Filtration modeling for cake flow:\nThe flow of filtrate through the cake in a plate-and-frame filter press is best described by which equation?

Introduction / Context:
Designing filtration cycles requires a model for flow through porous cakes. While Darcy’s law provides the macroscopic relation, permeability correlations such as Kozeny–Carman link permeability to particle size and porosity, enabling engineering estimates of pressure drop/flow relationships in cake filtration, especially in plate-and-frame presses.

Given Data / Assumptions:

• Laminar flow through a homogeneous porous cake.
• Filter cake properties characterised by porosity and specific surface.

Concept / Approach:
The Kozeny–Carman equation refines Darcy’s law by expressing permeability k in terms of particle size and porosity (k ∝ ε^3 / [S_v^2 * (1 − ε)^2]). This is appropriate for packed beds and filter cakes comprising small particles. Hagen–Poiseuille describes laminar flow in a single capillary tube, not a tortuous porous medium; Fanning and Bernoulli relate to pipe flows; Kremser is for gas absorption design, unrelated to filtration.

Step-by-Step Solution:

Verification / Alternative check:
Filtration textbooks derive cycle-time equations using Darcy’s law plus Kozeny–Carman to relate cake resistance to particle size and porosity.

Why Other Options Are Wrong:

• Hagen–Poiseuille/Fanning/Bernoulli: apply to pipe/capillary flows, not porous cakes.
• Kremser: absorption tower stage analysis, not filtration.

Common Pitfalls:
Treating a porous cake as a single large capillary; real cakes exhibit tortuosity that Kozeny–Carman accounts for empirically.

Final Answer:
Kozeny–Carman equation