VLE Flashing & The Rachford-Rice Equation

When a pressurized multi-component mixture flows through a valve into a separator drum, the sudden drop in pressure causes a portion of the liquid to boil ("flash") into vapor. Predicting exactly how much vapor is created, and what the chemical composition of that vapor will be, is the most fundamental calculation in chemical engineering. It is solved using the **Rachford-Rice Equation**. ## 1. Vapor-Liquid Equilibrium (K-Values) For any given component $i$, the ratio of its mole fraction in the vapor phase ($y_i$) to its mole fraction in the liquid phase ($x_i$) is known as the equilibrium constant, or **K-value**: $$ K_i = \frac{y_i}{x_i} $$ While complex equations of state (like Peng-Robinson) are often used, the **Wilson Equation** provides highly accurate K-values for light hydrocarbons based purely on their critical properties ($P_c, T_c$) and acentric factor ($\omega$): $$ K_i = \frac{P_{c,i}}{P} \exp \left[ 5.373 (1 + \omega_i) \left( 1 - \frac{T_{c,i}}{T} \right) \right] $$ ## 2. The Rachford-Rice Equation If we know the feed composition ($z_i$) and the K-values, we can set up a mass balance. The total vapor fraction ($V$) must satisfy the Rachford-Rice objective function: $$ \sum_{i=1}^{n} \frac{z_i (K_i - 1)}{1 + V (K_i - 1)} = 0 $$ Because this equation cannot be solved algebraically, engineers use iterative numerical methods (like Newton-Raphson or Bisection) to guess a value for $V$ (between 0 and 1) until the equation equals zero. Once $V$ is found, the exact phase compositions are calculated as: $$ x_i = \frac{z_i}{1 + V(K_i - 1)} \quad \text{and} \quad y_i = K_i x_i $$