Gas Choke & Pressure Relief / Depressurization Analyzer

When high-pressure natural gas flows through a constriction—such as a wellhead choke bean, restriction orifice, or pressure relief valve nozzle—its static pressure decreases while velocity increases due to conservation of momentum and energy.

Derivation of the Critical Pressure Ratio (\(r_c\))

Assuming 1D isentropic expansion of an ideal gas, the local Mach number (\(Ma = v / c\)) at the minimum constriction cross-section (throat or vena contracta) reaches unity (\(Ma = 1.0\)) when the downstream-to-upstream absolute pressure ratio drops to the critical pressure ratio (\(r_c\)):

\[ r_c = \frac{P^*}{P_1} = \left( \frac{2}{k+1} \right)^{\frac{k}{k-1}} \]

Where \(k = C_p / C_v\) is the ratio of specific heats of the natural gas mixture (typically 1.28 to 1.32 for methane-rich sweet gas). For \(k = 1.30\), \(r_c = 0.5457\). If upstream pressure is 70 barg (\(71.01 \text{ bar(a)}\)), flow becomes sonically choked whenever downstream pressure is equal to or below 38.7 bar(a).

Physical Significance of Choked Flow

When downstream pressure drops below the critical threshold (\(P_2 / P_1 \le r_c\)):

• The gas velocity at the orifice throat equals the local speed of sound (\(Ma = 1.0\)).
• Pressure disturbances or acoustic pressure waves in the downstream pipe cannot propagate upstream against the supersonic/sonic flow.
• Consequently, the mass flow rate becomes completely independent of downstream backpressure. Further reductions in downstream pressure cannot increase mass flow rate.
• Flow rate is governed exclusively by upstream stagnation pressure (\(P_1\)), upstream temperature (\(T_1\)), gas composition, and orifice throat area.

Orifice flow calculations follow two principal industrial conventions depending on whether the application is upstream wellhead production control or downstream relief/blowdown safety.

API Spec 6A & 14B Wellhead Choke Bean Sizing

In upstream petroleum engineering, positive and adjustable wellhead chokes are calibrated in 64ths of an inch (denoted as bean sizes, e.g., 16/64", 24/64", 32/64"):

\[ d_{in} = \frac{N_{64}}{64} \qquad d_{mm} = \frac{N_{64}}{64} \times 25.4 \text{ mm} \]

For example, a 24/64" choke bean has an internal throat diameter of 0.375 inches (9.525 mm). Wellhead chokes feature hardened ceramic or tungsten carbide sleeves to resist abrasive sand erosion. The typical discharge coefficient for a commercial wellhead choke bean ranges from \(C_d \approx 0.82 \text{ to } 0.88\).

API RP 520 Part I Relief Orifice Sizing

For safety relief valves (PSVs) and emergency depressuring blowdown valves, API RP 520 Part I provides certified discharge formulations:

\[ \dot{m}_{crit} = C_d \cdot A \cdot P_1 \sqrt{\frac{k \cdot M}{Z_1 \cdot R \cdot T_1} \left( \frac{2}{k+1} \right)^{\frac{k+1}{k-1}}} \qquad (\text{Critical Choked Flow}) \]

For subcritical subsonic flow (\(P_2 / P_1 > r_c\)):

\[ \dot{m}_{sub} = C_d \cdot A \cdot P_1 \sqrt{\frac{2 \cdot M}{Z_1 \cdot R \cdot T_1} \cdot \left( \frac{k}{k-1} \right) \left[ \left(\frac{P_2}{P_1}\right)^{\frac{2}{k}} - \left(\frac{P_2}{P_1}\right)^{\frac{k+1}{k}} \right]} \]

API certified nozzles exhibit high discharge coefficients (\(C_d = 0.975\)), while sharp-edged restriction orifices typically operate at \(C_d \approx 0.62\).

When natural gas expands across a restriction orifice or choke with zero external shaft work and negligible heat transfer with the surroundings, the process is isenthalpic (\(h_1 = h_2\)). Because real gases exhibit intermolecular attraction, expanding molecules must perform work against attractive van der Waals forces, which extracts thermal kinetic energy from the gas stream, inducing severe Joule-Thomson (JT) cooling.

Joule-Thomson Coefficient (\(\mu_{JT}\))

The Joule-Thomson coefficient is defined thermodynamically as the change in temperature with respect to pressure at constant enthalpy:

\[ \mu_{JT} = \left( \frac{\partial T}{\partial P} \right)_h = \frac{1}{C_p} \left[ T \left( \frac{\partial V}{\partial T} \right)_P - V \right] \]

For standard natural gas streams, \(\mu_{JT}\) averages 0.40°C to 0.55°C per bar of pressure drop (3.5°F to 5.0°F per 100 psi). For example, a high-pressure wellhead letdown from 150 barg to 10 barg (\(\Delta P = 140 \text{ bar}\)) produces a temperature drop exceeding 60°C (110°F). Feed gas entering at +20°C will plunge to -40°C downstream of the choke!

Minimum Design Metal Temperature (MDMT) & Metallurgy Alerts

Severe depressuring events (blowdown to flare) present catastrophic structural risks governed by ASME B31.3 Process Piping Code and API RP 521:

• Standard Carbon Steel (ASTM A106 Gr B / A53): Standard carbon steel exhibits a ductile-to-brittle transition temperature at -29°C (-20°F). Below -29°C, ordinary carbon steel loses impact toughness and is vulnerable to brittle fracture under flare discharge thrust forces.
• Low-Temperature Carbon Steel (LTCS, ASTM A333 Grade 6): Charpy V-notch impact-tested carbon steel is certified down to -46°C (-50°F).
• Austenitic Stainless Steel (304 / 316L) or Duplex Alloys: Required when downstream blowdown temperatures drop below -46°C, providing cryogenic ductile integrity down to -196°C.
• Hydrate Formation in Flare Lines: If free water or saturated water vapor is present, sub-zero downstream temperatures induce immediate hydrate freezing, which can plug blowdown orifice headers and overpressurize upstream process equipment.