In filtration engineering, the specific cake resistance for compressible sludges varies primarily as a function of the pressure drop across which part of the filter assembly?

Introduction / Context:
Specific cake resistance is a key parameter in solid–liquid separation. It quantifies how strongly a formed cake resists flow under a given pressure drop and is central for sizing filters, estimating cycle times, and predicting throughput. For compressible sludges (e.g., biological sludges, finely divided precipitates), the cake structure collapses as pressure increases, changing permeability significantly.

Given Data / Assumptions:

• The sludge is compressible, so cake voidage and permeability vary with pressure.
• Pressure drop exists across the cake and the filter medium.
• We are asked where the pressure dependence that controls specific cake resistance is concentrated.

Concept / Approach:
For compressible cakes, the specific cake resistance alpha is a function of effective stress within the cake skeleton. As deltaP increases, pores collapse, reducing permeability and raising alpha. The filter medium resistance is comparatively fixed and largely independent of the applied pressure (within normal operating ranges). Hence, the pressure acting across the cake controls compressibility effects and thus the pressure dependence of alpha.

Step-by-Step Solution:
Identify components: cake and medium; both share total deltaP.Recognise compressibility: cake structure changes with deltaP.Conclude that alpha(deltaP) is governed by the deltaP across the cake, not the medium.

Verification / Alternative check:
Design correlations often express alpha = alpha0 * (deltaP)^s for compressible cakes, where s > 0. This dependence arises from compaction of the cake, not the medium.

Why Other Options Are Wrong:
Across the filter medium only: medium resistance is mostly constant.Across the entire unit equally: the pressure dependence of alpha originates in the cake segment.Independent of pressure location: contradicts cake compressibility theory.

Common Pitfalls:
Assuming the medium contributes significant pressure-dependent resistance; in reality, cake compaction dominates.

Final Answer:
Across the filter cake only