Rotary drum vacuum filter (Rm ≪ Rc): the instantaneous filtrate flow rate is proportional to which viscosity term and to the available filtration segment of cycle time?

Introduction / Context:
Rotary drum vacuum filters (RDVFs) operate with a rotating sectoral cycle: submergence (cake formation), drainage, drying, and discharge. When the filter medium resistance Rm is negligible compared to cake resistance Rc, the classical cake filtration model governs the flow.

Given Data / Assumptions:

• Rm ≪ Rc, incompressible cake approximation during a short instant.
• Constant vacuum (approximately constant pressure drop during a sector).
• Filtrate viscosity μ is the dominant fluid property.

Concept / Approach:
Darcy’s law for cake filtration gives flux J ∝ ΔP / (μ Rc). Holding ΔP and Rc fixed in the short term, J ∝ 1/μ. Over a cycle, the total filtrate volume per revolution scales with the fraction of time the sector spends under vacuum; hence it is also proportional to the effective filtration segment of the cycle time.

Step-by-Step Solution:
Write J = ΔP / (μ (Rm + Rc)).With Rm ≪ Rc, J ≈ ΔP / (μ Rc) → J ∝ 1/μ.Total flow per cycle ∝ J * (time under vacuum) → proportional to the filtration time segment.

Verification / Alternative check:
Vendor performance curves show inverse proportionality to μ and near-linear increase with longer submergence or vacuum exposure.

Why Other Options Are Wrong:
μ or 1/μ^2: incorrect dependence; the law is first power inverse in μ.

Common Pitfalls:
Confusing filtrate volume per cycle (time dependent) with instantaneous flux; both scale with 1/μ but only the former reflects cycle fraction.

Final Answer:
1/μ (and proportional to the filtration time segment)