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Chemical, Process & Catalysis Engineering

Smith (1969) Annular-Mist Slip Ratio & Void Fraction Formula: Entrained Liquid e_k = 0.40

Calculate steam-water and refrigerant boiling cross-sectional void fraction alpha using S. L. Smith’s (1969) Momentum-Equalized Annular-Mist Slip Ratio S_Smith with entrained droplet fraction e_k = 0.40.

Model real annular-mist boiling — a homogeneous gas-droplet core surrounded by a slower liquid wall film!

Pure separated-flow models like Zivi (S = (rho_L / rho_G)^(1/3)) assume 100% of liquid travels along the pipe wall with zero droplets entrained in the gas core, underpredicting void fraction at moderate-to-high steam qualities! Conversely, Homogeneous Equilibrium (S = 1) assumes 100% entrainment! Modeling two-phase pipe flow as a homogeneous gas-plus-entrained-droplet core (carrying fraction e_k of total liquid mass at gas velocity u_G) exchanging equal dynamic momentum head (rho_core * u_core^2 = rho_film * u_film^2) with the remaining (1 - e_k) annular wall film, S. L. Smith (1969–1970, Proc. Inst. Mech. Eng.) derived the celebrated Annular-Mist Slip Ratio:

S_Smith = u_G / u_L,film = e_k + (1 - e_k) * SQRT(((rho_L / rho_G) + e_k * ((1 - x) / x)) / (1 + e_k * ((1 - x) / x))), with empirical universal entrainment constant e_k = 0.40 (40% of liquid entrained as mist droplets)! Notice how setting e_k = 1 recovers Homogeneous S = 1, setting e_k = 0 recovers Chawla/Chisholm separated momentum S = SQRT(rho_L / rho_G), and e_k = 0.40 matches boiling-water reactor, kettle reboiler, and refrigerant tube data within +/-5% across wide pressures!

Vapor Mass Quality x (at rho_L / rho_G = 50, Entrainment e_k = 0.40)Smith Slip Ratio S_SmithCross-Sectional Void Fraction alpha
Low Quality Bubbly/Slug (x = 0.05)2.21560.5489 (54.9% vapor area)
Mid Quality Annular-Mist (x = 0.20)3.07290.8027 (80.3% vapor area)
High Quality Mist Flow (x = 0.60)4.17900.9472 (94.7% vapor area)

Smith (1969) Annular-Mist Velocity Slip Ratio S_Smith (Default e_k = 0.40)

Evaluates e_k + (1 - e_k) * SQRT(((rho_L / rho_G) + e_k * ((1 - x) / x)) / (1 + e_k * ((1 - x) / x))) across vapor quality x (B2 in (0, 1)), density ratio rho_L / rho_G > 1 (B3), and entrained liquid fraction e_k (B4 in [0, 1], standard 0.40).

=IF(OR(B2 <= 0, B2 >= 1, B3 <= 1, B4 < 0, B4 > 1), "Requires 0 < x < 1, rho_L/rho_G > 1, and 0 <= e_k <= 1", ROUND(B4 + (1 - B4) * SQRT((B3 + B4 * ((1 - B2) / B2)) / (1 + B4 * ((1 - B2) / B2))), 4))

Smith Cross-Sectional Vapor Void Fraction alpha_Smith

Substitutes unrounded S_Smith into Fundamental Void Identity 1 / (1 + S_Smith * ((1 - x) / x) / (rho_L / rho_G)).

=IF(OR(B2 <= 0, B2 >= 1, B3 <= 1, B4 < 0, B4 > 1), "Requires 0 < x < 1, rho_L/rho_G > 1, and 0 <= e_k <= 1", ROUND(1 / (1 + (B4 + (1 - B4) * SQRT((B3 + B4 * ((1 - B2) / B2)) / (1 + B4 * ((1 - B2) / B2)))) * ((1 - B2) / B2) / B3), 4))

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