Packed beds under laminar conditions: for laminar liquid flow through a packed bed, the pressure drop varies proportionally to which combination (V_s = superficial velocity, D_p = particle diameter)?

Introduction:
Estimating pressure drop in packed columns is vital for absorber/stripper design. In the laminar regime, viscous forces dominate and the dependence on particle size and superficial velocity follows a specific scaling captured by the Ergun (or Kozeny–Carman) relations.

Given Data / Assumptions:

• Single-phase, incompressible liquid.
• Laminar flow through a uniformly packed bed.
• Constant porosity and sphericity for particles.

Concept / Approach:
The Ergun equation splits viscous and inertial contributions. In laminar flow the viscous term dominates: Δp/L ∝ (μ V_s)/(D_p^2) * function of porosity. Thus, pressure drop is directly proportional to superficial velocity and inversely proportional to the square of particle diameter.

Step-by-Step Solution:
Start with Ergun viscous term: Δp/L = 150 (1 − ε)^2 μ V_s / (ε^3 D_p^2).For fixed packing (ε constant), Δp ∝ μ V_s / D_p^2.Therefore, the proportionality is V_s / D_p^2.

Verification / Alternative check:
Kozeny–Carman form also yields the same D_p^−2 scaling in creeping flow through porous media.

Why Other Options Are Wrong:

• Terms with V_s^2 imply inertial effects (turbulent/pre-loading), not laminar regime.
• D_p^−3 scaling does not arise from the laminar viscous term.
• D_p / V_s^2 is dimensionally inconsistent with Δp scaling.

Common Pitfalls:
Using the full Ergun equation without checking Reynolds number; overestimating Δp by applying the inertial term in the laminar range.

Final Answer:
V_s / D_p^2