Transport analogies — For which case are mass diffusivity, thermal diffusivity, and eddy momentum diffusivity effectively equal (i.e., NPr = NSc = ______ )?

Introduction:Transport property analogies (Reynolds, Chilton–Colburn) connect momentum, heat, and mass transfer. When dimensionless groups align, similar boundary-layer behavior allows property analogies to be applied cleanly. This question asks the special condition under which momentum, heat, and mass diffusivities are equal in a normalized sense.

Given Data / Assumptions:

• Prandtl number NPr = ν/α (momentum diffusivity/thermal diffusivity).
• Schmidt number NSc = ν/D (momentum diffusivity/mass diffusivity).
• Fully turbulent regime with similar eddy diffusivities for momentum, heat, and mass.

Concept / Approach:

When NPr = 1 and NSc = 1, momentum, heat, and mass boundary layers have similar thicknesses because ν ≈ α ≈ D (or, in turbulent flow, their eddy counterparts are comparable). Under this condition, the Chilton–Colburn j-factor analogy reduces to direct equivalence, simplifying transfer coefficient estimation and permitting the use of a single Stanton number relation for all three modes.

Step-by-Step Solution:

Verification / Alternative check:

Textbook analogies yield j_H ≈ j_D when Pr = Sc, with best agreement near unity; many gases at moderate temperatures have Pr of order 1, supporting the approximation.

Why Other Options Are Wrong:

B, C, E: Values other than 1 imply unequal diffusivities and dissimilar boundary layers. D: NPr or NSc = 0 is non-physical for real fluids.

Common Pitfalls:

Assuming perfect equality is required; in practice, analogies work reasonably well for Pr and Sc not too far from unity.

Final Answer:

1