Question

Prandtl number in heat transfer is analogous to the following dimensionless number in mass transfer:

• A. Stanton number
• B. Schmidt number
• C. Sherwood number
• D. Peclet number

Hint:

Memorize the direct analogies between Heat and Mass Transfer dimensionless numbers: - Prandtl Number (\( \text{Pr} = \frac{\nu}{\alpha} \)) \( \Longleftrightarrow \) Schmidt Number (\( \text{Sc} = \frac{\nu}{D_{AB}} \)) - Nusselt Number (\( \text{Nu} = \frac{h L}{k} \)) \( \Longleftrightarrow \) Sherwood Number (\( \text{Sh} = \frac{k_c L}{D_{AB}} \)) - Biot Number (Bi) \( \Longleftrightarrow \) Mass Transfer Biot Number

Answer 1

The Correct Option is B

Concept: Transport phenomena establish clear mathematical analogies across momentum, heat, and mass transport. Boundary layer behaviors depend directly on specific dimensionless ratios that compare molecular diffusion properties.

• The Prandtl number (Pr) relates momentum transport to heat transport.

• The Schmidt number (Sc) relates momentum transport to mass transport.

Step 1: Analyzing the definition of the Prandtl Number.
The Prandtl number is a dimensionless metric that quantifies the relative ratio of molecular momentum diffusivity to molecular thermal diffusivity within a fluid medium: \[ \text{Pr} = \frac{\text{Kinematic Viscosity}}{\text{Thermal Diffusivity}} = \frac{\nu}{\alpha} = \frac{\mu / \rho}{k / (\rho \cdot C_p)} = \frac{\mu \cdot C_p}{k} \] This parameter dictates the relative thickness profiles of the velocity boundary layer compared to the thermal boundary layer.

Step 2: Finding the mass transfer analogue.
To translate this concept to mass transfer, we replace the thermal diffusivity term (\( \alpha \)) in the denominator with the mass diffusion coefficient (or molecular diffusivity, denoted as \( D_{AB} \)). The resulting dimensionless group that compares molecular momentum diffusivity directly to molecular mass diffusivity is known as the Schmidt number (Sc): \[ \text{Sc} = \frac{\text{Kinematic Viscosity}}{\text{Mass Diffusivity}} = \frac{\nu}{D_{AB}} = \frac{\mu}{\rho \cdot D_{AB}} \] The Schmidt number dictates the relative thickness profiles of the velocity boundary layer compared to the concentration (mass transfer) boundary layer.

Step 3: Comparing with other listed choices.
Let us review the remaining options to confirm our choice:

• Stanton number (St): Represents a ratio comparing heat transferred into a fluid to the thermal capacity of that fluid. Its mass transfer counterpart is the mass transfer Stanton number.

• Sherwood number (Sh): Represents the ratio of convective mass transport to molecular mass diffusion. It is directly analogous to the Nusselt number (Nu) from heat transfer.

• Peclet number (Pe): Represents the ratio of advective transport rates to diffusive transport rates.
Thus, the Prandtl number in heat transfer is directly analogous to the Schmidt number in mass transfer.