Identifying Incorrect Statement on Dimensionless Numbers

This question requires evaluating the typical applications of various dimensionless numbers in transport phenomena (momentum, heat, and mass transfer).

• Option 1: Reynolds number ($Re$) is relevant in forced convection and Grashof number ($Gr$) is relevant in natural convection.

  Correct. The Reynolds number ($Re = \frac{\rho v L}{\mu}$) characterizes the ratio of inertial forces to viscous forces, crucial for determining flow regimes (laminar vs. turbulent) in forced flows. The Grashof number ($Gr = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$) represents the ratio of buoyancy forces to viscous forces, dominating fluid motion in natural convection.

• Option 2: Prandtl number ($Pr$) is relevant in heat transfer and Schmidt number ($Sc$) is relevant in mass transfer.

  Correct. The Prandtl number ($Pr = \frac{\nu}{\alpha}$) relates momentum diffusivity to thermal diffusivity, linking fluid flow dynamics to heat transfer characteristics. The Schmidt number ($Sc = \frac{\nu}{D_{AB}}$) relates momentum diffusivity to mass diffusivity, important for mass transfer analysis.

• Option 3: Biot number ($Bi$) is relevant in heat transfer and Froude number ($Fr$) is relevant in mass transfer.

  Incorrect. The Biot number ($Bi = \frac{h_c L_c}{k_s}$) is indeed relevant in transient heat transfer, comparing internal conduction resistance to external convection resistance. However, the Froude number ($Fr = \frac{v}{\sqrt{gL}}$) relates inertial forces to gravitational forces and is typically used in fluid dynamics involving open channel flow, wave propagation, or situations where gravity significantly influences the flow, not primarily in mass transfer analysis.

• Option 4: Nusselt number ($Nu$) is relevant in heat transfer and Sherwood number ($Sh$) is relevant in mass transfer.

  Correct. The Nusselt number ($Nu = \frac{h L}{k}$) represents the ratio of convective heat transfer to conductive heat transfer across a boundary. The Sherwood number ($Sh = \frac{k_c L}{D_{AB}}$) is the mass transfer analogue, representing the ratio of convective mass transfer to diffusive mass transfer.

Conclusion: Identifying the Incorrect Statement

Option 3 incorrectly associates the Froude number with mass transfer processes. While the Biot number is correctly linked to heat transfer, the Froude number's primary relevance lies in gravity-driven flows.