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:

1. Input Processing & Unit Conversion — Convert user inputs to SI units; compute atmospheric pressure and air density at altitude.

2. Gas Properties (Cp polynomial, gamma) — Evaluate heat capacity polynomial and compute specific heat ratio.

3. Flow Regime (sonic vs subsonic) — Determine whether the discharge is choked or unchoked via critical pressure ratio.

4. Mass Flow Rate (vessel discharge) — Calculate mass flow from orifice conditions using Kakosimos discharge equations.

5. Exit Conditions (P, T, Mach, u, rho) — Compute pressure, temperature, Mach number, velocity, and density at the jet exit.

6. Equivalent Diameter (Ds) — Calculate the effective source diameter for flame length correlations.

7. Flame Length (Kalghatgi / CCPS) — Solve for flame length with the Chamberlain implicit equation (solid flame) or the CCPS correlation (point source).

8. Flame Geometry (solid flame only) — Compute tilt, lift-off, frustum length and base and tip widths (Yellow Book §6.5.3).

9. Surface Emissive Power (SEP) — Calculate radiated fraction and surface emissive power from flame area and heat of combustion.

10. Atmospheric Transmissivity — Yellow Book eq. 6.29 over the distance to the flame surface (solid flame) or CCPS 2.2.42 (point source).

11. Thermal Radiation at Distance — Compute incident radiation at a given distance using the frustum view factor (solid flame) or the point source formula.

12. Distance to Given Radiation (inverse Newton-Raphson) — Find the distance at which a specified radiation level occurs.

13. Probit Analysis (burns & fatality) — Convert thermal dose to burn probabilities and fatality using probit functions.

14. Domino Effect — TTF (Cozzani) — Estimate time-to-failure for nearby vessels under thermal radiation loading.

15. Fatality Estimation (concentric ring integration) — Integrate fatality probability over concentric rings to estimate total casualties.

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3. Principal Equations#

3.1 Input Processing and Unit Conversion#

Atmospheric pressure at altitude:

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

Variable	Description	Unit
$P_a$	Atmospheric pressure at altitude	Pa
$h$	Altitude above sea level	m

Reference: Standard barometric formula (ISO 2533)

Air density at altitude:

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

Variable	Description	Unit
$\rho_{air}$	Air density	kg/m³
$R_d$	Specific gas constant for dry air (287.05)	J/(kg·K)
$T_a$	Ambient temperature	K

3.2 Gas Properties#

Heat capacity polynomial (Cp):

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

Variable	Description	Unit
$C_{p,mol}$	Molar heat capacity at constant pressure	J/(mol·K)
$a$–$e$	Polynomial coefficients (cpga through cpge)	various
$T$	Internal gas temperature	K

Specific heat capacities and gamma:

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

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

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

Variable	Description	Unit
$C_p$	Specific heat at constant pressure	J/(kg·K)
$C_v$	Specific heat at constant volume	J/(kg·K)
$R$	Universal gas constant (8.31451)	J/(mol·K)
$MW$	Molecular weight	kg/mol
$\gamma$	Heat capacity ratio (must be > 1.0)	dimensionless

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):

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

Variable	Description	Unit
$\dot{m}$	Mass flow rate	kg/s
$C_d$	Discharge coefficient	dimensionless
$A$	Orifice area ($\pi d^2 / 4$)	m²
$P_0$	Internal pressure	Pa
$T_0$	Internal temperature	K

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):

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

Variable	Description	Unit
$D_s$	Equivalent diameter	m
$d$	Orifice diameter	m

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

3.7 Flame Length#

3.8 Solid Flame Geometry (Yellow Book §6.5.3)#

Expanded jet (steps 2–6). The jet velocity comes from the storage pressure and temperature, also with a known flow rate:

$\gamma = \left(1 - \frac{R_c}{C_p\,W_g}\right)^{-1}\qquad T_j = T_s\left(\frac{P_{air}}{P_{init}}\right)^{(\gamma-1)/\gamma}\qquad P_c = P_{init}\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}$

Choked flow ($P_{init}/P_{air} \geq ((\gamma+1)/2)^{\gamma/(\gamma-1)}$):

$M_j = \sqrt{\frac{(\gamma+1)\left(P_c/P_{air}\right)^{(\gamma-1)/\gamma} - 2}{\gamma-1}}$

Subsonic flow with orifice discharge:

$M_j = \sqrt{\frac{\sqrt{1 + 2(\gamma-1)F^2} - 1}{\gamma-1}}\qquad F = \frac{4\,\dot m}{\pi\,d_o^2\,P_{air}}\sqrt{\frac{R_c\,T_s}{\gamma\,W_g}}$

Subsonic flow with a known mass flow (isentropic expansion to $P_{air}$, consistent with the $T_j$ equation; with an orifice consistent with the flow it gives the same $M_j$):

$M_j = \sqrt{\frac{2}{\gamma-1}\left[\left(\frac{P_{init}}{P_{air}}\right)^{(\gamma-1)/\gamma} - 1\right]}$

$u_j = M_j\sqrt{\frac{\gamma\,R_c\,T_j}{W_g}}$

Effective source diameter $D_s$. By default («Expanded jet» block, Kalghatgi 1984 and Phast, engine 1.3.0) $D_s = \sqrt{4\dot m/(\pi\,\rho_{air}\,u_j)}$ (eq. 6.39) in any regime, with $u_j$ of the expanded jet from eq. 6.36. With the «Yellow Book literal» option, for choked flow the book's eq. 6.41 is used, $D_s = d\,\sqrt{\rho_j/\rho_{air}}$ with $\rho_j = P_c W_g/(R_c T_j)$; for subsonic flow, eq. 6.39, $D_s = \sqrt{4\dot m/(\pi\,\rho_{air}\,u_j)}$. With orifice discharge, $d$ is the orifice diameter. With a known mass flow, the diameter is not an input: mass flow, pressure and diameter are not independent (a given orifice at a given pressure passes a given flow). TekRisk uses the equivalent diameter of the ideal nozzle ($C_d = 1$) that discharges that flow:

$d_{eq} = \sqrt{\frac{4\,\dot m}{\pi\,P_{init}\sqrt{\dfrac{\gamma\,W_g}{R_c\,T_s}\left(\dfrac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}}}$

So with a known mass flow the flame grows with the flow and does not depend on any diameter the user might set.

Tilt (steps 7 and 13).

$R_w = \frac{u_w}{u_j}\qquad Ri(L_{b0}) = \left(\frac{g}{D_s^2\,u_j^2}\right)^{1/3} L_{b0}$

For $R_w \leq 0.05$ (jet-dominated flame):

$\alpha = (\Theta_{jv} - 90)\left(1 - e^{-25.6\,R_w}\right) + \frac{8000\,R_w}{Ri(L_{b0})}$

For $R_w > 0.05$ (wind-dominated):

$\alpha = (\Theta_{jv} - 90)\left(1 - e^{-25.6\,R_w}\right) + \frac{134 + 1726\sqrt{R_w - 0.026}}{Ri(L_{b0})}$

Lift-off and frustum length (steps 14–15).

$K = 0.185\,e^{-20\,R_w} + 0.015\qquad b = L_b\,\frac{\sin(K\alpha)}{\sin\alpha}$

In still air ($\alpha = 0$) $b = K\,L_b = 0.2\,L_b$; with $\alpha = 180°$, $b = 0.015\,L_b$.

$R_l = \sqrt{L_b^2 - b^2\sin^2\alpha} - b\cos\alpha$

Widths (steps 16–19).

$\frac{\rho_{air}}{\rho_j} = \frac{T_j\,W_{air}}{T_{air}\,W_g}\qquad Ri(D_s) = \left(\frac{g}{D_s^2\,u_j^2}\right)^{1/3} D_s\qquad C' = 1000\,e^{-100\,R_w} + 0.8$

$W_1 = D_s\left(13.5\,e^{-6R_w} + 1.5\right)\left\{1 - \left[1 - \frac{1}{15}\sqrt{\frac{\rho_{air}}{\rho_j}}\right]\exp\!\left(-70\,Ri(D_s)^{C' R_w}\right)\right\}$

$W_2 = L_b\left(0.18\,e^{-1.5\,R_w} + 0.31\right)\left(1 - 0.47\,e^{-25\,R_w}\right)$

Flame area (step 20). The frustum with both end discs (eq. 6.56):

$A = \frac{\pi}{4}\left(W_1^2 + W_2^2\right) + \frac{\pi}{2}\left(W_1 + W_2\right)\sqrt{R_l^2 + \left(\frac{W_2 - W_1}{2}\right)^2}$

The book offers the mean-width cylinder (eq. 6.57) as an alternative, and its example uses it; TekRisk publishes both and uses 6.56, because it is the surface over which the view factor is integrated.

Release orientation and 3D geometry. The book warns that its geometric transformation is only correct with the orifice in the vertical plane of the wind ($\Theta_j = \Theta_{jv}$). TekRisk offers the two orientations that satisfy it for any wind direction: vertical ($\Theta_j = 90°$: stack, vent) and horizontal downwind ($\Theta_j = 0°$: the worst case of a horizontal release of unknown heading). With the origin at the foot of the source, $x$ downwind and $z$ upwards:

$O = (0,\,0,\,h_s)\qquad P = O + b\,(\cos\Theta_j,\,0,\,\sin\Theta_j)\qquad T = P + R_l\,(\sin\varphi,\,0,\,\cos\varphi)$

with $\varphi = 90° - \Theta_j + \alpha$ the angle of the flame axis from the vertical. By construction $|T - O| = L_b$ (eq. 6.50 is the law of cosines of triangle $OPT$). The part of the flame below ground (low horizontal release) does not radiate and is discarded, with a warning.

Reference: Yellow Book (CPR 14E, 3rd ed.), §6.5.3, eqs. 6.31–6.57; Chamberlain (1987)

Radiation presets: Yellow Book, Phast and ALOHA#

The solid flame uses the Yellow Book by default. Under Advanced radiation parameters, the settings button next to «Calculation method», any user chooses a base methodology (which sets every block in the table) or combines the blocks as they see fit; it is the only place where the methodology is chosen. If you change any block from its methodology, the scenario becomes custom and is shown as such in the results, the PDF and the calculation report.

Block	Yellow Book (CPR 14E §6.5.3)	Phast-based (DNV cone)	ALOHA-based (5.4.4, §6.4)
Expanded jet and $D_s$	Eqs. 6.33–6.36 and $D_s$ from eq. 6.39 with the expanded jet (the book's and Kalghatgi 1984's definition); the book's eq. 6.41 shortcut remains as the «Yellow Book literal» option	Same as the Yellow Book	Pipe exit (§3.5.2 and §6.4.2): adiabatic expansion from the choked-flow interface $P_{ia}$ to atmospheric pressure, $T_j = T_s(P_{air}/P_{ia})^{(\gamma-1)/\gamma}$ and $D_s = d\,\sqrt{\rho_j/\rho_{air}}$ with the pipe diameter
Flame length	Kalghatgi: root $Y$ of eq. 6.42, $L_{b0} = Y\,D_s$ and eq. 6.44	Same as the Yellow Book	$L_b = 105.4\,D_s\left[1 - 6.07\cdot10^{-3}(\Theta_j - 90)\right]$ (Lees 16.18.21b); $L_{b0}$ is solved from eq. 6.44 for the tilt
Base width $W_1$	Eq. 6.54, power form	Eq. 6.54, product form	Power form
Radiated fraction	$F_s = 0.21\,e^{-0.00323\,u_j} + 0.11$	Same as the Yellow Book	$F_s = 0.21\,C_{MW}\,e^{-0.00323\,u_j} + 0.11$, with $C_{MW} = 1$ ($MW < 21$), $\sqrt{MW/21}$ (21–60) or 1.69 ($MW > 60$)
Transmissivity	Bagster–Pitblado, $\tau = 2.02\,(p_w x)^{-0.09}$	Wayne (1991): H₂O and CO₂ absorption	Cook, $\tau = 1.389 - 0.135\,\log_{10}(p_w x)$

The advanced settings also allow combining the blocks as you see fit and setting the receptor height. The default is 0 m: zones, receptors and the individual-risk field are evaluated at ground level, as in the Yellow Book and ALOHA.

What to expect from each preset with a small vertical leak#

With a small vertical leak the Yellow Book may not reach the 10 kW/m² zone (and with the literal eq. 6.41 shortcut not the 5 kW/m² one either) while the ALOHA-based preset reaches both. Nothing caps the Yellow Book: the engine's only cap is the black-body emissive power (SEP ≤ 300 kW/m², optional and off by default) and the Yellow Book stays below it. What changes is the effective diameter $D_s$ and the geometry. Example with propane, 1 in orifice at 8 bar abs and 25 °C, vertical release 1 m above ground, 4 m/s wind, mass flow 0.672 kg/s (the radiated heat is the same in every row; what changes is the surface it is spread over and where the receptor sees it from):

Configuration	Flame	Area	SEP (kW/m²)	Peak flux at receptor height	10 / 5 / 2 kW/m² zones
Yellow Book (expanded-jet Ds), receptor at 0 m	7.2 m	25 m²	199	8.3 kW/m² at 2.75 m	no / 7.5 / 13.3 m
Yellow Book, receptor at 1.5 m	7.2 m	25 m²	199	16.8 kW/m²	5.4 / 8.7 / 13.8 m
Yellow Book literal (eq. 6.41 shortcut), receptor at 0 m	12.6 m	77 m²	64	2.9 kW/m² at 4.5 m	no / no / 10.7 m
ALOHA-based	3.5 m	6 m²	799	27.2 kW/m² at 1.75 m	5.8 / 8.3 / 13.2 m
ALOHA-based, SEP cap on	3.5 m	6 m²	799 → 300	10.2 kW/m² at 1.75 m	2.1 / 4.9 / 8.1 m
Yellow Book, horizontal release	11.1 m	59 m²	84	84 kW/m² (inside the flame)	14.8 / 16.6 / 20.0 m

Reading:

1. The length is set by the effective diameter $D_s$. The book's eq. 6.41 shortcut (gas density at the exit plane) gives 12.6 m; the book's and Kalghatgi's (1984) general definition, the jet expanded to atmospheric pressure of eq. 6.39 that Phast uses, gives 7.2 m and is the preset's since engine 1.3.0. ALOHA starts from the gas expanded at the pipe exit and uses the laboratory correlation $105.4\,D_s$: 3.5 m and an emitter so concentrated (799 kW/m²) that it exceeds black-body emission; with the cap on TekRisk clips it to 300. For comparison, measured emissive powers of hydrocarbon flares (Chamberlain, 1987) range from 60 to 230 kW/m².
2. In a vertical release the ground receptor sees the flame edge-on. It starts lifted 1.2 m above the orifice, with a 3 cm base and a 2.2 m tip; from the ground the peak flux is 8.3 kW/m² and the 10 kW/m² zone is not reached. Raising the receptor to 1.5 m takes it to 16.8 kW/m² and the 10 kW/m² zone appears at 5.4 m. When no zone of 5 kW/m² or more is reached (literal-shortcut row), TekRisk flags it in Results with the vertical_flame_low_ground_flux warning.
3. The worst case of a small leak is the horizontal downwind orientation, not the vertical one: the same leak laid over gives 14.8, 16.6 and 20 m with the Yellow Book. For a risk study, evaluate that orientation whenever the leak direction is unknown.

Guide to the solid-flame blocks#

Each block under Advanced radiation parameters offers two or three options. For each one: what it is, when to use it and what its limitations are. When a calculation runs into one of those limitations, TekRisk flags it in Results (extrapolated τ, wind-sensitive W1, SEP above or capped at the black body, flame below ground, vertical release with no zone of 5 kW/m² or more at receptor height).

Expanded jet and effective diameter Ds#

Expanded jet (Kalghatgi 1984, Phast; eq. 6.39). The jet expands isentropically to atmospheric pressure (eqs. 6.33–6.36: $T_j$, $P_c$, $M_j$ from eq. 6.35 in choked flow) and $D_s$ follows from the book's general definition, $D_s = \sqrt{4\dot m/(\pi\,\rho_{air}\,u_j)}$: the diameter of the imaginary nozzle releasing air at ambient density with the same mass flow and the expanded-jet velocity. It is exactly what Kalghatgi (1984) does, who in choked flow "replaces the burner by an equivalent nozzle at whose exit the flow has expanded to atmospheric pressure", and what Phast does (for its report of an LPG release at 11.6 kg/s, this route reproduces its 45 m still-air flame and 24 m with wind). When: default, in any regime; it does not depend on the orifice, so with a known flow the diameter is irrelevant. Limitations: Kalghatgi's correlation was fitted with 1–10 mm burners and natural-gas flares; for choked jets of heavy gases the SEP rises (less area) and may approach the black-body cap.

Yellow Book literal (eq. 6.41, example §6.6.2). The eqs. 6.40–6.41 shortcut for choked flow: gas density at the exit plane, $\rho_j = P_c W_g/(R_c T_j)$, and "jet diameter ≈ orifice diameter", $D_s = d_o\sqrt{\rho_j/\rho_{air}}$ (with a known flow, the equivalent nozzle $d_{eq}$). When: reproducing the book's example §6.6.2 or studies made with that recipe. Limitations: it inflates $D_s$ by $\sqrt{P_c/P_{air}}$ relative to the general definition: ×2 at 5 bar, ×7 in the book's example at 100 bar (a 141 m flame for 30 kg/s of methane). Flames 1.5–2 times longer than Phast and a proportionally lower SEP; in a vertical release it underestimates the flux at ground level.

ALOHA pipe exit (§3.5.2 and §6.4.2). ALOHA's pipeline model (Wilson) assumes nearly isothermal flow up to the last 200 diameters, where the flow is choked, and adiabatic expansion from there to atmospheric pressure. TekRisk takes the pressure at that interface, $P_{ia} = \frac{4\,\dot m}{\pi d^2}\sqrt{R_c T_s/W_g}$, the exit temperature $T_j = T_s\,(P_{air}/P_{ia})^{(\gamma-1)/\gamma}$, the density $\rho_j$ at atmospheric pressure and the velocity of the expanded gas at the exit section, $u_j = \dot m/(\rho_j\,\pi d^2/4)$; with it eq. 6.39 gives exactly $D_s = d\,\sqrt{\rho_j/\rho_{air}}$, ALOHA's form. With the mass flow ALOHA prints it reproduces its flame length in three real runs (propane at 4.7 and 4.8 atm: 6.9 m versus 7; methane at 20 atm: 5.4 m versus 5). It asks for the pipe diameter also with a known flow. $P_{ia}$ is limited to the range between atmospheric and storage pressure: below it the gas leaves without expanding; above it the mass flow is not consistent with the diameter and the pressure, the calculation continues with the storage pressure and Results flags it (pipe_flow_exceeds_storage_pressure). When: comparing with ALOHA and full-bore pipe ruptures. Limitations: the exit velocity it implies is higher than an isentropic jet's (several times the speed of sound at high pressure), but it only enters the radiated fraction, which at those velocities is already at its 0.11 minimum; ALOHA 5.4.7's zones fall below what these equations give with its own mass flow (see the SEP cap).

Flame length#

Kalghatgi (Yellow Book, eqs. 6.42–6.44). Solves the root $Y$ of Kalghatgi's implicit equation: the length depends on the fuel (molar mass, jet temperature and stoichiometric fraction), on the expanded jet velocity and on the wind through the Richardson number. When: risk analysis with either orientation; it is the correlation Chamberlain validated against industrial natural-gas and propane flares. Limitations: assumes a subsonic expanded gas jet; $\beta = 2.85$ is fitted for paraffins (non_paraffin_beta warning for other gases); at very low wind the length barely changes but the base width does (see W1).

L = 105.4·Ds (Lees 16.18.21b, ALOHA). Laboratory correlation: the length is proportional to the expanded source diameter, with a tilt factor $[1 - 6.07\cdot10^{-3}(\Theta_j - 90)]$; $L_{b0}$ is then back-solved from eq. 6.44. When: comparing with ALOHA or with studies based on Lees; quick screening. Limitations: independent of fuel and wind, so methane and propane get the same flame for the same diameter; it was derived from small-scale vertical jets and for large releases gives shorter flames than Kalghatgi (10 m versus the Yellow Book's 15.7 m in the LPG example above, both with the expanded jet); ALOHA only models vertical releases.

Base width W1#

Power form (Yellow Book, ALOHA). Eq. 6.54 as printed by the Yellow Book and by ALOHA, which states it was checked against Chamberlain: the term $Ri(D_s)^{C'R_w}$ is a power. When: Yellow Book preset; reproduces example §6.6.2. Limitations: except in very strong wind the base comes out narrow, of the order of $D_s$; in near-calm wind ($R_w < 0.002$) it jumps to $15\,D_s$ and the result is sensitive to small wind changes (w1_low_wind_sensitive warning).

Product form (Lees, Casal, Phast). The transcription in Lees (2001), followed by Casal and Kakosimos: $Ri(D_s)\cdot C'\cdot R_w$. It is the one DNV Phast uses: with it TekRisk reproduces the $W_1 = 2.59$ m of a real Phast report (2.58 m). When: Phast-based preset and calculations made with Lees or Casal. Limitations: in normal wind it gives a wide base, of the order of 10–20 $D_s$, very different from the power form (2.6 m versus 0.12 m in the Phast case); it does not reproduce the Yellow Book example.

Radiated fraction Fs#

Eq. 6.59 (Yellow Book). $F_s = 0.21\,e^{-0.00323\,u_j} + 0.11$: the fraction of the heat released that leaves as radiation, decreasing with jet velocity. When: default; it is Chamberlain's. Limitations: fitted with natural-gas flares; for heavier gases (propane, butane, LPG), which burn with more luminous flames, it tends to underestimate the emissive power.

With molecular-weight factor C_MW (Cook, ALOHA). Multiplies the exponential term by $C_{MW} = 1$ if $MW < 21$ g/mol, $\sqrt{MW/21}$ between 21 and 60 g/mol and 1.69 above. When: gases heavier than natural gas and comparisons with ALOHA. Limitations: an empirical scaling from Cook et al. (1990), not in the Yellow Book; for heavy gases the SEP may exceed black-body emission (sep_above_black_body warning); with the cap on it is clipped to 300 kW/m² (sep_capped_black_body warning) and the flux no longer grows with $F_s$.

Atmospheric transmissivity τ#

Bagster–Pitblado (Yellow Book, eq. 6.29). $\tau = 2.02\,(p_w\,x)^{-0.09}$, with $x$ measured to the flame surface. When: default for risk analysis. Limitations: published only for $10^4 \le p_w x \le 10^5$ N/m; at 25 °C and 50 % humidity that is path lengths of about 6 to 63 m. Below the range the formula exceeds 1 (TekRisk caps it) and above it is extrapolated, although there it practically matches Cook. The transmissivity_extrapolated warning means some zone or receiver fell outside the range.

Wayne (1991), Phast's. Absorption by water vapour and CO₂ along the optical path, with $X_{H_2O} = HR\,x\,p_{sat}[\text{mmHg}]\cdot 288.651/T_{air}$ and $X_{CO_2} = 273\,x/T_{air}$. When: Phast-based preset; with it TekRisk reproduces the ground-level radiation profile of a Phast report within ±1.5 % (with Bagster–Pitblado or Cook it comes out 1–2 % low). Limitations: published for 10 to 1000 m; below that it is extrapolated, although in the near field τ ≈ 1. It accounts for water vapour and CO₂, not smoke or fog.

Cook (ALOHA). $\tau = 1.389 - 0.135\,\log_{10}(p_w x)$, from NOAA OR&R 43. When: comparing with ALOHA and near field (under 6 m), where it does not exceed 1. Limitations: no published validity range; beyond about 60 m it gives the same as eq. 6.29; like it, it only accounts for water vapour and ambient CO₂, not smoke or fog.

Receptor height#

Height above ground at which zones, receivers and the individual-risk field are evaluated. 0 m is the Yellow Book and ALOHA convention. 1.5 m approximates a standing person: with a lifted vertical flame, raising the receptor increases the flux beneath the flame (in the LPG example, the 10 kW/m² zone goes from not reached to about 11 m). Limitations: a single height for the whole scenario, not per receiver; it does not replace an elevated-equipment calculation nor account for obstacles.

3.9 Surface Emissive Power (SEP)#

Solid flame (steps 21–23).

$Q' = \dot m\,\Delta H_c\qquad F_s = 0.21\,e^{-0.00323\,u_j} + 0.11\qquad SEP = \frac{F_s\,Q'}{A}$

No molecular-weight correction factor (it is not in eq. 6.59) and no soot ($\varsigma = 0$, as in the book's example). By default the SEP is not capped; if it exceeds 300 kW/m² (black-body emission of a hydrocarbon flame), Results flags it. The cap is optional (see below).

Black-body SEP cap#

A hydrocarbon flame at about 1500 K cannot radiate more than a black body at that temperature ($\sigma T^4 \approx 300$ kW/m²), and emissive powers measured on flares (Chamberlain, 1987) range from 60 to 230 kW/m². The $F_s\,Q'/A$ balance, however, can give much more when the computed flame is short and its area small: with propane and LPG the ALOHA-based preset reaches 700–800 kW/m², and with high-pressure methane more than 2000. The Black-body SEP cap option, under Advanced radiation parameters, decides what to do with that excess. It is not a preset block: changing it does not turn the methodology into "Custom".

• Off (default since engine 1.4.0). The SEP is published as it comes out of the balance and Results flags it (sep_above_black_body). This is what ALOHA does, which does not cap the emissive power. The result stays on the conservative side.
• On. The SEP is capped at 300 kW/m² and Results flags it (sep_capped_black_body). Above the cap the flux no longer grows with $F_s$ or with the mass flow.

It only has an effect when the balance exceeds 300 kW/m². In the examples on this page the Yellow Book and Phast stay below it (199–236 kW/m²); the ALOHA-based preset with heavy gases almost always exceeds it.

When to leave it off: conservative consequence and risk studies, and comparisons with ALOHA. Comparison with three real ALOHA 5.4.7 runs (2 in pipeline, vertical release, ALOHA's mass flow as a known flow; propane unless stated), 10 / 5 / 2 kW/m² zones:

Run	ALOHA	Uncapped	Capped
4 m/s, 2.10 kg/s	10 / 12 / 19 m	11.4 / 15.7 / 24.5 m	7.1 / 10.5 / 16.1 m
1.5 m/s, 2.03 kg/s	10 / 12 / 18 m	10.5 / 15.1 / 23.5 m	5.2 / 9.0 / 14.8 m
Methane 20 atm, 1.5 m/s, 6.62 kg/s	10 / 12 / 18 m	16.2 / 22.7 / 35.1 m	5.3 / 8.2 / 13.4 m

With ALOHA's «Max Burn Rate» neither option reproduces its zones: the flame length matches in all three runs, but uncapped TekRisk comes out +4 to +30 % above with propane (on the conservative side) and capped −12 to −48 %. The most likely cause is not the cap: in a pipeline release that value is the initial depressurization peak, and ALOHA appears to compute the radiation with a time-averaged flow rate. With a flow rate of 1.05–1.36 times the hourly mean (the one given by the «Total Amount Burned»), uncapped TekRisk reproduces the 5 and 2 kW/m² zones of all three runs within ALOHA's rounding. ALOHA's 10 kW/m² zone always comes out at 10 m and is drawn as a regular polygon: it looks like an ALOHA minimum display distance, not a result. Details in Model validation.

When to turn it on: when the study calls for a realistic rather than conservative estimate of the near field (for example, design distances to equipment, where overestimating also has a cost), or to reproduce TekRisk results computed before engine 1.4.0, which always capped. Results saved before that change keep the cap in Results and in the calculation reports; when recalculated, the scenario's option applies.

Variable	Description	Unit
$Q'$	Heat released by combustion	kW
$F_s$	Fraction of the heat radiated from the surface	dimensionless
$SEP$	Surface emissive power	kW/m²
$\Delta H_c$	Heat of combustion	kJ/kg

Reference: Yellow Book, eqs. 6.58–6.60; Chamberlain (1987)

Point source. It does not use a SEP: the radiated fraction $f_r$ is chosen by the user (section 3.11).

3.10 Atmospheric Transmissivity#

Solid flame. Yellow Book eq. 6.29 (Bagster and Pitblado), with the partial water vapour pressure from the book's appendix 2.1 and the optical path measured to the flame surface ($x$ in eq. 6.63):

$\tau = 2.02\,(p_w\,x)^{-0.09}\qquad p_w = RH\cdot p_w^{sat}(T_a)$

TekRisk clamps $\tau$ to $[0,\,1]$ and warns when a published point falls outside the book's validity range ($10^4 \leq p_w\,x \leq 10^5$ N/m). The book's example uses the absorption charts 6.2 and 6.3 (eq. 6.24) instead.

Point source (CCPS Eqs. 2.2.42–2.2.43).

$p_w = 1013.25 \times RH \times \exp(14.4114 - 5328 / T_a)\qquad \tau = 2.02 \times (p_w \times x)^{-0.09}$

Variable	Description	Unit
$\tau$	Atmospheric transmissivity	dimensionless
$p_w$	Partial pressure of water vapour	Pa
$RH$	Relative humidity (as a 0–1 fraction)	dimensionless
$x$	Optical path	m

References: Yellow Book, eq. 6.29; 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. The solid flame does not use this inverse calculation: it solves the isolines per azimuth over the ground-level field (section 3.11).

3.13 Probit Analysis#

Thermal dose:

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

Variable	Description	Unit
$D$	Thermal dose	$(W/m^2)^{4/3} \cdot s$
$t_{exp}$	Exposure time	s
$q$	Thermal radiation (converted from kW to W)	W/m²

Probit equations:

Effect	Equation	Reference
First degree burn	$Pr = -39.83 + 3.0186 \ln(D)$	TNO Green Book, Eq. 3.4, p. 20
Second degree burn	$Pr = -43.14 + 3.0186 \ln(D)$	TNO Green Book, Eq. 3.7, p. 20
Fatality (CCPS)	$Pr = -14.9 + 2.56 \ln(D / 10000)$	CCPS, p. 269
Fatality (TNO)	$Pr = -36.38 + 2.56 \ln(D)$	TNO Green Book, Eq. 3.5, p. 20

Probit to probability conversion:

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

Variable	Description	Value
$f_k$	Protection factor (no protective clothing)	1.0
$Pr$	Probit value	dimensionless
$\text{erf}$	Error function (Taylor series, 50 terms)	dimensionless

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

3.14 Domino Effect — Time to Fail (Cozzani)#

TTF correlations by vessel type:

Vessel Type	Equation	Reference
Atmospheric	$TTF = \exp(-1.13 \ln(q) - 2.667 \times 10^{-5} V + 9.877)$	Cozzani et al.
Pressurized	$TTF = \exp(-0.95 \ln(q) + 8.845 V^{0.032})$	Cozzani et al.
Full engulfment	$TTF = \exp(-1.29 \ln(q) + 10.97 V^{0.026})$	Cozzani et al.

Variable	Description	Unit
$TTF$	Time to fail	s
$q$	Thermal radiation received	kW/m²
$V$	Vessel volume	m³

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).

Parameter	Default Value
Ring increment	5 m
Maximum radius	10 km
Minimum probability threshold	0.1%
TNO methodology	Used for ring-based fatalities

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

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4. Justification of Methodology Selection#

4.1 Chamberlain Solid Flame Model (Yellow Book)#

The Chamberlain (1987) model is the one the Yellow Book selects for jet fires (§6.4.2) because:

• It represents the flame as a cone frustum radiating as a solid body, with shape correlations validated in wind tunnels and field trials onshore and offshore.
• It reproduces the wind effects on the flame tilt, lift-off and widths, which the point source cannot see.
• Its surface allows a physical view factor in the near field, where the point source is no longer applicable.
• It is the reference model of ALOHA (NOAA/EPA) and the basis of the frustum models in SAFETI and Phast.

TekRisk follows the letter of the book in eqs. 6.30–6.60 (including the power form of eq. 6.54, see section 3.8) and applies the complete method for radiation: view factor of the actual frustum and optical path to the flame surface.

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#

Parameter	Typical Range	Notes
Internal pressure	1–200 atm	Higher pressures may violate ideal gas assumption
Orifice diameter	1–500 mm	Very large diameters may produce non-jet behavior
Molecular weight	2–150 g/mol	MW correction factor applied for SEP
Wind speed	0–30 m/s	Model validated primarily for moderate winds
Gas temperature	> boiling point	Must be gas phase at release conditions
Heat capacity ratio	$\gamma > 1.0$	Strictly gas-phase releases only
Hole angle (solid flame)	90° or 0°	90° = vertical; 0° = horizontal downwind. The point source does not depend on the angle

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7. References#