The mass transfer co-efficient for a solid sphere of radius 'a' dissolving in a large volume of quiescent liquid, in which D is the diffusivity of solute, is
A. D/a
B. D/2a
C. Proportional to $$\sqrt {\text{D}} $$
D. Dependent on the Reynolds number
Answer: Option A
Solution (By Examveda Team)
For a solid sphere dissolving in a large volume of quiescent (stationary) liquid, the system is governed by diffusion alone since there is no convective flow.Under these conditions, the mass transfer process is described by a steady-state diffusion model.
According to the solution of Fick’s second law for a stationary medium around a dissolving sphere, the mass transfer coefficient (kc) is given by:
kc = D/a, where
D = diffusivity of the solute in the liquid
a = radius of the solid sphere
This expression assumes that the diffusion is radial and the concentration at the surface of the sphere remains constant.
Option B (D/2a) is incorrect because it does not match the standard solution for diffusion from a sphere.
Option C (Proportional to √D) applies in some unsteady-state or film-type approximations, not for steady diffusion from a sphere.
Option D (Dependent on Reynolds number) is incorrect here, since the liquid is quiescent and there is no bulk motion, making Reynolds number irrelevant.
Therefore, the correct answer is D/a.
Correct Answer: Option A: D/a
This Question Belongs to Chemical Engineering >> Mass Transfer
The correct answer is:
✅ A. D / a
🔍 Explanation:
For a solid sphere dissolving in a large volume of stagnant (quiescent) liquid, the mass transfer occurs purely by molecular diffusion, since there is no convective flow (i.e., Reynolds number is essentially zero).
Under these conditions, the mass transfer coefficient
𝑘
𝐿
k
L
is given by the diffusion-controlled model:
𝑘
𝐿
=
𝐷
𝑎
k
L
=
a
D
Where:
𝐷
D = diffusivity of the solute
𝑎
a = radius of the sphere
𝑘
𝐿
k
L
= mass transfer coefficient
This is derived from the steady-state diffusion equation in spherical coordinates with appropriate boundary conditions.
❌ Why not the other options?
B. D/2a → Incorrect. This form does not match the standard diffusion solution.
C. Proportional to
𝐷
D
→ Applies in unsteady diffusion or convective mass transfer, not in this case.
D. Dependent on Reynolds number → Not applicable here; fluid is quiescent (Re ≈ 0).
✅ Final Answer:
A. D / a