Jet Fire

1. Introduction and General Description

A jet fire is a turbulent diffusion flame that results from the combustion of a pressurized gas or volatile liquid released continuously through an orifice or pipe failure. Unlike pool fires, jet fires are characterized by high momentum discharge, directional flame geometry, and intense localized thermal radiation.

1.1 Implemented Models

TekRisk PRO implements two complementary methodologies for jet fire consequence analysis:

1.2 Source Types

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2. Calculation Sequence

The algorithm follows 15 sequential stages:

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3. Principal Equations

3.1 Input Processing and Unit Conversion

Atmospheric pressure at altitude:

$P_a = 101325 \times (1 - 2.5577 \times 10^{-5} \times h)^{5.25588}$

Reference: Standard barometric formula (ISO 2533)

Air density at altitude:

$\rho_{air} = \frac{P_a}{R_d \times T_a}$

3.2 Gas Properties

Heat capacity polynomial (Cp):

$C_{p,mol} = a + bT + cT^2 + dT^3 + eT^4$

Specific heat capacities and gamma:

$C_p = \frac{C_{p,mol}}{MW} \times 1000$

$C_v = C_p - \frac{R}{MW} \times 1000$

$\gamma = \frac{C_p}{C_v}$

3.3 Flow Regime Determination

Critical pressure ratio (sonic threshold):

$\frac{P_0}{P_a} \leq \left(\frac{\gamma + 1}{2}\right)^{\gamma / (\gamma - 1)}$

If the pressure ratio is less than or equal to the critical value, the flow is subsonic (unchoked). Otherwise, the flow is sonic/supersonic (choked).

Reference: Kakosimos, Eq. B2.14, p. 36

3.4 Mass Flow Rate from Vessel Discharge

General discharge equation (Kakosimos B2.13):

$\dot{m} = C_d \times A \times P_0 \times K \times \sqrt{\frac{MW}{\gamma \times R \times T_0}}$

Factor K for sonic flow (Kakosimos B2.14):

$K = \gamma \times \left(\frac{2}{\gamma + 1}\right)^{(\gamma + 1) / (2(\gamma - 1))}$

Factor K for subsonic flow (Kakosimos B2.15):

$K = \sqrt{\frac{2\gamma^2}{\gamma - 1} \times \left(\frac{P_a}{P_0}\right)^{2/\gamma} \times \left(1 - \left(\frac{P_a}{P_0}\right)^{(\gamma-1)/\gamma}\right)}$

Reference: Kakosimos, K.E., "Complex Hazardous Activities", Eqs. B2.13–B2.15, p. 36

3.5 Exit Conditions

Exit pressure for known flow (adiabatic, Kakosimos C2.55):

$P_{exit} = P_0 \times \left(\frac{2}{1 + \gamma}\right)^{\gamma / (\gamma - 1)}$

Exit pressure for orifice discharge (Kakosimos C2.52):

$P_{exit} = \frac{4\dot{m}}{\pi d^2} \times \sqrt{\frac{R \times T_o}{\gamma \times MW}}$

where $T_o = 2T_0 / (1 + \gamma)$ (Kakosimos C2.54)

Exit temperature (adiabatic, Kakosimos C2.56):

$T_j = T_0 \times \left(\frac{P_a}{P_0}\right)^{(\gamma - 1) / \gamma}$

Mach number at exit (sonic flow):

$M_j = \sqrt{\frac{(\gamma + 1)(P_{exit}/P_a)^{(\gamma-1)/\gamma} - 2}{\gamma - 1}}$

Exit velocity (Kakosimos C2.50):

$u_j = M_j \times \sqrt{\frac{\gamma \times R \times T_j}{MW}}$

Exit density (ideal gas):

$\rho_j = \frac{P_{exit} \times MW}{R \times T_j}$

References: Kakosimos, Eqs. C2.50, C2.52, C2.54–C2.56, pp. 108–109; TNO Yellow Book, Eqs. 6.33, 6.36

3.6 Equivalent Diameter

For known mass flow (Kakosimos C2.59):

$D_s = \sqrt{\frac{4\dot{m}}{\pi \times \rho_{air} \times u_j}}$

For orifice discharge (Kakosimos C2.60):

$D_s = d \times \sqrt{\frac{\rho_j}{\rho_{air}}}$

Reference: Kakosimos, Eqs. C2.59–C2.60, p. 110

3.7 Flame Length

3.8 Flame Geometry (Solid Plume Only)

Flame tilt angle (Kakosimos C2.68):

$R_1 = u_w / u_j$

Richardson number:

$Ri = L_{Bo} \times \left(\frac{g}{D_s^2 \times u_j^2}\right)^{1/3}$

For $R_1 \leq 0.05$:

$\alpha = (\theta - 90)(1 - e^{-25.6 R_1}) + \frac{8000 R_1}{Ri}$

For $R_1 > 0.05$:

$\alpha = (\theta - 90)(1 - e^{-25.6 R_1}) + \frac{134 + 1726\sqrt{R_1 - 0.026}}{Ri}$

Reference: Kakosimos, Eq. C2.68, p. 111; TNO Yellow Book

Lift-off distance:

For zero wind: $b = 0.2 \times L_f$

For flame angle > 175 deg: $b = 0.015 \times L_f$

Otherwise (Chamberlain):

$K_{lo} = 0.187 \exp(-20 R_1) + 0.015$

$b = L_f \times \frac{\sin(K_{lo} \times \alpha)}{\sin(\alpha)}$

Reference: TNO Yellow Book, Eq. 6.49, p. 6.57; Chamberlain (1987)

Frustum length:

$R_L = \sqrt{L_f^2 - b^2 \sin^2(\alpha)} - b \cos(\alpha)$

Base width (Chamberlain):

$C = 1000 \exp(-100 R_1) + 0.8$

$Ri_{Ds} = D_s \times \left(\frac{g}{u_j^2 \times D_s^2}\right)^{1/3}$

$W_1 = D_s \times (13.5 e^{-6R_1} + 1.5) \times \left(1 - e^{-70 Ri_{Ds}^{CR_1}} \left(1 - \frac{1}{15}\sqrt{\rho_r}\right)\right)$

where $\rho_r = T_j MW_{air} / (T_a MW)$ is the relative density.

Reference: Chamberlain (1987); ALOHA Technical Documentation, p. 75; Kakosimos Eq. C2.73

Top width:

$W_2 = L_f \times (0.18 e^{-1.5 R_1} + 0.31) \times (1 - 0.47 e^{-25 R_1})$

Reference: Chamberlain (1987)

Flame surface area (frustum):

$A_1 = \sqrt{R_L^2 + \left(\frac{W_2 - W_1}{2}\right)^2}$

$A_{flame} = \frac{\pi}{4}(W_1^2 + W_2^2) + \frac{\pi}{2}(W_1 + W_2) A_1$

3.9 Surface Emissive Power (SEP)

Molecular weight correction factor (ALOHA):

Reference: ALOHA Technical Documentation, NOAA/EPA, p. 72

Radiated fraction (Chamberlain 1987):

$F_s = 0.21 \times C_{mw} \times \exp(-0.00323 \times u_j) + 0.11$

Surface Emissive Power:

$SEP = F_s \times \frac{\dot{m}}{A_{flame}} \times \Delta H_c$

Reference: Chamberlain, G.A. (1987), Chem. Eng. Res. Des., 65; ALOHA Technical Documentation, p. 72

3.10 Atmospheric Transmissivity

Wayne model (CCPS Eqs. 2.2.42–2.2.43):

Partial vapor pressure of water:

$p_w = 1013.25 \times RH \times \exp(14.4114 - 5328 / T_a)$

Atmospheric transmissivity:

$\tau = 2.02 \times (p_w \times x)^{-0.09}$

Reference: CCPS, "Guidelines for CPQRA", 2nd Ed., Eqs. 2.2.42–2.2.43, p. 209

3.11 Thermal Radiation at Distance

3.12 Distance to Given Radiation (Inverse Calculation)

The distance $x$ at which a specified thermal radiation $q^*$ is received is found by solving:

$q(x) - q^* = 0$

This is solved using the Newton-Raphson method (npm package newton-raphson-method), with the flame top width as the initial guess. A fallback iterative method with 0.1 m increments is also implemented.

3.13 Probit Analysis

Thermal dose:

$D = t_{exp} \times (q \times 1000)^{4/3}$

Probit equations:

Probit to probability conversion:

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

References: TNO Green Book (CPR 16E), p. 20; CCPS, p. 269

3.14 Domino Effect — Time to Fail (Cozzani)

TTF correlations by vessel type:

Full engulfment criterion: Equipment is considered fully engulfed when its distance from the jet fire source is less than $1.1 \times L_f$ (10% safety margin).

Domino probit (Cozzani):

$Pr_{domino} = 9.25 - 1.847 \times \ln(TTF / 60)$

Reference: Cozzani, V. et al., "The assessment of risk caused by domino effect in quantitative area risk analysis", Journal of Hazardous Materials, p. 300

3.15 Fatality Estimation

Fatalities are estimated using concentric ring integration:

1. The area around the source is divided into concentric rings of width $\Delta r = 5$ m
2. For each ring at distance $r_i$:
   • Thermal radiation $q(r_i)$ is calculated
   • Thermal dose $D_i = t_{exp} \times (q_i \times 1000)^{4/3}$ is computed
   • Probit value is converted to fatality probability $P_i$
   • Ring area: $A_i = \pi (r_{out}^2 - r_{in}^2)$ where $r_{in} = r_i - \Delta r/2$, $r_{out} = r_i + \Delta r/2$
   • Fatalities in ring: $F_i = A_i \times \rho_{pop} \times P_i / 100$
3. Total fatalities: $F = \sum F_i$ for all rings where $P_i \geq 0.1\%$
4. If $F > 0.6$, result is $\lceil F \rceil$; otherwise result is 0

Polygon receiver exclusion: When polygon receivers are defined, their areas are subtracted from the concentric rings to avoid double-counting population (polygon populations are calculated separately with distributed spatial discretization).

Reference: CCPS, "Guidelines for CPQRA", 2nd Ed., p. 273

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4. Justification of Methodology Selection

4.1 Chamberlain Frustum Model (Solid Plume)

The Chamberlain (1987) model was selected for the Solid Plume approach because:

• It provides a validated frustum geometry for the flame shape, allowing accurate near-field radiation calculations
• The model has been extensively validated against full-scale natural gas jet fire experiments
• It accounts for wind effects on flame tilt, lift-off, and width variation
• The view factor calculation captures the directional nature of radiation from an extended source
• It is the standard model used by ALOHA (NOAA/EPA) and recommended in the TNO Yellow Book

4.2 CCPS Point Source Model

The Point Source model was selected as an alternative because:

• It provides conservative estimates suitable for preliminary hazard assessment
• It requires fewer input parameters (no flame geometry needed for known flow)
• The model is computationally simpler and avoids view factor convergence issues
• It is recommended by CCPS for far-field radiation estimates where flame geometry is less critical

4.3 Newton-Raphson Method

Newton-Raphson iteration is used for two purposes:

1. Flame length calculation (solving the non-linear $Y$ equation) — Provides rapid convergence (typically 3–5 iterations) for the implicit Chamberlain equation
2. Inverse distance calculation — Finding the distance at which a given thermal radiation level occurs

The newton-raphson-method npm package is used for the inverse distance calculation, with the flame top width as the initial guess.

4.4 Dual Probit Methodologies

Two probit approaches are available:

• TNO (default for ring-based fatalities): Standard European methodology, widely used in QRA
• CCPS (default for JetFire probit): Standard American methodology, mathematically equivalent when properly normalized

4.5 Cozzani Domino Correlations

The Cozzani correlations are the only published empirical correlations specifically developed for estimating time-to-failure of industrial vessels under thermal radiation loading. They are supported by experimental data and distinguish between atmospheric and pressurized vessel behavior.

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5. Model Limitations

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6. Range of Applicability

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7. References