Chemical, Process & Catalysis Engineering
Zivi Slip Ratio & Two-Phase Void Fraction Formula: Annular vs Homogeneous Holdup
Calculate cross-sectional vapor void fraction alpha, liquid holdup (1 - alpha), and static mixture density rho_TP in gas-liquid pipes using S. M. Zivi’s (1964) minimum-entropy slip ratio S = (rho_L / rho_G)^(1/3) versus Homogeneous No-Slip (S = 1).
Why lighter gas outruns liquid film (S > 1) — drastically lowering vapor void fraction below homogeneous predictions!
In gas-liquid boiling tubes and risers, mass vapor quality x = m_dot_G / m_dot_total = 0.05 (only 5% vapor by mass!) can occupy > 65% of the pipe’s cross-sectional area because vapor density rho_G is far smaller than liquid density rho_L! Under fundamental continuity, cross-sectional Vapor Void Fraction alpha = A_G / A_total relates to Slip Ratio S = u_G / u_L via exact Fundamental Void Identity: alpha = 1 / (1 + S * ((1 - x) / x) * (rho_G / rho_L))!
Assuming equal phase velocities (Homogeneous Equilibrium Model, S = 1) overpredicts void fraction whenever faster-moving gas core slips past slower wall liquid! Minimizing kinetic-energy entropy generation in separated/annular flow without empirical tuning constants, S. M. Zivi (1964, ASME J. Heat Transfer) derived the celebrated analytical slip ratio S_Zivi = (rho_L / rho_G)^(1/3), yielding Two-Phase Static Gravitational Density rho_TP = alpha * rho_G + (1 - alpha) * rho_L!
| Two-Phase Quantity (x = 0.10, rho_L = 800 kg/m^3, rho_G = 10 kg/m^3 -> Ratio = 80) | Homogeneous Model (S = 1.00) | Zivi Minimum-Entropy Model (S = 80^(1/3) = 4.3089) |
|---|---|---|
| Velocity Slip Ratio S = u_G / u_L | 1.0000 | 4.3089 |
| Cross-Sectional Vapor Void Fraction alpha | 0.8989 (89.9% gas area) | 0.6735 (67.4% gas area) |
| Cross-Sectional Liquid Holdup (1 - alpha) | 0.1011 (10.1% liquid) | 0.3265 (32.7% liquid) |
| Static Gravitational Head Density rho_TP | 89.89 kg/m^3 | 267.95 kg/m^3 (3x heavier static head!) |
Zivi (1964) Separated-Flow Vapor Void Fraction alpha_Zivi
Substitutes S = (rho_L / rho_G)^(1/3) into 1 / (1 + S * ((1 - x) / x) * (rho_G / rho_L)) across vapor mass quality B2 (0 < x < 1), liquid density B3 (kg/m^3), and gas density B4 (< B3, kg/m^3).
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B4 >= B3), "Requires 0 < Quality x < 1 and 0 < rho_G < rho_L", ROUND(1 / (1 + ((B3 / B4) ^ (1 / 3)) * ((1 - B2) / B2) * (B4 / B3)), 4))Homogeneous No-Slip Void Fraction alpha_H (S = 1) for Comparison
Evaluates upper-bound no-slip void fraction 1 / (1 + ((1 - x) / x) * (rho_G / rho_L)).
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B4 >= B3), "Requires 0 < Quality x < 1 and 0 < rho_G < rho_L", ROUND(1 / (1 + ((1 - B2) / B2) * (B4 / B3)), 4))Try the example
Use the original values or change them to test your understanding.