For constant-pressure filtration of a slurry forming a cake, the qualitative plot of filtrate volume V versus time t most closely resembles which curve?

Introduction / Context:
Filtration under constant pressure features increasing cake thickness over time, which raises resistance and slows the filtrate rate. Recognising the V–t relationship aids in data reduction and scale-up.

Given Data / Assumptions:

• Constant transmembrane pressure.
• Cake-forming slurry with negligible medium compressibility.
• Standard cake filtration model applies.

Concept / Approach:
The filtration equation for constant pressure can be expressed as: t = a * V + b * V^2, where the linear term represents medium resistance and the quadratic term represents cake resistance. Rearranged, V vs t shows a concave-down increase with diminishing slope (rate dV/dt decreases). While mathematically V ≈ k * sqrt(t) at long times (cake-dominated), in qualitative MCQs this is often described as a “parabolic” trend in the V–t plane (since t scales with V^2).

Step-by-Step Solution:
Write t = k1 * V + k2 * V^2.As t grows, the V^2 term dominates → t ∝ V^2.Hence V grows sublinearly with t (concave-down), commonly labelled parabolic.

Verification / Alternative check:
Plotting typical filtration test data yields a straight line for t/V vs V and a curved V vs t with diminishing slope.

Why Other Options Are Wrong:
Straight line: corresponds to constant-rate filtration, not constant pressure.Hyperbola/Exponential: do not reflect the t ∝ V^2 dependence.

Common Pitfalls:
Confusing the diagnostic plot t/V vs V (linear) with the raw V vs t curve (curved).

Final Answer:
Parabola (V increases with t but with diminishing slope)