📑 Table of Contents (Tap to view sections)
1. Problem Statement & Facility Architecture
An onshore natural gas processing terminal must design an elevated emergency blowdown flare stack. The flare system must safely dispose of the maximum credible emergency relief load: an instantaneous total platform depressurization initiated by a fire detection interlock.
Flame Center (xc, H + yc)
( * ) <- Flame deflected by wind uw
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/ | \ Thermal Radiation K (kW/m²)
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=================[ Tip ]==+========\========================== Grade Level
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Stack | H |
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[ Base ]======[ R_sterile ]======[ Boundary Fence ]
Relieving Stream & Environmental Parameters
- Relief Gas Flow Rate ($Q$): $120.0\text{ MMSCFD}$ ($38.54\text{ kg/s} = 138,750\text{ kg/h}$)
- Gas Composition: Rich natural gas ($90.0\%\text{ CH}_4$, $6.0\%\text{ C}_2\text{H}_6$, $3.0\%\text{ C}_3\text{H}_8$, $1.0\%\text{ N}_2$)
- Gas Molecular Weight ($M_w$): $17.65\text{ kg/kmol}$
- Gas Lower Heating Value ($\text{LHV}$): $48.20\text{ MJ/kg}$ ($20,720\text{ Btu/lb}$)
- Relief Temperature at Flare Tip ($T_{\text{tip}}$): $45.0\text{ }^\circ\text{C}$ ($318.15\text{ K}$)
- Flare Tip Diameter ($D_{\text{tip}}$): $0.6096\text{ m}$ ($24.0\text{ inches}$)
- Design Crosswind Velocity ($u_w$): $9.0\text{ m/s}$ ($32.4\text{ km/h} \approx 20.1\text{ mph}$)
- Relative Humidity ($\text{RH}$): $60\%$ at $25\text{ }^\circ\text{C}$ ambient
- Solar Radiation Flux ($K_{\text{solar}}$): $0.79\text{ kW/m}^2$ ($250\text{ Btu/hr}\cdot\text{ft}^2$)
Engineering Permissible Radiation Thresholds (API 521 Table 10)
| Zone / Exposure Condition | Maximum Allowable Incident Radiation ($K_{\text{total}}$) | Personnel / Structural Exposure Condition |
|---|---|---|
| Zone 1: Flare Base & Stack Access | $9.46\text{ kW/m}^2$ ($3000\text{ Btu/hr}\cdot\text{ft}^2$) | Short escape only (max 30 seconds with appropriate clothing) |
| Zone 2: Process Area & Operators | $4.73\text{ kW/m}^2$ ($1500\text{ Btu/hr}\cdot\text{ft}^2$) | Action required within 2 to 3 minutes; escape path |
| Zone 3: Property Boundary / Fence | $1.58\text{ kW/m}^2$ ($500\text{ Btu/hr}\cdot\text{ft}^2$) | Continuous personnel occupancy; no protective clothing |
Sizing Objectives
- Calculate the total heat release ($Q_h$) and radiant emissive fraction ($F$).
- Compute the flare tip exit velocity and Mach number ($M$).
- Determine the flame length ($L_f$) and flame center deflection coordinates $(x_c, y_c)$ using the Brzustowski-Sommer plume model.
- Calculate the minimum stack height ($H_{\text{stack}}$) required to keep radiation at the process area boundary ($50\text{ m}$ from base) below $4.73\text{ kW/m}^2$.
- Determine the ground-level sterile exclusion zone radius ($R_{\text{sterile}}$) for continuous boundary exposure ($1.58\text{ kW/m}^2$).
2. Governing Equations & Methodology
2.1 Total Heat Release Rate ($Q_h$)
The heat release rate of the flare represents the total thermodynamic energy rate discharged to the atmosphere:
where $\dot{m}$ is mass flow rate ($\text{kg/s}$) and $\text{LHV}$ is in $\text{kJ/kg}$ or $\text{kW}/(\text{kg/s})$.
2.2 Radiant Fraction ($F$)
Per API Standard 521 Section 5.7.2.3, the fraction of liberated heat radiated to the surrounding environment depends on the molecular weight, burner velocity, and soot-producing tendencies of the gas:
For $M_w = 17.65\text{ kg/kmol}$:
2.3 Flame Length ($L_f$) - API 521 / Brzustowski Correlation
The flame length represents the continuous combustion envelope from the tip to the plume burnout point:
2.4 Wind Deflection & Flame Center Coordinates
Under a crosswind $u_w$, the flame bends in the downwind direction. Brzustowski and Sommer define the dimensionless velocity momentum ratio:
The coordinates of the flame center relative to the flare tip are:
where the flame tilt angle $\theta$ relative to the vertical axis is given by:
2.5 API 521 Simple Point Source Radiation Equation
At any grade observation point located at distance $R$ from the flame center $(x_c, H + y_c)$:
where: * $K$ = Incident thermal radiation at the receiver ($\text{kW/m}^2$) * $\tau$ = Atmospheric transmissivity factor ($0.85\text{--}0.95$ depending on humidity and distance) * $R$ = True 3D straight-line distance from the flame center to the receiver ($1.8\text{ m}$ above grade):
3. Step-by-Step Engineering Calculations
Step 1: Heat Release & Gas Tip Exit Velocity
Mass Flow Rate:
Total Heat Release ($Q_h$):
In imperial units:
Volumetric Flow Rate at Tip Conditions ($1.013\text{ bar abs}, 45\text{ }^\circ\text{C}$):
Using real gas density ($\rho_{\text{tip}} = \frac{P M_w}{Z R T}$):
Tip Area ($D_{\text{tip}} = 0.6096\text{ m} = 24\text{''}$):
Tip Exit Velocity ($v_{\text{tip}}$):
Sonic Velocity and Mach Number:
For natural gas ($\gamma \approx 1.28$, $M_w = 17.65$):
API 521 Check: Mach number is $0.445 \le 0.50$. The flare tip operates within the recommended subsonic velocity envelope, avoiding extreme acoustic vibration and flame lift-off instability.
Step 2: Flame Length & Center Coordinates
Flame Length ($L_f$):
The burning flame envelope extends $67.0\text{ meters}$ into the air!
Flame Center Displacement under $9.0\text{ m/s}$ Crosswind:
Using the Brzustowski momentum ratio:
The flame center coordinates relative to the tip are:
Step 3: Determining Required Flare Stack Height ($H_{\text{stack}}$)
To protect personnel in the process unit ($50\text{ m}$ downwind from stack base, receiver height $1.8\text{ m}$), the total radiation must satisfy:
Subtracting solar radiation:
Required Distance from Flame Center ($R_{\text{req}}$):
Using average atmospheric transmissivity $\tau \approx 0.88$:
Solving for Stack Height $H_{\text{stack}}$:
At the process boundary point ($x = 50.0\text{ m}$, $y = 0\text{ m}$ downwind):
Adding a standard $10\%$ engineering safety margin for wind gust turbulence:
Step 4: Ground Radiation Profile & Sterile Exclusion Zone Sizing
With the stack height fixed at $H_{\text{stack}} = 60.0\text{ m}$, the flame center elevation above grade is:
Effective vertical distance to person ($h = 1.8\text{ m}$):
Sizing the Sterile Exclusion Zone ($K_{\text{total}} = 1.58\text{ kW/m}^2$ at Boundary Fence):
Allowable flame radiation:
Required distance from flame center:
Now solving for the downwind ground radius ($x_{\text{fence}}$):
4. Summary of Thermal Radiation Contours
The complete radial profile from the flare stack base downwind is summarized below:
| Ground Distance from Stack Base ($x$) | Distance to Flame Center ($R$) | Flame Radiation ($K_{\text{flame}}$) | Solar Flux ($K_{\text{solar}}$) | Total Incident Radiation ($K_{\text{total}}$) | Permissible Occupancy / API Status |
|---|---|---|---|---|---|
| $0.0\text{ m}$ (Stack Base) | $83.9\text{ m}$ | $3.72\text{ kW/m}^2$ | $0.79\text{ kW/m}^2$ | $4.51\text{ kW/m}^2$ | Permissible for short maintenance (< 3 min) |
| $18.8\text{ m}$ (Directly Under Flame) | $81.8\text{ m}$ | $3.92\text{ kW/m}^2$ | $0.79\text{ kW/m}^2$ | $4.71\text{ kW/m}^2$ | Peak grade radiation point |
| $50.0\text{ m}$ (Process Boundary) | $87.5\text{ m}$ | $3.43\text{ kW/m}^2$ | $0.79\text{ kW/m}^2$ | $4.22\text{ kW/m}^2$ | COMPLIANT ($\le 4.73\text{ kW/m}^2$) |
| $100.0\text{ m}$ | $115.3\text{ m}$ | $1.97\text{ kW/m}^2$ | $0.79\text{ kW/m}^2$ | $2.76\text{ kW/m}^2$ | Restricted access area |
| $150.0\text{ m}$ | $154.9\text{ m}$ | $1.09\text{ kW/m}^2$ | $0.79\text{ kW/m}^2$ | $1.88\text{ kW/m}^2$ | Transition zone |
| $182.0\text{ m}$ (Sterile Fence) | $182.2\text{ m}$ | $0.79\text{ kW/m}^2$ | $0.79\text{ kW/m}^2$ | $\mathbf{1.58\text{ kW/m}^2}$ | STERILE BOUNDARY (Continuous public/worker access) |
5. Engineering Observations & Best Practices
- Wind Dominance on Sterile Zone: A crosswind of $9.0\text{ m/s}$ shifts the flame center downwind by nearly $19\text{ m}$, elongating the downwind exclusion perimeter from $163\text{ m}$ (calm) to $182\text{ m}$. In plant layouts, the flare must be positioned downwind of prevailing seasonal wind directions relative to offices and control rooms.
- Tip Mach Number Safety: Sizing the tip at $24\text{ inches}$ ($610\text{ mm}$) yields a Mach number of $0.445$. While API 521 allows Mach numbers up to $0.50$ for emergency peak flaring, higher velocities cause severe acoustic resonance ($> 115\text{ dBA}$) and risk lifting the flame off the burner rim.
- Smoke Suppression: Rich hydrocarbon gas ($M_w = 17.65$) burns with moderate soot formation. For smokeless capacity up to $20\%$ of peak relief, high-pressure steam injection ($0.3\text{ kg steam/kg gas}$) or an air-assisted blower shroud should be integrated.
Flare Stack Thermal Radiation & Dispersion Safety Screener
Apply this methodology directly in the ChemProCal calculator.
Open Calculator →References
- API Standard 521: Pressure-relieving and Depressuring Systems (7th Edition, American Petroleum Institute, 2020) - Standard [Link ↗]
- ISO 23251: Petroleum, petrochemical and natural gas industries - Pressure-relieving and depressuring systems (2nd Edition, International Organization for Standa) - Standard [Link ↗]
- GPSA Engineering Data Book, Section 5: Relief Systems and Flares (14th Edition, Gas Processors Suppliers Association) - Standard [Link ↗]
- Brzustowski, T.A. and Sommer, E.C. - Predicting Radiant Heating from Flares (API Proceedings, Division of Refining, 1973) - Paper
Apply Methodology in Flare Stack Thermal Radiation & Dispersion Safety Screener
Execute complete engineering simulations using the governing mathematical equations from this guide:
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