Flash Calculations

Overview

A flash calculation determines the equilibrium phase split of a mixture at specified pressure and temperature. Given a feed composition $z_i$ , it finds:

Vapor fraction $V$ (moles of vapor per mole of feed)
Liquid composition $x_i$ (mole fractions in liquid phase)
Vapor composition $y_i$ (mole fractions in vapor phase)

Flash calculations are the computational core of all compositional reservoir simulation and process engineering.

Theory

Equilibrium Conditions

At thermodynamic equilibrium, the fugacity of each component must be equal in all phases:

$f_i^L = f_i^V \quad \text{for all } i = 1, \ldots, N$

This is expressed using the K-value (equilibrium ratio):

$K_i = \frac{y_i}{x_i} = \frac{\hat{\phi}_i^L}{\hat{\phi}_i^V}$

Where $\hat{\phi}_i$ is the fugacity coefficient calculated from the equation of state.

Material Balance

For a feed with $N$ components at mole fractions $z_i$ :

$z_i = x_i(1 - V) + y_i V$

Substituting $y_i = K_i x_i$ :

$x_i = \frac{z_i}{1 + V(K_i - 1)}$

$y_i = \frac{K_i z_i}{1 + V(K_i - 1)}$

The Rachford-Rice Equation

Since $\sum x_i = \sum y_i = 1$ , subtracting gives:

$\sum_{i=1}^{N} (y_i - x_i) = \sum_{i=1}^{N} \frac{z_i(K_i - 1)}{1 + V(K_i - 1)} = 0$

This is the Rachford-Rice equation. It has a single unknown $V$ and is solved by Newton-Raphson iteration.

Physical Bounds

The vapor fraction must satisfy $0 \le V \le 1$ for two-phase equilibrium:

$V = 0$ : all liquid (bubble point)
$V = 1$ : all vapor (dew point)
$V < 0$ or $V > 1$ : single phase (no flash needed)

K-Value Estimation

Wilson Correlation (Initial Estimate)

$K_i = \frac{P_{c,i}}{P} \exp\left[5.37(1 + \omega_i)\left(1 - \frac{T_{c,i}}{T}\right)\right]$

The Wilson equation provides a reasonable starting point for the iterative flash procedure. Convergence to the EoS solution typically requires 5-15 iterations.

Converged K-Values

After solving the flash problem, the converged K-values from the EoS are:

$K_i = \frac{\hat{\phi}_i^L(T, P, x)}{\hat{\phi}_i^V(T, P, y)}$

These are far more accurate than the Wilson estimates and account for composition-dependent interactions.

Flash Algorithm

Successive Substitution

1. Initialize K-values (Wilson correlation)
2. Solve Rachford-Rice for V
3. Calculate x_i, y_i from K-values and V
4. Calculate fugacity coefficients φ_i^L(x) and φ_i^V(y) from EoS
5. Update K-values: K_i = φ_i^L / φ_i^V
6. Check convergence: |ln(K_i^new/K_i^old)| < tolerance
7. If not converged, go to Step 2

Convergence Criteria

Criterion	Typical Tolerance
K-value change	$\sum (\ln K_i^{new} - \ln K_i^{old})^2 < 10^{-10}$
Fugacity equality	$\max \\|f_i^L/f_i^V - 1\\| < 10^{-8}$
Rachford-Rice residual	$\\|g(V)\\| < 10^{-12}$

Stability Analysis

Why Stability Testing?

Before performing a flash calculation, it is important to determine whether the mixture actually splits into two phases. A stability test answers: "Is the single-phase state stable, or will the system spontaneously separate?"

Tangent Plane Distance (TPD)

The mixture is unstable (two phases will form) if there exists any trial composition $w_i$ such that:

$TPD(w) = \sum_i w_i [\ln w_i + \ln \hat{\phi}_i(w) - \ln z_i - \ln \hat{\phi}_i(z)] < 0$

The stability test is performed with two trial phases:

Vapor-like trial (using Wilson K-values: $w_i = K_i z_i$ )
Liquid-like trial (using inverse Wilson: $w_i = z_i / K_i$ )

If either trial gives $TPD < 0$ , the mixture is unstable and a flash calculation is warranted.

Saturation Point Calculations

Bubble Point Pressure

At the bubble point, $V = 0$ (incipient vapor formation):

$\sum_{i=1}^{N} K_i z_i = 1$

The bubble point pressure $P_b$ is found iteratively by adjusting $P$ until this condition is satisfied.

Dew Point Pressure

At the dew point, $V = 1$ (incipient liquid formation):

$\sum_{i=1}^{N} \frac{z_i}{K_i} = 1$

The dew point pressure $P_{dew}$ is found iteratively by adjusting $P$ .

Applicability and Limitations

Valid Conditions

The Rachford-Rice formulation assumes two phases (vapor + liquid)
Three-phase flash (VLLE) requires extended formulations
Components must have known $T_c$ , $P_c$ , $\omega$ for K-value initialization

Common Issues

Problem	Cause	Solution
No convergence	Near critical point	Switch to Newton's method with Hessian
Trivial solution ( $K_i = 1$ )	Poor initialization	Use different initial K-values
Negative flash ( $V < 0$ or $V > 1$ )	Single phase at this P, T	Perform stability test first
Slow convergence	Successive substitution limit	Apply GDEM acceleration

Peng-Robinson EoS — Fugacity coefficient calculation
Phase Envelope — Series of saturation point calculations
C7+ Characterization — Component properties for flash
EoS Overview — When to use EoS modeling

References

Rachford, H.H. and Rice, J.D. (1952). "Procedure for Use of Electronic Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium." Journal of Petroleum Technology, 4(10), 19-3.
Wilson, G.M. (1969). "A Modified Redlich-Kwong Equation of State, Application to General Physical Data Calculations." 65th National AIChE Meeting, Paper No. 15C, Cleveland, Ohio.
Michelsen, M.L. (1982). "The Isothermal Flash Problem. Part I. Stability." Fluid Phase Equilibria, 9(1), 1-19.
Michelsen, M.L. (1982). "The Isothermal Flash Problem. Part II. Phase-Split Calculation." Fluid Phase Equilibria, 9(1), 21-40.
Whitson, C.H. and Brule, M.R. (2000). Phase Behavior. SPE Monograph Vol. 20.