End-of-cycle washing in constant-pressure filtration: when washing begins immediately after filtration at the same pressure, how does the initial washing rate compare to the filtration rate?

Introduction / Context:
In constant-pressure filtration, cake forms and thickens with time, raising flow resistance. At the end of filtration, a washing step may begin at the same applied pressure to displace mother liquor from pores. Understanding the relation between the washing rate and the preceding filtrate rate helps estimate washing time and solvent usage.

Given Data / Assumptions:

• Cake is already formed; filtration was at constant pressure.
• Washing starts immediately at the same pressure differential.
• Fluid properties during wash are similar, unless otherwise noted.

Concept / Approach:
At the instant washing begins, the hydraulic resistance of the cake and medium has not changed from the end of filtration. Therefore, with the same pressure drop, the volumetric flow rate initially equals the final filtration rate. As washing proceeds, composition and viscosity may change and concentration gradients disappear, often causing a gradual rate decrease as flow paths equilibrate.

Step-by-Step Solution:
Recall Darcy's law: Q = ΔP / (μ * R_total) * A.At switch-over, ΔP, μ (approximately), and R_total are initially the same as at end of filtration.Therefore, initial wash rate equals the rate at end of filtration; then it decreases as conditions evolve (e.g., cake consolidation, viscosity differences).

Verification / Alternative check:
Batch filtration wash curves commonly start at the end-of-filtration rate and then decline, matching industrial experience and standard design heuristics.

Why Other Options Are Wrong:
“At time zero” without context is ambiguous; the key is equality at the transition point.“At the end of filtration” alone omits the subsequent decrease.“Half the filtrate” has no basis in Darcy's law for this comparison.

Common Pitfalls:
Ignoring cake compressibility and fluid property changes during washing; both can accentuate the decline in rate compared with the end-of-filtration value.

Final Answer:
at the end of filtration, but decreases with time subsequently.