Specific cake resistance for incompressible sludges:\nHow does the specific cake resistance of an incompressible sludge depend on pressure drop ΔP across the cake?

Introduction / Context:
In filtration theory, the specific cake resistance alpha characterises how difficult it is for fluid to pass through a unit mass (or thickness) of cake. Whether alpha depends on pressure distinguishes incompressible from compressible cakes and greatly affects design equations and scale-up predictions.

Given Data / Assumptions:

• Sludge/cake is incompressible over the operating pressure range.
• Fluid properties (viscosity) are constant.

Concept / Approach:
An incompressible cake maintains its structure under increased pressure; pore geometry does not collapse, so alpha remains constant with respect to ΔP. Hence, the pressure drop affects the volumetric rate linearly through Darcy’s law, but alpha itself does not change. In contrast, compressible cakes densify under pressure, making alpha an increasing function of ΔP.

Step-by-Step Solution:

Verification / Alternative check:
Plotting α versus ΔP for rigid granular cakes yields a flat trend; for compressible sludges, α rises with ΔP.

Why Other Options Are Wrong:

• Proportionalities (a–c,e) describe compressible or nonphysical behaviors for incompressible cakes.

Common Pitfalls:
Applying compressible-cake correlations to rigid cakes, leading to overdesign.

Final Answer:
Independent of ΔP