Flare Thermal Radiation & Dispersion: API 521 Point Source, Brzustowski Plume Physics & Sterile Zone Design Guide

Executive Summary & System Philosophy

In hydrocarbons and chemical processing facilities, the flare system acts as the ultimate line of defense for emergency overpressure disposal. During catastrophic plant disruptions—such as total electrical power failure, cooling water loss, emergency blowdown, or external fire exposure—enormous inventories of flammable natural gas must be relieved and incinerated instantly.

While flaring safely converts volatile hydrocarbons to carbon dioxide and water vapor, it produces intense thermal radiation, extreme noise, and toxic combustion products. If a flare stack is undersized in height or placed too close to operating units, radiant heat can inflict fatal burns on operating personnel, ignite paint or insulation on neighboring storage tanks, and cause structural steel to lose mechanical yield strength.

This design guide presents the fundamental thermodynamics, flame kinematics, and radiant heat transfer formulations specified by API Standard 521 and ISO 23251, guiding process and safety engineers through the rigorous design of flare stacks, sterile exclusion zones, and dispersion screening.

1. Thermodynamic Heat Release & Radiant Emissive Fractions

1.1 Total Heat Release Rate ($Q_h$)

The total thermal energy liberated by the flare per unit time is the product of the mass relief rate and the gas Lower Heating Value ($\text{LHV}$):

$$Q_h = \dot{m} \cdot \text{LHV} = \left( \frac{Q_{\text{scfd}} \cdot M_w}{379.48 \times 24 \times 3600} \right) \cdot \text{LHV}$$

where: * $Q_h$ = Total heat release rate ($\text{kW}$ or $\text{Btu/hr}$) * $\dot{m}$ = Mass flow rate ($\text{kg/s}$ or $\text{lb/hr}$) * $\text{LHV}$ = Lower Heating Value of the gas mixture ($\text{kJ/kg}$ or $\text{Btu/lb}$) * $Q_{\text{scfd}}$ = Standard volumetric flow rate at $60\text{ }^\circ\text{F}, 14.7\text{ psia}$

1.2 Radiant Emissive Fraction ($F$)

Not all liberated combustion heat radiates outward; a large portion is carried aloft by the buoyant convective rise of hot flue gases. The fraction of total heat converted into radiant energy is the radiant fraction ($F$).

Per API Standard 521 Section 5.7.2.3, the empirical fraction is primarily a function of gas molecular weight:

$$F = 0.048 \sqrt{M_w}$$

2. Flame Geometry & Kinematics under Crosswind

2.1 Flame Length ($L_f$)

The length of the turbulent diffusion flame envelope depends on the heat release and burner momentum. The API 521 empirical correlation defines flame length as:

$$L_f = 0.00678 \cdot \left( Q_h \cdot 1000 \right)^{0.478} \quad \text{[with } Q_h \text{ in kW, } L_f \text{ in meters]}$$

In US Customary units: $$L_f = 0.006 \cdot \left( Q_h \right)^{0.478} \quad \text{[with } Q_h \text{ in Btu/hr, } L_f \text{ in feet]}$$

2.2 Flame Tilt & Center Coordinates (The Brzustowski-Sommer Method)

Under calm wind conditions, the flame rises vertically above the tip. However, crosswinds bend the flame envelope downwind, dramatically bringing the radiant heat source closer to ground personnel and adjacent equipment.

       Wind uw --->
                   Tip
                   [ ]=========-..
                    |             ``--..
                    |                   `\ (xc, H + yc)  <- Flame Radiant Center
                    |                      \
              Stack | H                     \  Flame Envelope Length Lf
                    |                        \
                    |                         v Flame Tip Burnout
       =============+======================================= Grade

The deflection of the flame is governed by the momentum ratio of the crosswind velocity ($u_w$) to the flare tip discharge velocity ($v_{\text{tip}}$):

$$u_r = \frac{u_w}{v_{\text{tip}}}$$

The coordinates of the flame midpoint $(x_c, y_c)$ relative to the tip are determined from empirical trajectory integration:

$$\tan(\theta) = \frac{u_w}{v_{\text{tip}} \cdot 0.35 + 0.01}$$

$$\Delta x_c = 0.45 \cdot L_f \cdot \sin(\theta)$$

$$\Delta y_c = 0.45 \cdot L_f \cdot \cos(\theta)$$

The absolute elevation of the flame center above grade is:

$$Z_c = H_{\text{stack}} + \Delta y_c$$

3. Incident Thermal Radiation Modeling

3.1 The API 521 Simple Point Source Model

API Standard 521 assumes that the entire radiant heat output emanates from a single point located at the flame center $(x_c, Z_c)$. At any grade receiver located at coordinate $(x, y, h_{\text{person}})$:

$$K_{\text{flame}} = \frac{\tau \cdot F \cdot Q_h}{4 \pi R^2}$$

$$K_{\text{total}} = K_{\text{flame}} + K_{\text{solar}}$$

where: * $K_{\text{total}}$ = Total incident radiation intensity ($\text{kW/m}^2$ or $\text{Btu/hr}\cdot\text{ft}^2$) * $R$ = True 3D Euclidean distance from flame center to the observer: $$R = \sqrt{(x - \Delta x_c)^2 + y^2 + (H_{\text{stack}} + \Delta y_c - h_{\text{person}})^2}$$ * $\tau$ = Atmospheric transmissivity factor ($0.75 \le \tau \le 1.0$) * $K_{\text{solar}}$ = Ambient solar radiation flux ($0.79\text{--}1.05\text{ kW/m}^2$ or $250\text{--}330\text{ Btu/hr}\cdot\text{ft}^2$)

3.2 Atmospheric Transmissivity ($\tau$)

As infrared radiation traverses the air, atmospheric water vapor and carbon dioxide absorb a fraction of the thermal energy. Brzustowski and Sommer established the transmissivity correlation based on absolute humidity ($e_w$, $\text{mm Hg}$) and distance ($R$, $\text{meters}$):

$$\tau = 0.79 \cdot \left( \frac{100}{RH} \right)^{1/16} \cdot \left( \frac{100}{R} \right)^{1/16}$$

In practice, for humid tropical or marine environments, $\tau$ typically ranges from $0.80$ to $0.88$, whereas in dry desert environments, $\tau \approx 0.95\text{--}0.98$. Sizing assuming $\tau = 1.0$ provides a conservative margin.

4. API 521 Permissible Thermal Radiation Limits

Table 10 of API Standard 521 establishes mandatory maximum allowable radiation intensities based on human physiological pain thresholds and equipment integrity:

5. Flare Tip Hydraulics & Operational Limits

5.1 Tip Velocity & Mach Number Limits

The flare tip diameter must balance two competing hydraulic constraints: 1. Low Velocity (Undersized Pressure Drop): If the tip is too large, the exit velocity drops below $3\text{ m/s}$, allowing crosswinds to force the flame down inside the tip (internal burning / burn-back), destroying the tip metal within hours. 2. High Velocity (Acoustic Vibration & Blowout): If the tip is too small, extreme gas velocities create deafening noise ($> 120\text{ dBA}$) and risk blowing the flame off the tip rim (flame lift-off).

Per API Standard 521, the tip Mach number during peak emergency relief is constrained by:

$$M = \frac{v_{\text{tip}}}{c_{\text{sonic}}} = \frac{v_{\text{tip}}}{\sqrt{\gamma \cdot Z \cdot R \cdot T / M_w}} \le \mathbf{0.50}$$

For continuous routine flaring, velocities are generally limited to $M \le 0.20$.

5.2 Smoke-Less Technologies

Clean, smokeless combustion requires rapid mixing of atmospheric air into the hydrocarbon core before thermal cracking produces soot particles: * Steam-Assisted Flares: High-pressure steam jets inject into the flame base, inducing rapid turbulence and reacting chemically ($\text{C} + \text{H}_2\text{O} \to \text{CO} + \text{H}_2$) to consume carbon precursor rings. Typical ratio: $0.25\text{--}0.40\text{ kg steam / kg hydrocarbon}$. * Air-Assisted Flares: High-capacity electric blowers direct secondary air into an annular duct surrounding the tip. Ideal for remote onshore fields lacking steam boilers. * High-Pressure Sonic Tips: Multipoint sonic burner nozzles utilize raw reservoir pressure to achieve intimate acoustic mixing without external steam or air utilities.

6. Cold Unignited Flammable Dispersion Screening

If the flare pilots fail during an emergency release, cold unburned natural gas discharges into the atmosphere, creating a massive, dense flammable cloud.

Dispersion Screening Criteria:

1. Lower Flammability Limit (LFL): The cloud boundary is defined by the $50\%\text{ LFL}$ contour (typically $2.5\text{ mol}\%$ for natural gas).
2. Jet Momentum vs Buoyant Plume: At high exit velocities ($M \ge 0.2$), momentum jets dilute the gas rapidly within $10\text{--}20$ tip diameters.
3. Plume Touchdown Check: The heavy or cold gas plume must not touch down at ground level before dispersing below $10\%\text{ LFL}$ or toxic thresholds ($10\text{ ppmv H}_2\text{S}$).