Filtration plotting: a straight line is obtained by plotting reciprocal filtration rate (dV/dt)^{-1} versus filtrate volume V for which combination of cake compressibility and flow regime?

Introduction / Context:
Filtration analysis often linearizes the governing equation to estimate specific cake resistance and medium resistance. One common linear plot uses the reciprocal filtration rate as a function of filtrate volume.

Given Data / Assumptions:

• Constant-pressure filtration.
• Darcy’s law applies (laminar flow through porous media).
• Cake is incompressible (resistance proportional to thickness).

Concept / Approach:
For incompressible cakes under laminar flow, the Ruth equation gives t = (μ * α * C / (2 * A^2 * ΔP)) * V^2 + (μ * Rm / (A * ΔP)) * V, where symbols have their usual meanings. Differentiating, the reciprocal filtration rate (dV/dt)^{-1} varies linearly with V with slope proportional to μ * α * C / (A^2 * ΔP) and intercept proportional to μ * Rm / (A * ΔP).

Step-by-Step Solution:
Start from constant-ΔP filtration with incompressible cake: t = aV^2 + bV.Differentiate: dt/dV = 2aV + b → (dV/dt)^{-1} = 2aV + b, which is linear in V.Therefore, a straight line in the indicated plot implies laminar flow with an incompressible cake.

Verification / Alternative check:
With compressible cakes, α depends on pressure and cake thickness, introducing nonlinearity; turbulent flow also violates Darcy’s law linearity.

Why Other Options Are Wrong:
Compressible cake: slope/intercept vary with V and ΔP → nonlinearity.Turbulent conditions: Darcy’s law no longer holds; empirical exponents distort linear plots.

Common Pitfalls:
Plotting t/V vs V and (dV/dt)^{-1} vs V interchangeably; both give linear forms for incompressible cakes but with different coefficients—ensure the correct plot is used.

Final Answer:
incompressible cake and laminar flow