In liquid–solid filtration, which factors affect the filtration rate under constant-pressure operation?

Introduction / Context:
Filtration rate quantifies how quickly liquid passes through a filter medium while solids are retained as a cake. In chemical and environmental engineering, understanding rate control helps select equipment size, cycle times, and operating pressures.

Given Data / Assumptions:

• Constant-pressure filtration of a compressible or incompressible cake.
• Newtonian filtrate; steady operating conditions during a given cycle segment.
• Uniform medium with known clean-medium resistance.

Concept / Approach:
The classical filtration equation relates filtrate volume to time considering pressure drop, viscosity, cake resistance (which grows with cake thickness), and medium resistance. Larger area reduces superficial velocity for a given flow, improving throughput or lowering pressure drop.

Step-by-Step Solution:
Rate ∝ ΔP / (μ * (R_cake + R_medium)) for a given area.Increasing ΔP increases rate, within mechanical limits.Higher viscosity μ reduces rate.Greater area increases capacity at fixed flux.Cake resistance grows with time as solids accumulate.

Verification / Alternative check:
Plot filtrate volume vs time; the slope decreases as cake builds, demonstrating dependence on R_cake and μ at fixed ΔP and area.

Why Other Options Are Wrong:

• Only pressure drop: ignores viscosity and resistances.
• Only cake or only medium resistance: incomplete.
• Only area/viscosity: incomplete without ΔP and resistances.
• Only cake at early times: early-time dynamics still include medium resistance and viscosity.

Common Pitfalls:
Neglecting medium resistance at the start; assuming unlimited benefit from increasing pressure without considering cake compressibility.

Final Answer:
All of the above (a, b, and c)