Question

The penetration theory relates average mass transfer coefficient ($ k $) with diffusivity ($ D $)

• A. \( k \propto D \)
• B. \( k \propto \sqrt{D} \)
• C. \( k \propto D^{1.5} \)
• D. \( k \propto D^2 \)

Hint:

Penetration theory predicts \( k \propto \sqrt{D} \), reflecting the unsteady-state diffusion of mass into a fluid element during short contact times.

Answer 1

The Correct Option is B

Step 1: Understand the penetration theory.
The penetration theory, proposed by Higbie, models mass transfer at a gas-liquid or liquid-liquid interface. It assumes that a fluid element is exposed to the interface for a short time (\( t \)), during which mass transfer occurs by unsteady-state diffusion into the fluid element. The theory is often applied to processes like absorption or bubble flow, where fresh fluid elements are periodically brought to the interface. Step 2: Derive the mass transfer coefficient using penetration theory.
According to the penetration theory:
Mass transfer occurs by diffusion into a semi-infinite medium for a short contact time \( t \).
The concentration profile in the fluid element is governed by the diffusion equation: \[ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}, \] where \( c \) is concentration, \( D \) is the diffusivity, and \( x \) is the distance from the interface.
Boundary conditions: At \( t = 0 \), the concentration in the fluid is uniform; at \( x = 0 \), the interface concentration is \( c_s \); at \( x \to \infty \), the concentration is the bulk value \( c_\infty \). The solution to this diffusion problem gives the concentration profile, and the mass flux at the interface (\( x = 0 \)) is: \[ N = -D \left( \frac{\partial c}{\partial x} \right)_{x=0}. \] The average mass flux over the contact time \( t \) is proportional to: \[ N \propto \sqrt{\frac{D}{\pi t}} (c_s - c_\infty). \] The mass transfer coefficient \( k \) is defined as: \[ N = k (c_s - c_\infty), \] so: \[ k \propto \sqrt{\frac{D}{\pi t}}. \] Since \( t \) is a constant for a given system (related to the exposure time of the fluid element), the mass transfer coefficient scales as: \[ k \propto \sqrt{D}. \] Step 3: Evaluate the options.
(1) \( k \propto D \): Incorrect, as penetration theory predicts a square root dependence, not linear. Incorrect.
(2) \( k \propto \sqrt{D} \): Correct, as derived from the penetration theory. Correct.
(3) \( k \propto D^{1.5} \): Incorrect, as the exponent is 0.5, not 1.5. Incorrect.
(4) \( k \propto D^2 \): Incorrect, as the exponent is 0.5, not 2. Incorrect. Step 4: Select the correct answer.
The penetration theory relates the average mass transfer coefficient \( k \) with diffusivity \( D \) as \( k \propto \sqrt{D} \), matching option (2).