According to penetration theory (Higbie model), turbulent eddies intermittently contact the interface for an exposure time t_e. Which correlation relates the liquid-side mass transfer coefficient K_L to diffusivity D_AB and exposure time?

Introduction:
Penetration theory explains unsteady diffusion into fluid elements that periodically renew the interface. It is a foundational concept for estimating K_L when interfacial renewal controls mass transfer in agitated or turbulent systems.

Given Data / Assumptions:

• Fluid elements contact the interface for time t_e before being replaced.
• Diffusion is unsteady during each contact.
• Transport is characterized by molecular diffusivity D_AB.

Concept / Approach:
Solving the transient diffusion equation for exposure time t_e yields K_L proportional to the square root of D_AB / t_e. The constant 2 / √π arises from the analytical solution for a semi-infinite medium initially free of solute and suddenly exposed at the interface.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis confirms K_L units of length/time when D_AB has length^2/time and t_e is time, consistent with the square-root dependence.

Why Other Options Are Wrong:

• Exponents 0.25 or 0.75 do not follow from the transient solution.
• Linear dependence on D_AB / t_e is inconsistent with diffusion scaling.

Common Pitfalls:
Confusing penetration theory (unsteady) with film theory (steady) or surface renewal theory; while related, they use different characteristic parameters.

Final Answer:
K_L = 2 * (D_AB / (π * t_e))^0.5