Superficial Gas Velocity Calculation

Reference ID: MET-CAB0 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Superficial gas velocity, v_S, is the volumetric gas flow rate divided by the cross-sectional area of the vessel. It is the single most influential parameter for characterising gas–liquid contactors such as stirred-tank fermenters, bubble columns, and packed towers. A correct value of v_S is required to:

predict gas hold-up, interfacial area, and mass-transfer coefficients (k_La);
check the risk of flooding or impeller over-loading;
scale-up or scale-down reactors while keeping hydrodynamic regimes similar.

Methodology & Formulas

1. Convert operating conditions to actual gas flow

Standard flow, Q_G,std (Nm³ h^-1), is corrected to actual temperature and pressure using the ideal-gas law:

\[ Q_{G,\text{act}} = Q_{G,\text{std}}\,\frac{T_{\text{act}}}{T_{\text{std}}}\,\frac{P_{\text{std}}}{P_{\text{act}}}\,(1-\phi\,\frac{p_{\text{v}}}{P_{\text{act}}}) \]

where T_act and P_act are absolute values. Relative humidity φ and water vapour pressure p_v may be set to zero for dry-gas duty.

2. Superficial gas velocity

\[ v_{S} = \frac{Q_{G,\text{act}}}{A_{T}},\qquad A_{T}=\frac{\pi D_{T}^{2}}{4} \]

3. Flooding (impeller loading) limit

The middle-bracket correlation gives the flooding velocity for a radial impeller in a baffled tank:

\[ v_{S,\text{flood}} = 0.8\,(ND)^{0.7}\left(\frac{\sigma}{\rho_{L}}\right)^{0.3}(PV)^{-0.2}\left(\frac{D}{D_{T}}\right)^{0.3} \]

where ND is the tip speed, σ surface tension, ρ_L liquid density, and PV power input per unit volume.

4. Regime validity

Parameter	Threshold	Remarks
Impeller Reynolds number	Re_imp > 10,000	Correlation derived for fully turbulent regime
Safety factor	SF = v_S,flood/v_S > 1.3	Prevents flooding and flow instabilities

5. Derived hydrodynamic quantities

Bubble residence time: t_res = H_L/v_S
Gas hold-up (empirical): ε_G = v_S/(v_S + 0.25)
Volumetric mass-transfer coefficient: k_La = 0.015 (PV)^0.7 (v_S)^0.6

Superficial gas velocity (U_g) is the volumetric flow rate of gas divided by the cross-sectional area of the empty vessel or pipe. It is not the actual velocity of the gas, because it ignores the volume occupied by liquid or solid phases. Engineers use it to:

Predict flow regimes (bubbly, slug, annular, etc.) in multiphase systems
Size bubble columns, fluidized beds, and scrubbers
Estimate pressure drop and mass-transfer coefficients

Convert mass flow to volumetric flow at operating conditions, then divide by area:

Step 1: Volumetric flow Q_g = ṁ_g / ρ_g, where ρ_g is gas density at P, T
Step 2: U_g = Q_g / A, with A = πD²/4 for circular pipes
Always use actual gas density, not standard or normal density, to avoid 5–15% error

Use the empty-column cross-section, not the void area between particles. This convention keeps U_g independent of packing type, particle size, or bed expansion, so correlations for flooding, minimum fluidization, or regime maps remain valid across different internals.

Rough guidelines at ambient pressure:

Bubble columns: 0.05–0.3 m s^-1
Fluidized beds: 0.1–1 m s^-1 (depending on particle size and density)
Tray columns: 0.3–1.2 m s^-1 based on active bubbling area
Venturi scrubbers: 30–120 m s^-1 in the throat

Always verify against flooding or entrainment correlations for your specific system.

Worked Example – Estimating Superficial Gas Velocity in an Aerobic Fermenter

A brewery is scaling-up a 700 mm ID stirred-tank for a high-gravity lager fermentation. The vessel is fitted with a single Rushton turbine (N = 150 rpm) and sparged with 90 Nm³ h⁻¹ of sterile air. Process engineers need the superficial gas velocity, v_S, to check if the impeller is flooded and to estimate the mass-transfer coefficient k_La.

Knowns

Column inside diameter, D = 0.700 m
Impeller speed, N = 150 rpm = 2.5 rps
Gas flow-rate (standard), Q_G,std = 90 Nm³ h⁻¹ = 0.025 m³ s⁻¹
Temperature in tank, T_act = 30 °C = 303.15 K
Absolute pressure in tank, P_act = 1.2 bar = 120,000 Pa
Standard conditions: T_std = 273.15 K, P_std = 101,325 Pa
Liquid density, ρ_L = 1000 kg m⁻³
Liquid viscosity, μ_L = 0.001 Pa s
Surface tension, σ = 0.072 N m⁻¹
Gravitational acceleration, g = 9.81 m s⁻²

Step-by-step calculation

Convert the gas flow to actual tank conditions using the ideal-gas law:

\[ Q_{G,act} = Q_{G,std}\frac{P_{std}}{P_{act}}\frac{T_{act}}{T_{std}} = 0.025\frac{101,325}{120,000}\frac{303.15}{273.15} = 0.023\ \text{m}^3\ \text{s}^{-1} \]
Calculate the tank cross-sectional area:

\[ A_T = \frac{\pi D^2}{4} = \frac{\pi(0.7)^2}{4} = 0.385\ \text{m}^2 \]
Compute superficial gas velocity:

\[ v_S = \frac{Q_{G,act}}{A_T} = \frac{0.023}{0.385} = 0.060\ \text{m s}^{-1} \]
Check for impeller flooding. First evaluate the impeller Reynolds number:

\[ Re_{imp} = \frac{\rho_L N D^2}{\mu_L} = \frac{1000 \cdot 2.5 \cdot 0.7^2}{0.001} = 1,225,000 \]

For a Rushton turbine, the flooding correlation gives:

\[ v_{S,flood} = 0.16\,\frac{N}{D^{0.3}}\left(\frac{\sigma}{\rho_L}\right)^{0.5} = 0.16\,\frac{2.5}{0.7^{0.3}}\left(\frac{0.072}{1000}\right)^{0.5} = 0.011\ \text{m s}^{-1} \]
Determine the safety factor:

\[ SF = \frac{v_{S,flood}}{v_S} = \frac{0.011}{0.060} = 0.18 \]

Because SF < 1, the impeller operates well below the flooding limit.
Estimate the volumetric mass-transfer coefficient k_La (for water-air) using the empirical correlation for stirred tanks:

\[ k_La = 2.0 \cdot 10^{-3}\left(\frac{P}{V}\right)^{0.7}v_S^{0.6} \quad \text{with} \quad \frac{P}{V} = 1500\ \text{W m}^{-3} \]

\[ k_La = 2.0 \cdot 10^{-3}(1500)^{0.7}(0.060)^{0.6} = 0.133\ \text{s}^{-1} \]

Final Answer

The superficial gas velocity in the fermenter is 0.060 m s⁻¹, corresponding to a flooding safety factor of 0.18 and a predicted k_La of 0.133 s⁻¹.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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