Axial mixing analogy: Analogous to molecular diffusion, the x-directional flux Jn of suspended microorganisms due to axial mixing can be written as (cn is the concentration of species n):

Introduction / Context:
When modeling non-ideal flow in bioreactors and chromatographic columns, axial mixing is often represented with a Fickian flux term. This treats hydrodynamic dispersion similarly to molecular diffusion, using an effective axial dispersion coefficient D (or D_ax). Correctly writing the constitutive relation for flux is essential for mass balance closure.

Given Data / Assumptions:

• Continuum representation of suspended cells or tracer species.
• One-dimensional axial coordinate x.
• Effective dispersion coefficient D captures mixing intensity.

Concept / Approach:
The constitutive law mirrors Fick’s first law: flux is proportional to the negative gradient of concentration. The minus sign indicates transport down-gradient, from high concentration to low concentration. Replacing molecular diffusivity with effective dispersion yields the axial-mixing form.

Step-by-Step Solution:
Write Fick-like relation: Jn = − D * dcn/dx.Note the sign convention: negative because flux is down the concentration gradient.Recognize D as an effective parameter that lumps hydrodynamic eddies and velocity variations.Use this in the axial species balance along with convection and reaction terms.

Verification / Alternative check:
Tracer RTD fitting yields D (or Péclet number Pe = uL/D). Consistency across flow rates supports the Fickian dispersion analogy in many systems.

Why Other Options Are Wrong:
Expressions without D omit the proportionality constant and are dimensionally incorrect.

Positive sign would imply flux up-gradient, contrary to physical diffusion.

Common Pitfalls:

• Confusing D with molecular diffusivity; D can be orders of magnitude larger due to hydrodynamics.
• Applying 1D dispersion where strong radial gradients are present without justification.

Final Answer:
Jn = − D * dcn/dx