For aerated mixing vessels handling aqueous solutions, which empirical dependence best represents the liquid-film mass transfer coefficient kL on specific power input (Pm/V)?

Introduction / Context:
The liquid-side mass transfer coefficient (kL) reflects the rate at which dissolved species cross the liquid film surrounding bubbles or particles. In stirred, aerated tanks with aqueous media, kL is often correlated with turbulence intensity, which in turn scales with specific power input (Pm/V).

Given Data / Assumptions:

• Newtonian, low-to-moderate viscosity aqueous solutions.
• Standard Rushton or axial-flow impellers with typical aeration rates.
• kL considered separately from interfacial area a (thus not kLa).

Concept / Approach:
Classical surface-renewal and turbulence theories predict that film thickness decreases with increasing energy dissipation, giving a sublinear power-law dependence of kL on (Pm/V). Empirical studies commonly report an exponent near 0.33 in water-like liquids, while the overall kLa includes an additional dependence on gas flow that governs interfacial area (a).

Step-by-Step Solution:

Verification / Alternative check:
Log–log plots of measured kL against (Pm/V) yield slopes close to 1/3 in water and dilute broths; deviations occur at high viscosity or with non-Newtonian rheology.

Why Other Options Are Wrong:

• Exponents 0.50 or 0.66: too strong for typical aqueous kL scaling (more typical for other regimes or parameters).
• Exponent 1.00: unrealistically strong sensitivity.
• Independence: contradicts turbulence-controlled film dynamics.

Common Pitfalls:
Confusing kL with kLa; the latter depends on both kL and interfacial area a, which also varies with gas flow rate and dispersion quality.

Final Answer:
kL ∝ (Pm/V)^0.33