Flow through filter cakes: for laminar filtrate flow through the deposited cake on a septum, which relation is applicable?

Introduction / Context:
Modeling flow through porous media such as filter cakes requires empirical–theoretical correlations that link pressure drop, viscosity, velocity, and structural parameters (porosity, specific surface). Identifying the appropriate equation is foundational in filtration calculations.

Given Data / Assumptions:

• Laminar flow regime within the cake pores.
• Porous cake characterized by porosity and specific surface.
• Incompressible or slowly compressible cake so that parameters are meaningful over small ranges.

Concept / Approach:
The Kozeny–Carman equation, derived from Darcy’s law with a capillary bundle model, relates pressure drop to superficial velocity and structural factors: ΔP/L = (180 * μ * (1 − ε)^2 / (ε^3 * d_p^2)) * v for beds of approximately spherical particles. The same form underpins the specific cake resistance used in filtration analysis.

Step-by-Step Solution:
Assume laminar flow through tortuous pores → Darcy-type behavior applies.Use Kozeny–Carman to connect ΔP, v, μ, ε, and particle size/specific surface.Conclude that Kozeny–Carman is the valid relation among the options.

Verification / Alternative check:
Other named equations are less standard for cake-flow laminar modeling; practical filtration textbooks reduce cake resistance to a Kozeny–Carman-based specific resistance α.

Why Other Options Are Wrong:
Leva’s and Blake–Plummer forms are not the canonical choice for laminar cake flow and are less widely used in filtration design.

Common Pitfalls:
Confusing bulk packed-bed pressure drop correlations with turbulent or transition-regime expressions; for cakes and low Reynolds numbers, the Kozeny–Carman framework is appropriate.

Final Answer:
Kozeny–Carman equation