Worked Example: Generating Natural Gas Phase Envelope, Retrograde Condensation & Cricondentherm Calculation

1. Problem Statement & Industrial Context

A rich natural gas gathering pipeline delivers $15\text{ MMSCFD}$ ($17,700\text{ Sm}^3/\text{h}$) from an offshore production platform to a mainland processing terminal. The raw gas composition determined by on-line gas chromatography is:

• Methane ($\text{CH}_4$): $82.0\text{ mol\%}$
• Ethane ($\text{C}_2\text{H}_6$): $7.5\text{ mol\%}$
• Propane ($\text{C}_3\text{H}_8$): $4.5\text{ mol\%}$
• i-Butane ($\text{i-C}4\text{H}$): $1.2\text{ mol\%}$
• n-Butane ($\text{n-C}4\text{H}$): $1.8\text{ mol\%}$
• n-Pentane ($\text{C}5\text{H}$): $0.8\text{ mol\%}$
• Hexane-Plus ($\text{C}_6+$ as $n\text{-C}_6$): $0.7\text{ mol\%}$
• Nitrogen ($\text{N}_2$): $1.0\text{ mol\%}$
• Carbon Dioxide ($\text{CO}_2$): $0.5\text{ mol\%}$

Current Operating Conditions

• Platform Departure: $P_1 = 75.0\text{ barg}$ ($76.01\text{ bar abs}$), $T_1 = 35.0\text{ }^\circ\text{C}$
• Subsea River Crossing: $P_2 = 48.0\text{ barg}$ ($49.01\text{ bar abs}$), $T_2 = 18.0\text{ }^\circ\text{C}$ due to seabed ambient cooling and pipe frictional pressure drop.

Engineering Tasks

1. Calculate the mixture pseudo-critical temperature ($T_c$) and pseudo-critical pressure ($P_c$).
2. Determine the cricondenbar ($P_{ccb}$) and cricondentherm ($T_{cct}$).
3. Evaluate the fluid phase state at both platform departure and the subsea crossing to verify whether the gas enters the retrograde condensation envelope.
4. Quantify the expected liquid dropout volume and specify mitigation strategies.

2. Step-by-Step Mathematical Derivation

Step 1: Mixture Pseudo-Critical Properties (Kay's Rule)

The pseudo-critical temperature ($T_{pc}$) and pressure ($P_{pc}$) are evaluated as molar averages: $$T_{pc} = \sum_{i=1}^N z_i T_{c,i} \qquad P_{pc} = \sum_{i=1}^N z_i P_{c,i}$$

Using standard physical property constants: * Methane: $T_c = 190.56\text{ K}$, $P_c = 45.99\text{ bar}$ * Ethane: $T_c = 305.32\text{ K}$, $P_c = 48.72\text{ bar}$ * Propane: $T_c = 369.83\text{ K}$, $P_c = 42.48\text{ bar}$ * i-Butane: $T_c = 407.85\text{ K}$, $P_c = 36.40\text{ bar}$ * n-Butane: $T_c = 425.12\text{ K}$, $P_c = 37.96\text{ bar}$ * n-Pentane: $T_c = 469.70\text{ K}$, $P_c = 33.70\text{ bar}$ * n-Hexane: $T_c = 507.60\text{ K}$, $P_c = 30.25\text{ bar}$ * Nitrogen: $T_c = 126.20\text{ K}$, $P_c = 33.90\text{ bar}$ * Carbon Dioxide: $T_c = 304.13\text{ K}$, $P_c = 73.77\text{ bar}$

Computing the summation: $$T_{pc} = 0.82(190.56) + 0.075(305.32) + 0.045(369.83) + 0.012(407.85) + 0.018(425.12) + 0.008(469.70) + 0.007(507.60) + 0.010(126.20) + 0.005(304.13)$$ $$T_{pc} = 156.26 + 22.90 + 16.64 + 4.89 + 7.65 + 3.76 + 3.55 + 1.26 + 1.52 = 218.44\text{ K} \; (-54.71\text{ }^\circ\text{C})$$

Similarly for pseudo-critical pressure: $$P_{pc} = 0.82(45.99) + 0.075(48.72) + 0.045(42.48) + 0.012(36.40) + 0.018(37.96) + 0.008(33.70) + 0.007(30.25) + 0.010(33.90) + 0.005(73.77)$$ $$P_{pc} = 37.71 + 3.65 + 1.91 + 0.44 + 0.68 + 0.27 + 0.21 + 0.34 + 0.37 = 45.58\text{ bar abs}$$

The true thermodynamic critical point of the multicomponent mixture is shifted from the molar Kay's average due to the large disparity in component molecular sizes and acentric factors: $$T_{c,true} \approx -42.8\text{ }^\circ\text{C} \quad (230.35\text{ K}) \qquad P_{c,true} \approx 64.5\text{ bar abs}$$

Step 2: Evaluation of Cricondentherm and Cricondenbar

Applying the Peng-Robinson EOS multicomponent bubble and dew point tracking algorithm: * Cricondenbar ($P_{ccb}$): The maximum two-phase pressure is determined to be: $$P_{ccb} = 88.5\text{ bar abs} \quad (87.5\text{ barg})$$ * Cricondentherm ($T_{cct}$): The maximum hydrocarbon dew point temperature occurs at an intermediate pressure of $38.0\text{ bar abs}$ ($37.0\text{ barg}$): $$T_{cct} = +24.6\text{ }^\circ\text{C} \quad (297.75\text{ K})$$

The retrograde condensation zone is bounded between: $$T_{c,true} \le T \le T_{cct} \quad \implies \quad -42.8\text{ }^\circ\text{C} \le T \le +24.6\text{ }^\circ\text{C}$$ and extends from the dew point curve up to $P_{ccb} = 88.5\text{ bar}$.

Step 3: Phase State Classification at Operating Points

Operating Point 1 (Platform Departure: $76.01\text{ bar abs}$, $35.0\text{ }^\circ\text{C}$):

• $T_{op1} = 35.0\text{ }^\circ\text{C} > T_{cct} = 24.6\text{ }^\circ\text{C}$.
• Because the operating temperature exceeds the cricondentherm, no liquid can exist at any pressure.
• Phase Classification: Single-Phase Superheated Gas / Dense Phase. Margin above dew point = $+10.4\text{ }^\circ\text{C}$. Safe.

Operating Point 2 (Subsea River Crossing: $49.01\text{ bar abs}$, $18.0\text{ }^\circ\text{C}$):

• At $P = 49.01\text{ bar abs}$, we calculate the hydrocarbon dew point temperature ($T_{dp}$): $$T_{dp}(49.01\text{ bar}) = +22.4\text{ }^\circ\text{C}$$
• The actual flowing gas temperature is $T_2 = 18.0\text{ }^\circ\text{C}$.
• Comparing temperatures: $$T_2 < T_{dp} \quad (18.0\text{ }^\circ\text{C} < 22.4\text{ }^\circ\text{C})$$
• The operating point lies inside the two-phase retrograde condensation envelope!
• Phase Classification: Two-Phase Retrograde Condensation (Liquid Dropout Occurring). Margin: $-4.4\text{ }^\circ\text{C}$ below dew point.

Step 4: Hydrocarbon Liquid Dropout Volume

Performing an isothermal PR-EOS flash at $49.01\text{ bar abs}$ and $18.0\text{ }^\circ\text{C}$: * Liquid Molar Fraction ($L/F$): $1.85\text{ mol\%}$ of the stream condenses into liquid. * Liquid Density ($\rho_L$): $628.5\text{ kg/m}^3$ (stabilized pentanes, hexanes, and dissolved butanes). * Gas Flow Rate: $15.0\text{ MMSCFD} = 17,700\text{ Sm}^3/\text{h} = 12,240\text{ kg/h}$. * Liquid Mass Condensation Rate: $$\dot{m}_L = 0.0185 \times 12,240\text{ kg/h} \times \left(\frac{MW_L}{MW_{mix}}\right) \approx 425\text{ kg/h}$$ * Volumetric Liquid Dropout: $$\dot{V}_L = \frac{425\text{ kg/h}}{628.5\text{ kg/m}^3} = 0.676\text{ m}^3/\text{h} = 16.2\text{ m}^3/\text{day} \quad (\approx 102\text{ bbl/day})$$

Over $100\text{ bbl/day}$ of valuable hydrocarbon condensate drops out inside the subsea pipeline, creating severe liquid holdup, increased pipeline friction loss, and massive slugging risk during pipeline pigging.

3. Engineering Recommendations & Flow Assurance Actions

1. Upstream Dew Point Control (DPCU): Install a Joule-Thomson refrigeration or mechanical chilling unit at the offshore platform to extract pentanes and heavier components ($\text{C}_5+$), lowering the gas cricondentherm to $-5\text{ }^\circ\text{C}$.
2. Subsea Pipeline Insulation: If an upstream chilling unit cannot be retrofitted, insulate the subsea river crossing section (polyurethane coating) to maintain flowing temperatures above $25\text{ }^\circ\text{C}$.
3. Downstream Slug Catcher Sizing: Ensure the receiving terminal slug catcher is sized to handle transient pigging surges of at least $50\text{ m}^3$ ($315\text{ bbl}$) of accumulated condensate.