The Rachford-Rice Equation and Isothermal Flash

What is a Flash Calculation?

A "Flash" occurs when a pressurized liquid is routed through a valve into a lower-pressure vessel (like a Flash Drum or 2-Phase Separator). The sudden drop in pressure causes a portion of the liquid to instantaneously vaporize (or "flash") into a gas.

The fundamental goal of a flash calculation is to answer two questions:

1. How much vapor and how much liquid are produced?
2. What is the exact chemical composition of the resulting vapor ($y_i$) and liquid ($x_i$)?

The Mass Balance Problem

Let $F$ be the total moles of feed entering the drum, with overall composition $z_i$. It splits into a Vapor stream ($V$) and a Liquid stream ($L$).

Overall Mass Balance:

Component Mass Balance:

We also know from the definition of the K-value that $y_i = K_i \cdot x_i$. Finally, we know that the sum of all mole fractions in the vapor phase must equal 1.0 ($\sum y_i = 1$), and the sum in the liquid phase must equal 1.0 ($\sum x_i = 1$). Therefore, the difference between their sums must be zero:

The Rachford-Rice Equation

In 1952, H.H. Rachford Jr. and J.D. Rice took the mass balance equations above and algebraically combined them into a single, elegant equation. It calculates the Vapor Fraction ($\frac{V}{F}$, often denoted as $\alpha$ or $\psi$).

The Rachford-Rice Equation is universally written as:

Where:

• $z_i$ = Overall mole fraction of component $i$ in the feed
• $K_i$ = Equilibrium constant (K-value) for component $i$
• $\frac{V}{F}$ = The molar Vapor Fraction (the unknown variable to solve for)

Why is Rachford-Rice so Brilliant?

The Rachford-Rice equation is mathematically beautiful because its derivative with respect to $\frac{V}{F}$ is strictly negative (it is monotonically decreasing).

Because it is monotonic, it has exactly one unique root between $\frac{V}{F} = 0$ (the bubble point) and $\frac{V}{F} = 1$ (the dew point). This guarantees that numerical solvers (like the Newton-Raphson method) will converge extremely fast and will never get stuck in an infinite loop or find a fake mathematical solution.

Every single process simulator in the world (HYSYS, Aspen Plus, Pro/II) uses the Newton-Raphson method applied to the Rachford-Rice equation millions of times per second to solve flash drums, distillation columns, and pipe phase envelopes.

Step-by-Step Flash Algorithm

If you need to code a flash calculator in Python or Excel, here is the algorithm:

1. Calculate the K-values for all components at the given T and P using an Equation of State.
2. Use a root-finding algorithm (like Newton-Raphson or Bisection) to solve the Rachford-Rice equation for $\frac{V}{F}$ such that the sum equals zero.
3. Once $\frac{V}{F}$ is found, calculate the liquid composition for each component: $x_i = \frac{z_i}{1 + \frac{V}{F} \cdot (K_i - 1)}$
4. Calculate the vapor composition for each component: $y_i = K_i \cdot x_i$

Try the Calculator

Want to see Rachford-Rice in action? Use the Separator Calculator on this page. It executes this exact algorithm to determine the vapor fraction and liquid compositions dynamically!