Rachford-Rice flash

The vapor-fraction objective every isothermal two-phase flash solves — one nonlinear equation in the vapor fraction ψ, monotonic so Newton always converges.

Governing equation

$\psi$

Vapor fraction (mol vapor / mol feed) — the unknown.

$z_i$

Feed mole fraction of component i.

$K_i$

Vapor-liquid equilibrium ratio y_i/x_i of component i (fixed for this flash).

Derivation

1. 1
   $$z_i = x_i\,(1-\psi) + y_i\,\psi$$

   Component material balance: the feed splits into liquid (fraction 1−ψ) and vapor (fraction ψ).

2. 2
   $$x_i = \frac{z_i}{1 + \psi\,(K_i - 1)}$$

   Substitute the equilibrium relation y_i = K_i x_i and solve the balance for the liquid mole fraction x_i.

3. 3
   $$\sum_i y_i - \sum_i x_i = \sum_i (K_i - 1)\,x_i = 0$$

   Both phase mole fractions must sum to one; subtracting the two summation constraints removes the trivial ψ-independent root and leaves a monotonic function.

4. 4
   $$F(\psi) = \sum_{i} \frac{z_i\,(K_i - 1)}{1 + \psi\,(K_i - 1)} = 0$$

   Insert x_i to obtain the Rachford-Rice form — strictly decreasing in ψ, so a single root is bracketed and Newton converges globally.

Jacobian

The matrix $J_{ij} = \partial F_i / \partial x_j$ of the residual vector with respect to the unknowns, at the representative operating point. Every value below is computed by the exact forward-mode autodiff engine the solver uses — click a non-zero cell to see the symbolic derivative, each one verified against that engine in our test suite. This is the block a Newton step factorizes.

References

• Rachford, H.H. & Rice, J.D. (1952). Procedure for use of electronic digital computers in calculating flash vaporization hydrocarbon equilibrium. J. Petroleum Technology 4(10).
• Smith, Van Ness & Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., §10.4.