z \cdot P_w^{-0.09}}{4\pi} \cdot x^{-3.09}, \qquad x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Convergence tolerance: 0.01 m. Code: PoolFire.js, lines 681–737.

15. Thermal Dose — `dose(x)`

D_{dose} = t_{exp} \cdot \left[q(x) \times 10^3\right]^{4/3} \quad \left[(\text{W/m}^2)^{4/3} \cdot \text{s}\right]

The factor $10^3$ converts $q$ from kW/m² to W/m². Code: PoolFire.js, line 758.

N_{fatal} = \sum_{r_i} P_{death}(r_i) \cdot \rho_{pop} \cdot A_{ring}(r_i)

Symbol	Description	Unit
$P_{death}(r_i)$	Probability of death at distance $r_i$ (CCPS method)	%
$\rho_{pop}$	Population density	persons/m²
$A_{ring}(r_i)$	Annular ring area	m²

For polygon receivers (zones with known population), FatalityUtils.js uses a 10 m grid to distribute the population within the polygon and excludes that area from the uniform density calculation.

Code: PoolFire.js, lines 818–860; delegated to FatalityUtils.js.

Pool Fire

1. Introduction and Physical Phenomenon

1.1 Pool Fire

1.2 Industrial Context

Continuous Spill

Massive (Instantaneous)

Circular Dike

Rectangular Dike

1.3 Scope of This Model

2. Calculation Sequence

3. Chemical Properties (YAWS Correlations)

3.1 Heat of Vaporization at $T_{amb}$

4. Burning Rate — `burningRate()`

4.1 Special Case: Gasoline

5. Pool Diameter — `poolDiameter()`

5.1 Continuous Spill (CCPS p. 228)

6. Time Calculations

6.1 Continuous Spill — `timeToReachPoolSize()`

6.2 Massive Spill

7. Dimensionless Wind Speed — `ux()`

8. Flame Height — `alturaFlama()`

8.1 Thomas Method — No Wind ( $u_w = 0$ )

9. Surface Emissive Power (SEP) — `SEP()`

10. Flame Tilt Angle — `anguloFlama()`

11. View Factor — `viewFactor(x)`

11.1 Point Source Model

12. Atmospheric Transmissivity — `ta(x)`

13. Thermal Radiation — `qTermAtX(x)`

14. Distance to a Given Radiation Level — `xTerm(q_target)`

15. Thermal Dose — `dose(x)`

16. Effects — Probit Functions

17. Time to Failure for Vessels (Domino Effect) — Cozzani Correlations

18. Fatality Calculation — `fatalidades()`

19. Model Limitations

20. Bibliographic References

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Pool Fire

Continuous Spill

Massive (Instantaneous)

Circular Dike

Rectangular Dike

1st Degree Burns (TNO Green Book, eq. 3.4)

2nd Degree Burns (TNO Green Book, eq. 3.7)

Fatalities — TNO Methodology (eq. 3.5)

Fatalities — CCPS Methodology (Eisenberg)

Probit to Percentage Conversion

Atmospheric Vessels

Pressurized Vessels

Pressurized Vessels — Full Engulfment

Domino Effect Probit (Cozzani, p. 300)

Cylindrical flame geometry

Uniform SEP distribution

Unidirectional wind

Flat horizontal pool

Steady-state assumption

No flame-spill feedback

Cozzani correlations scope

Atmospheric transmissivity range

Point Source near-field

View complete reference list

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