Compressible cakes: how does the specific cake resistance of a compressible sludge change as the pressure drop across the cake increases?

Introduction / Context:
Filtration performance depends strongly on the structure of the cake. Many sludges are compressible: their pore structure compacts under higher pressure, changing permeability and thus the specific cake resistance.

Given Data / Assumptions:

• Compressible cake (e.g., biological sludges, fine clays).
• Pressure drop across the cake is increased during operation or by control.
• Darcy-type laminar flow through porous media applies.

Concept / Approach:
For compressible cakes, the specific cake resistance α typically follows α = α0 * (ΔP)^s with a positive compressibility index s (often 0.1–1). As ΔP rises, particles pack more tightly; porosity falls; permeability decreases; α increases.

Step-by-Step Solution:
Recognize compressibility: porosity is pressure dependent.Higher ΔP → greater compaction → lower permeability.Therefore, specific cake resistance increases with ΔP.

Verification / Alternative check:
Filtration curves (time vs. filtrate volume) steepen under higher ΔP for compressible cakes, indicating rising resistance per unit mass captured.

Why Other Options Are Wrong:
Decreases/constant: valid only for rigid (incompressible) cakes.Increases linearly: the relationship is commonly non-linear and captured via an exponent s.

Common Pitfalls:
Blindly increasing pressure to speed filtration without considering cake compressibility can slow the process and worsen washing efficiency.

Final Answer:
increases