Filtration modes: in constant-rate filtration of a slurry, how does the inlet (driving) pressure typically vary as cake builds?

Introduction / Context:
Industrial pressure and vacuum filters can be operated in different modes. In constant-rate filtration, the objective is to maintain a fixed volumetric flow rate despite the growing cake resistance. Understanding the pressure behavior is essential for pump and vessel sizing.

Given Data / Assumptions:

• Darcy’s law applies: ΔP = μ * (Rm + α * w/A) * (V̇/A).
• V̇/A is held constant (constant-rate mode).
• Cake resistance increases with cake mass per area w.

Concept / Approach:
As filtration proceeds, solids accumulate, increasing the cake resistance term α * w/A. With V̇/A fixed, the only way to satisfy Darcy’s law is for ΔP to rise to overcome the higher resistance, assuming viscosity and medium resistance remain roughly constant during the run.

Step-by-Step Solution:
Start: thin cake, low resistance → low ΔP.Progress: w increases → α·w/A increases.To keep V̇/A constant, ΔP must increase over time.

Verification / Alternative check:
Constant-pressure filtration shows the opposite behavior: with ΔP fixed, V̇ decreases as cake thickens. This inverse pair confirms the constant-rate trend.

Why Other Options Are Wrong:
Decreasing or constant ΔP contradicts Darcy’s law under rising resistance.“None of these” is unnecessary because (a) is correct.

Common Pitfalls:
Confusing operation modes; always distinguish constant-rate vs constant-pressure filtration.

Final Answer:
Increases continuously