In a two-phase gas–liquid system with stationary (non-flowing) liquid, express the gas hold-up (ε) in terms of the superficial gas velocity Vs and the bubble rise velocity Vt.

Introduction / Context:Gas hold-up (ε) is the volumetric fraction of gas in a gas–liquid dispersion and is a key parameter for mass transfer, residence time, and hydrodynamics in bubble columns and aerated bioreactors. A simple kinematic relation links ε to the superficial gas velocity (Vs) and the bubble rise velocity (Vt) when the liquid phase is not flowing on average.

Given Data / Assumptions:

• Continuous liquid phase is stationary on average (no net bulk liquid flow).
• Dispersed gas phase ascends as bubbles with average rise velocity Vt relative to the liquid.
• Superficial gas velocity Vs equals volumetric gas flux divided by column cross-section.

Concept / Approach:The volumetric gas flux Jg equals the product of gas volume fraction and bubble rise speed: Jg = ε * Vt. By definition, Vs = Jg. Therefore, ε = Vs / Vt. This relation captures how more gas flux or slower-rising bubbles (e.g., due to smaller size or higher viscosity) increase hold-up.

Step-by-Step Solution:Define gas flux: Jg = volumetric gas flow per area = Vs.Relate phases: Jg = ε * Vt (gas fraction times slip velocity).Equate: Vs = ε * Vt.Solve: ε = Vs / Vt.

Verification / Alternative check:Dimensional check: Vs and Vt both have units of length/time, so their ratio is dimensionless as required for ε. Empirically, increased aeration (higher Vs) at fixed bubble rise speed raises ε, consistent with observations.

Why Other Options Are Wrong:Forms with Vs ± Vt in numerator/denominator do not follow from the flux balance in a static liquid.ε = (Vs + Vt)/Vs would exceed 1 for typical conditions, which is non-physical for hold-up.

Common Pitfalls:Confusing bubble rise velocity relative to liquid (Vt) with superficial gas velocity (Vs); applying this relation when there is significant liquid co-current flow, which requires modified expressions.

Final Answer:ε = Vs / Vt