Cake filtration (modified Darcy): Under laminar conditions, the filtration flux through a growing cake is given by which relation between rate, pressure drop Δp, viscosity μ, cake permeability K, cake thickness L, and area A?

Difficulty: Medium

Correct Answer: (1/A) * dVf/dt = (K/μL) * Δp

Explanation:


Introduction:
Pressure-driven cake filtration is central to harvesting cells and removing fines. The modified Darcy equation relates filtration rate to pressure drop, fluid viscosity, cake permeability, and cake thickness. Correctly identifying the proportionalities is essential for scale-up and cycle-time estimation.


Given Data / Assumptions:

  • Laminar flow through a homogeneous cake on a porous support.
  • Darcy regime with negligible inertial effects.
  • Cake permeability K (m^2), cake thickness L (m), viscosity μ (Pa·s), area A (m^2).


Concept / Approach:
Darcy’s law states superficial velocity v = (K/μ) * (Δp/L). Since flux J = v = (1/A) * dVf/dt, the rate per area is proportional to Δp and K and inversely proportional to μ and L. Rearrangements that multiply L and K in the numerator are dimensionally and physically incorrect.


Step-by-Step Solution:
Start: v = (K/μ) * (Δp/L).Identify J ≡ v = (1/A) * dVf/dt.Substitute to obtain (1/A) * dVf/dt = (K/μL) * Δp.Interpretation: increasing Δp or K increases flux; increasing μ or L decreases flux.


Verification / Alternative check:
Plotting (A/Δp) * dVf/dt versus 1/L should yield a straight line with slope K/μ for constant cake properties, a common diagnostic in filtration tests.


Why Other Options Are Wrong:
(1/A)dVf/dt = (Δp/μ)(L/K): inverts K/L, giving the wrong dependence.

Terms with L*K in the numerator: dimensionally incorrect and contradict Darcy scaling.


Common Pitfalls:

  • Confusing permeability K with specific cake resistance α (where α = 1/K in some texts).
  • Ignoring that L often increases with time, making the filtration rate decline during the run.


Final Answer:
(1/A) * dVf/dt = (K/μL) * Δp

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