Worked Example: 3-Stage Gas Condensate Separation Optimization, Intermediate Pressure Tuning & Vapor Recovery Economics (Campbell Rule & Peng-Robinson)

1. Problem Statement & Separation Flowsheet

A high-pressure shale gas production facility produces $10,000\text{ BPD}$ ($66.2\text{ m}^3/\text{h}$) of raw unstabilized liquid hydrocarbon condensate at $95.0\text{ barg}$ ($1392\text{ psia}$) and $40.0\text{ }^\circ\text{C}$.

If this high-pressure stream is flashed directly into atmospheric storage tanks in a single stage, violent vaporization occurs: the rapidly boiling methane and ethane carry away high fractions of valuable propane, butanes, and pentanes into the tank vent, causing massive liquid shrinkage, severe product gravity degradation, and extreme vapor recovery compressor loads.

To maximize liquid barrel yield and stabilize stock tank Reid Vapor Pressure (RVP), the facility designs a 3-stage separation train:

 Raw Condensate
 [95 barg, 40°C] ──> [ 1st Stage HP Separator ] ──(Gas to HP Sales / Grid)
                           | (Liquid)
                           v
                     [ 2nd Stage IP Separator ] ──(Gas to Medium Compression)
                           | (Liquid)
                           v
                     [ Atmospheric Stock Tank ] ──(Flash Gas to VRU Compressor)
                           |
                           v
                     [ Stabilized Condensate Sales ]

Feed Fluid Composition & Operating Conditions

• Total Feed Liquid Flow: $10,000\text{ BPD}$ ($1,325\text{ kmol/h}$)
• Feed Pressure ($P_{\text{feed}}$): $95.0\text{ barg}$ ($96.01\text{ bar abs}$)
• Operating Temperature ($T$): $40.0\text{ }^\circ\text{C}$ ($104.0\text{ }^\circ\text{F}$) across all separator stages
• Atmospheric Tank Pressure ($P_3$): $0.35\text{ barg}$ ($1.363\text{ bar abs} \approx 5.0\text{ psig}$)
• Feed Stream Composition (mol%):

Optimization Objectives

1. Determine the optimal intermediate pressures for Stage 1 ($P_1$) and Stage 2 ($P_2$) using the Campbell Equal Pressure Ratio Rule.
2. Solve the stage-by-stage vapor-liquid equilibrium (VLE) flash using the Rachford-Rice equation.
3. Compare the final stabilized liquid yield ($\text{BPD}$) of the 3-stage train against a direct single-stage atmospheric dump.
4. Verify stock tank liquid Reid Vapor Pressure ($\text{RVP} \le 9.0\text{ psi}$).
5. Size the Vapor Recovery Unit (VRU) compressor power ($\text{kW}$) required to recompress the tank overhead vapors.

2. Governing Flash Equations & Thermodynamic Rules

2.1 The Campbell Equal Pressure Ratio Rule

Per Dr. John M. Campbell (Gas Conditioning and Processing, Vol. 1), maximum liquid recovery in a multi-stage separation train occurs when the stage pressure ratios are approximately equal:

$$R_p = \left( \frac{P_1}{P_n} \right)^{\frac{1}{n-1}}$$

where $n$ is the total number of separation stages, $P_1$ is the first-stage separator pressure, and $P_n$ is the final stock tank pressure.

For a 3-stage train ($n = 3$) where the 1st stage is tied to field gathering constraints ($P_1 = 32.0\text{ barg} = 33.01\text{ bar abs}$) and the stock tank operates at $P_3 = 0.35\text{ barg} = 1.363\text{ bar abs}$:

$$R_p = \left( \frac{33.01}{1.363} \right)^{\frac{1}{3-1}} = \left( 24.218 \right)^{0.5} = 4.921$$

The optimal intermediate Stage 2 pressure ($P_2$) is:

$$P_2 = \frac{P_1}{R_p} = \sqrt{P_1 \cdot P_3} = \sqrt{33.01 \times 1.363} = \sqrt{44.99} = 6.707\text{ bar abs} \approx 5.69\text{ barg}$$

In field practice, rounding to a standard operating point gives $P_2 = 6.0\text{ barg}$ ($7.01\text{ bar abs}$).

2.2 Rachford-Rice Vapor-Liquid Equilibrium

At each stage, the feed stream flashes into equilibrium vapor ($V$) and liquid ($L$) streams satisfying the material balance:

$$\sum_{i=1}^C \frac{z_i (K_i - 1)}{1 + \psi (K_i - 1)} = 0$$

where: * $\psi = V/F$ is the vapor fraction ($0 \le \psi \le 1$) * $K_i = y_i / x_i$ is the vapor-liquid equilibrium ratio of component $i$ * $z_i$ is the feed mole fraction to that stage

Equilibrium liquid and vapor compositions are recovered via:

$$x_i = \frac{z_i}{1 + \psi (K_i - 1)}, \quad y_i = K_i \cdot x_i$$

2.3 Wilson Thermodynamic K-Value Correlation

In the moderate pressure range ($P \le 50\text{ bar}$), the semi-empirical Wilson formulation provides reliable K-values:

$$K_i = \frac{P_{c,i}}{P} \cdot \exp\left[ 5.373 \cdot (1 + \omega_i) \left( 1 - \frac{T_{c,i}}{T} \right) \right]$$

3. Stage-by-Stage Flash Calculations

Stage 1 Flash: HP Separator ($P_1 = 33.01\text{ bar abs}, T_1 = 40.0\text{ }^\circ\text{C} = 313.15\text{ K}$)

At $P_1 = 33.01\text{ bar abs}$, the Wilson K-values and Rachford-Rice flash solution yield: * Vapor Fraction ($\psi_1 = V_1/F_1$): $0.3842$ (flashing mostly methane and ethane) * Liquid Fraction ($L_1/F_1$): $0.6158$

Component Split in Stage 1:

• HP Offgas: $509.1\text{ kmol/h}$ ($10.87\text{ MMSCFD}$), rich in methane ($69.3\%$).
• HP Liquid Outlet: $815.9\text{ kmol/h}$ routed to Stage 2.

Stage 2 Flash: IP Separator ($P_2 = 7.01\text{ bar abs}, T_2 = 40.0\text{ }^\circ\text{C}$)

Feeding the Stage 1 liquid ($815.9\text{ kmol/h}$) into the intermediate separator at $6.0\text{ barg}$: * Vapor Fraction ($\psi_2 = V_2/F_2$): $0.1685$ * Liquid Fraction ($L_2/F_2$): $0.8315$

Component Split in Stage 2:

• IP Offgas: $137.5\text{ kmol/h}$ ($2.94\text{ MMSCFD}$), sent to field medium-pressure compressor.
• IP Liquid Outlet: $678.4\text{ kmol/h}$ routed to atmospheric stock tank.

Stage 3 Flash: Atmospheric Stock Tank ($P_3 = 1.363\text{ bar abs}, T_3 = 40.0\text{ }^\circ\text{C}$)

Flashing the IP liquid into the stock tank at $0.35\text{ barg}$ ($5\text{ psig}$): * Vapor Fraction ($\psi_3 = V_3/F_3$): $0.0782$ * Final Stock Tank Liquid Fraction ($L_3/F_3$): $0.9218$ * Final Stabilized Liquid Flow: $625.4\text{ kmol/h}$

Final Stabilized Liquid & Tank Vent Composition:

4. Multi-Stage Separation vs Single-Stage Flash Comparison

To prove the economic justification for the 3-stage system, we compare it against flashing the same $95\text{ barg}$ raw condensate directly into the atmospheric tank in a single step:

5. Vapor Recovery Unit (VRU) Compressor Power Sizing

The overhead vapor flashing from the Stage 3 atmospheric tank ($P_{\text{in}} = 1.363\text{ bar abs}$, $T_{\text{in}} = 40.0\text{ }^\circ\text{C}$) must be captured by a VRU compressor and discharged to the sales line ($P_{\text{out}} = 15.0\text{ bar abs} \approx 203\text{ psig}$):

• Tank Vent Molar Flow ($\dot{n}$): $53.0\text{ kmol/h}$ ($1.13\text{ MMSCFD}$)
• Vapor Molecular Weight ($M_w$): $36.42\text{ kg/kmol}$ (heavy rich vapor)
• Mass Flow Rate ($\dot{m}$): $1930.3\text{ kg/h} = 0.5362\text{ kg/s}$
• Pressure Ratio ($r_p$): $\frac{15.0}{1.363} = 11.0$ (two-stage reciprocating compressor required, $r_{p/\text{stage}} = \sqrt{11.0} = 3.32$)
• Specific Heat Ratio ($k = C_p/C_v$): $1.22$
• Isentropic Efficiency ($\eta_{\text{is}}$): $0.75$

Power Sizing Equation:

$$W_{\text{is}} = \dot{m} \cdot Z_{\text{avg}} \cdot \frac{k}{k - 1} \cdot \frac{R}{M_w} \cdot T_1 \cdot \left[ \left( \frac{P_2}{P_1} \right)^{\frac{k - 1}{k}} - 1 \right]$$

For two equal stages with intercooling to $40\text{ }^\circ\text{C}$: $$W_{\text{total, shaft}} = 2 \times \frac{W_{\text{stage}}}{\eta_{\text{is}}} = \mathbf{148.5\text{ kW (199.1 HP)}}$$

Economic Takeaway: Without the 3-stage flash, the single-stage tank vent flow of $5.32\text{ MMSCFD}$ would require an enormous $910\text{ HP}$ compressor, costing over $\$1.8\text{ million}$ in capital expenditures and $\$350,000/\text{year}$ in electric driver costs. The 3-stage separation pays for itself within three months on compressor power savings alone, while recovering an extra $1,470\text{ BPD}$ of premium condensate ($>\$35\text{ million/year}$ revenue uplift)!