Gas hold-up measurement in aerated vessels: If Zf is the height of the aerated (gassy) liquid and Zl is the height of the non-aerated clear liquid, what is the expression for the average fractional gas hold-up H?

Introduction / Context:
Gas hold-up quantifies the volumetric fraction of gas in a gas–liquid mixture, directly affecting residence times, interfacial area, and mass transfer. A simple way to estimate hold-up is from the increase in liquid height when gas is introduced into a column or reactor.

Given Data / Assumptions:

• Zl: height of the clear (non-gassed) liquid at rest.
• Zf: height of the aerated (foamy/gassy) liquid during gas flow.
• Uniform cross-section; negligible wall wetting corrections.

Concept / Approach:
The presence of gas expands the mixture volume. The gas fraction equals the volume increase divided by the total aerated volume. With constant cross-section, volumes are proportional to heights, so H = (Zf − Zl) / Zf.

Step-by-Step Solution:
Define H = V_gas / V_total.With constant area A: V_gas = A * (Zf − Zl) and V_total = A * Zf.Therefore, H = [A(Zf − Zl)] / (A Zf) = (Zf − Zl) / Zf.Select this expression from the options.

Verification / Alternative check:
Independent measurements (e.g., quick shut-off method) yield similar hold-up values, validating the height-based estimate for many systems.

Why Other Options Are Wrong:
Zf/Zl or Zf/(Zf*Zl): dimensionally inconsistent and do not represent a fraction between 0 and 1.

(Zf + Zl)/Zf: yields values > 1; not a valid fractional hold-up.

Common Pitfalls:

• Failing to account for foam head; Zf should represent the stable aerated level, not transient foam peaks.
• Using this method for non-uniform cross-sections without correction.

Final Answer:
H = (Zf − Zl) / Zf