Question

If \( k \) is the mass transfer coefficient and \( D_v \) is the molecular diffusivity, which one of the following statements is NOT CORRECT with respect to mass transfer theories?

• A. For Film theory, \( k \propto D_v \)
• B. For Penetration theory, \( k \propto D_v^{1/3} \)
• C. For Surface Renewal theory, \( k \propto D_v^{1/2} \)
• D. For Boundary Layer theory, \( k \propto D_v^{2/3} \)

Hint:

It's essential to accurately understand and apply the theoretical relationships in mass transfer to ensure correct engineering calculations and designs.

Answer 1

The Correct Option is B

Step 1: Examine Mass Transfer Theories.
Each mass transfer theory proposes a specific relationship between the mass transfer coefficient \( k \) and the molecular diffusivity \( D_v \), often involving power-law dependencies that reflect the underlying physical phenomena: Film theory generally approximates \( k \) to be proportional to \( D_v^{1/2} \), factoring in the film thickness and diffusivity.
Penetration theory actually suggests \( k \propto D_v^{1/2} \) as well, based on the transient penetration of solute into a stagnant fluid layer. The incorrect statement (B) wrongly attributes \( k \propto D_v^{1/3} \) to Penetration theory.
Surface Renewal theory supports \( k \propto D_v^{1/2} \), consistent with its conceptualization of continuous renewal of surface elements.
Boundary Layer theory correctly describes \( k \propto D_v^{1/2} \) for a laminar boundary layer, again reflecting diffusion across a boundary layer of defined thickness. 
Step 2: Identify the Incorrect Statement.
From the theories discussed, the statement (B) incorrectly specifies the relationship for Penetration theory, which should correctly reflect a \( D_v^{1/2} \) dependency, not \( D_v^{1/3} \).