Axial dispersion in turbulent flow: the effective longitudinal dispersion coefficient (mixing in the flow direction) in turbulent pipe/bed flows is commonly correlated primarily as a function of which dimensionless group?

Introduction:
Axial (longitudinal) dispersion accounts for spreading of solute along the direction of bulk flow due to the combined action of velocity profile nonuniformity and turbulent eddy mixing. In turbulent systems, dispersion strongly depends on flow intensity, which is typically characterized by the Reynolds number.

Given Data / Assumptions:

• Turbulent regime where eddy diffusivity dominates molecular diffusion.
• Geometry such as pipes or packed/structured beds under high Re.
• Empirical correlations relate dispersion coefficient to operating and geometric variables.

Concept / Approach:
Reynolds number Re encapsulates the ratio of inertial to viscous forces and governs the strength and scale of turbulence. In turbulent flow, axial dispersion coefficients increase with Re due to enhanced mixing by eddies. While Schmidt number (ratio of momentum to mass diffusivity) can appear in detailed models, the dominant scaling for turbulent dispersion is with Re, and Sherwood number pertains to mass transfer at interfaces, not axial mixing. Grashof relates to natural convection, not forced turbulent dispersion.

Step-by-Step Solution:
Identify the phenomenon: axial dispersion in forced turbulent flow.Recognize that turbulence intensity scales with Re.Select Reynolds number as the primary correlating dimensionless group.

Verification / Alternative check:
Classic correlations for pipe dispersion (e.g., Taylor–Aris extended to turbulence) and packed bed models show dispersion increasing with Re, confirming the emphasis on Reynolds number.

Why Other Options Are Wrong:

• Sherwood number: Interfacial mass transfer, not axial dispersion.
• Schmidt number: Secondary parameter; important but not the primary flow-intensity descriptor.
• Grashof number: Natural convection, not applicable here.
• Fourier number: Transient diffusion time scaling, not a flow-intensity parameter.

Common Pitfalls:
Using laminar dispersion theory in turbulent regimes; ensure the correct regime and scaling are applied.

Final Answer:
Reynolds number