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?
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AK_L = 2 * (D_AB / (π * t_e))^0.25
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BK_L = 2 * (D_AB / (π * t_e))^0.5
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CK_L = 2 * (D_AB / (π * t_e))^0.75
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DK_L = 2 * (D_AB / (π * t_e))
Answer
Correct Answer: K_L = 2 * (D_AB / (π * t_e))^0.5
Explanation
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:
1) Assume elements renew at the interface every t_e seconds.2) Transient diffusion thickness grows as (D_AB * t)^0.5.3) Average flux over t_e leads to K_L ∝ (D_AB / t_e)^0.5.4) The Higbie solution gives K_L = 2 * (D_AB / (π * t_e))^0.5.5) Select the option with exponent 0.5 and coefficient form 2 / √π.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