Worked Example: Wellhead Bean Choke Sizing, Critical Sonic Flow, and Joule-Thomson Depressurization

A high-pressure natural gas wellhead choke is the primary barrier controlling production deliverability and protecting downstream separation facilities from overpressure. When high-pressure gas expands across a fixed choke bean or emergency depressuring blowdown valve, two critical physical phenomena govern system integrity:

1. Sonic Choking: As downstream pressure drops below the critical pressure ratio ($P_2 / P_1 < r_c$), gas velocity at the choke throat reaches the local speed of sound ($\text{Mach } 1.0$). Beyond this threshold, flow is choked—mass flow is strictly determined by upstream pressure and temperature, completely independent of downstream fluctuations.
2. Joule-Thomson (JT) Refrigeration: High-pressure gas experiences a severe isenthalpic temperature drop across the restriction. If downstream temperatures plunge below the Minimum Design Metal Temperature (MDMT), carbon steel piping undergoes brittle fracture, risking cataclysmic containment loss.

This worked example models an API Spec 6A #24 bean choke operating across a severe $175\text{ bar}$ differential.

1. Wellhead Operating & Geometry Data

A high-pressure offshore gas well produces through an API Spec 6A fixed bean choke into a production header: * Upstream Wellhead Pressure ($P_1$): $240.0\text{ barg}$ ($241.01\text{ bar(a)} = 3,495.6\text{ psia}$) * Upstream Wellhead Temperature ($T_1$): $50.0^\circ\text{C}$ ($122.0^\circ\text{F} = 323.15\text{ K}$) * Downstream Header Pressure ($P_2$): $65.0\text{ barg}$ ($66.01\text{ bar(a)} = 957.4\text{ psia}$) * Choke Bean Size: $24/64\text{ inch}$ ($d_o = 0.375\text{ in} = 9.525\text{ mm}$) * Orifice Discharge Coefficient ($C_d$): $0.85$ (calibrated for API fixed bean) * Gas Properties: Specific Gravity $G = 0.65$ ($M_{mix} = 18.83\text{ kg/kmol}$) * Isentropic Expansion Ratio ($k = c_p/c_v$): $1.30$ * Upstream Compressibility ($Z_1$): $0.85$

2. Step 1: Critical Pressure Ratio & Sonic Flow Verification

The theoretical critical pressure ratio ($r_c$) at which throat velocity reaches Mach 1 is:

$$r_c = \left( \frac{2}{k + 1} \right)^{\frac{k}{k - 1}}$$

For $k = 1.30$:

$$\frac{2}{k + 1} = \frac{2}{2.30} = 0.86957$$

$$\frac{k}{k - 1} = \frac{1.30}{0.30} = 4.3333$$

$$r_c = (0.86957)^{4.3333} = \mathbf{0.5457}$$

Critical Choking Pressure ($P_{crit}$):

$$P_{crit} = r_c \cdot P_1 = 0.5457 \times 241.01\text{ bar(a)} = \mathbf{131.52\text{ bar(a)}} \quad (130.51\text{ barg})$$

Actual Pressure Ratio ($r_{act}$):

$$r_{act} = \frac{P_2}{P_1} = \frac{66.01\text{ bar(a)}}{241.01\text{ bar(a)}} = \mathbf{0.2739}$$

Flow Regime Determination:

Since $r_{act} = 0.2739 < r_c = 0.5457$: * The gas is in CRITICAL (SONIC / CHOKED) FLOW. * Acoustic waves cannot propagate upstream against supersonic gas expansion. Any pressure disturbance downstream of the choke will have zero effect on wellhead flow rate.

graph LR
    A[Upstream P1 = 240 barg] --> B[Bean Choke Throat: Mach 1.0, Pcrit = 131.5 barg]
    B --> C[Isenthalpic Expansion]
    C --> D[Downstream P2 = 65 barg, T2 = -11.3°C]

3. Step 2: Choke Throat Area & Sonic Mass Flow Rate

A. Bean Orifice Cross-Sectional Area ($A_o$):

$$d_o = 9.525\text{ mm} = 0.009525\text{ m}$$

$$A_o = \frac{\pi \cdot d_o^2}{4} = \frac{\pi \times (0.009525)^2}{4} = 7.1256 \times 10^{-5}\text{ m}^2 = \mathbf{71.26\text{ mm}^2}$$

B. Sonic Mass Flow Rate via API 520 / ISO 5167 Formulation:

Under critical choked flow, the mass flux is:

$$\dot{m} = C_d \cdot A_o \cdot P_1 \cdot \sqrt{\frac{k \cdot M_{mix}}{Z_1 \cdot R \cdot T_1} \cdot \left( \frac{2}{k + 1} \right)^{\frac{k + 1}{k - 1}}}$$

Evaluating component terms: 1. $P_1 = 24,101,000\text{ Pa}$ 2. $Z_1 \cdot R \cdot T_1 = 0.85 \times 8,314.46 \times 323.15 = 2,283,780\text{ J/kmol}$ 3. $\frac{k \cdot M_{mix}}{Z_1 R T_1} = \frac{1.30 \times 18.826}{2,283,780} = 1.0716 \times 10^{-5}\text{ kg}^2/\text{J}$ 4. $\left( \frac{2}{k+1} \right)^{\frac{k+1}{k-1}} = (0.86957)^{\frac{2.30}{0.30}} = (0.86957)^{7.6667} = 0.3424$ 5. Square root term $= \sqrt{1.0716 \times 10^{-5} \times 0.3424} = \sqrt{3.669 \times 10^{-6}} = 0.0019155$

Substituting into the mass rate formula:

$$\begin{aligned} \dot{m} &= 0.85 \times (7.1256 \times 10^{-5}\text{ m}^2) \times 24,101,000\text{ Pa} \times 0.0019155 \\ &= \mathbf{6.3596\text{ kg/s}} \end{aligned}$$

Converting to hourly and imperial mass rates:

$$\dot{m}_{hr} = 6.3596 \times 3,600 = \mathbf{22,894.6\text{ kg/h}} \quad (\mathbf{50,474\text{ lb/h}})$$

4. Step 3: Volumetric Gas Flow Rate at Standard Conditions

Using standard gas density $\rho_{std} = 0.7980\text{ kg/Sm}^3$:

$$\dot{V}_{Sm^3/d} = \frac{22,894.6\text{ kg/h} \times 24\text{ h/d}}{0.7980\text{ kg/Sm}^3} = \mathbf{688,320\text{ Sm}^3/\text{day}}$$

Converting to MMSCFD:

$$Q = \frac{688,320\text{ Sm}^3/\text{d}}{28,316.85\text{ Sm}^3/\text{MMSCF}} = \mathbf{24.31\text{ MMSCFD}}$$

5. Step 4: Joule-Thomson Depressurization Cooling ($\Delta T_{JT}$)

During rapid expansion across a restriction with zero shaft work and negligible external heat transfer, the expansion is isenthalpic ($\Delta H = 0$).

The Joule-Thomson cooling coefficient ($\mu_{JT}$) is:

$$\mu_{JT} = \left( \frac{\partial T}{\partial P} \right)_H$$

For lean natural gas ($G = 0.65$) expanding across $240\text{ barg} \to 65\text{ barg}$: $$\mu_{JT} \approx 0.350^\circ\text{C/bar} \quad (0.432^\circ\text{F/psi})$$

Total Differential Pressure ($\Delta P$):

$$\Delta P = P_1 - P_2 = 240.0 - 65.0 = \mathbf{175.0\text{ bar}}$$

Temperature Drop:

$$\Delta T_{JT} = \mu_{JT} \times \Delta P = 0.350^\circ\text{C/bar} \times 175.0\text{ bar} = \mathbf{61.25^\circ\text{C}} \quad (110.25^\circ\text{F})$$

Downstream Gas Temperature ($T_2$):

$$T_2 = T_1 - \Delta T_{JT} = 50.0^\circ\text{C} - 61.25^\circ\text{C} = \mathbf{-11.25^\circ\text{C}} \quad (\mathbf{11.75^\circ\text{F}})$$

6. Step 5: Low-Temperature Metallurgy & MDMT Assessment

Under ASME Section VIII Div. 1 (UCS-66) and ASME B31.3 (Process Piping), all pressure-containing materials must be rated for the Minimum Design Metal Temperature (MDMT) to prevent catastrophic brittle fracture.

graph TD
    A[Normal Upstream: +50°C] --> B[Choke Depressurization: dT = 61.3°C]
    B --> C[Downstream Temperature: -11.3°C]
    C --> D[Standard Carbon Steel A106 Gr B MDMT = -29°C]
    D --> E[Acceptable for Normal Production Choking]
    B --> F[Emergency Full Depressurization to Flare at 1 barg]
    F --> G[Downstream T drops to -45°C!]
    G --> H[MANDATES LTCS ASTM A333 Gr 6 or 316L SS]

Material Selection Verdict:

1. Steady-State Production Choking ($P_2 = 65\text{ barg}$):
2. Operating temperature is $-11.25^\circ\text{C}$.
3. Standard Normalized Carbon Steel (ASTM A106 Gr B / A350 LF2) has a rated MDMT of $-29.0^\circ\text{C}$.
4. Status: ACCEPTABLE without impact testing.
5. Emergency Blowdown / Full Depressurization ($P_2 \to 1.5\text{ barg}$):
6. If the choke discharges during a platform ESD emergency blowdown to flare ($\Delta P \approx 238\text{ bar}$): $$\Delta T_{JT} \approx 0.35 \times 238 = 83.3^\circ\text{C} \implies T_{downstream} = 50.0 - 83.3 = \mathbf{-33.3^\circ\text{C}}$$
7. At $-33.3^\circ\text{C}$, standard carbon steel falls below its Charpy V-notch ductile-to-brittle transition temperature.
8. Requirement: Downstream piping must be specified in Low-Temperature Carbon Steel (LTCS - ASTM A333 Grade 6) with mandatory Charpy impact testing at $-46^\circ\text{C}$.

7. Performance Curve across Standard Bean Sizes

The table below illustrates flow capacity and JT refrigeration across API Spec 6A bean sizes at $240\text{ barg}$ upstream:

Takeaways:

• Mass flow scales strictly with the square of the bean diameter ($d_o^2$). Doubling the bean size from $16/64$ to $32/64$ quadruples capacity from $10.81$ to $43.22\text{ MMSCFD}$.
• Temperature drop is determined by fluid thermodynamics and pressure drop ($\Delta P$), remaining constant across bean diameters, but high mass rates dramatically increase Acoustic-Induced Vibration (AIV) risks on downstream pipe tees and reducers.