Chemical, Process & Catalysis Engineering
Rachford-Rice Isothermal VLE Flash Formula: Bubble/Dew Check & Objective f(V/F)
Test multicomponent feed phase stability (Subcooled Liquid SUM(z_i * K_i) <= 1, Superheated Vapor SUM(z_i / K_i) <= 1, or Two-Phase Flash) and evaluate H. H. Rachford and J. D. Rice’s (1952) monotonic flash residual f(psi).
Why subtracting liquid sum from vapor sum made Rachford-Rice the universal flash equation!
In refinery and gas-plant flash drums, a multicomponent feed of mole fractions z_i (SUM(z_i) = 1) enters at fixed Temperature T and Pressure P where each component has equilibrium K-value K_i = y_i / x_i! Before solving for vaporized mole fraction psi = V / F in (0, 1), engineers check two exact phase boundaries: if Bubble-Point Sum SUM(z_i * K_i) <= 1, the feed remains 100% Subcooled/Bubble Liquid (psi = 0); if Dew-Point Sum SUM(z_i / K_i) <= 1, the feed flashes to 100% Superheated/Dew Vapor (psi = 1)!
When BOTH SUM(z_i * K_i) > 1 AND SUM(z_i / K_i) > 1, two phases coexist! Rather than solving SUM(y_i) = 1 or SUM(x_i) = 1 alone (which exhibit severe pole curvature), Henry H. Rachford Jr. and John D. Rice (1952, J. Pet. Technol.) subtracted the two identities to obtain the strictly monotonic decreasing objective f(psi) = SUM((z_i * (K_i - 1)) / (1 + psi * (K_i - 1))) = 0 — guaranteeing a unique root psi in (0, 1) and yielding equilibrium compositions x_i = z_i / (1 + psi * (K_i - 1)) and y_i = K_i * x_i!
| Ternary Hydrocarbon Component | Feed Mole Fraction z_i | VLE K-Value K_i at Drum T, P | At Root psi = V/F = 0.50: x_i & y_i |
|---|---|---|---|
| Propane (Light Key) | z_1 = 0.30 | K_1 = 2.50 | x_1 = 0.1714 | y_1 = 0.4286 |
| n-Butane (Mid Key) | z_2 = 0.40 | K_2 = 1.00 | x_2 = 0.4000 | y_2 = 0.4000 |
| n-Pentane (Heavy Key) | z_3 = 0.30 | K_3 = 0.40 | x_3 = 0.4286 | y_3 = 0.1714 |
Multicomponent Flash Phase Regime Classifier (Subcooled, Two-Phase, or Superheated)
Checks mole-fraction normalization ABS(SUM(B2:B4) - 1) <= 0.001, Bubble-Point Sum SUMPRODUCT(B2:B4, C2:C4), and Dew-Point Sum B2/C2 + B3/C3 + B4/C4.
=IF(OR(ABS(SUM(B2:B4) - 1) > 0.001, MIN(C2:C4) <= 0), "Requires Normalized Feed SUM(z_i) = 1 and K_i > 0", IF(SUMPRODUCT(B2:B4, C2:C4) <= 1, "Subcooled / Bubble-Point Liquid (Vapor Fraction V/F = 0)", IF((B2 / C2 + B3 / C3 + B4 / C4) <= 1, "Superheated / Dew-Point Vapor (Vapor Fraction V/F = 1)", "Two-Phase VLE Flash Drum Regime (0 < V/F < 1)")))Rachford-Rice Monotonic Flash Residual f(psi) at Trial Vapor Fraction E2
Evaluates SUM((z_i * (K_i - 1)) / (1 + psi * (K_i - 1))) across the 3-component feed at trial vapor fraction psi = V/F in E2 (0 <= E2 <= 1); f(psi) = 0 identifies exact thermodynamic equilibrium!
=IF(OR(ABS(SUM(B2:B4) - 1) > 0.001, MIN(C2:C4) <= 0, E2 < 0, E2 > 1), "Invalid Feed, K-Values, or Vapor Fraction psi (0..1)", ROUND((B2 * (C2 - 1)) / (1 + E2 * (C2 - 1)) + (B3 * (C3 - 1)) / (1 + E2 * (C3 - 1)) + (B4 * (C4 - 1)) / (1 + E2 * (C4 - 1)), 6))Try the example
Use the original values or change them to test your understanding.