Superficial vs. interstitial velocity in fibrous beds: if packing density α is defined as the fiber volume per unit filter-bed volume (so porosity ε = 1 − α), what is the relationship between the interstitial velocity V in the void space and the superficial (undisturbed) velocity V0?

Introduction:
In flow through porous or fibrous media, engineers distinguish between superficial velocity (based on total cross-sectional area) and interstitial velocity (actual fluid speed within the voids). The relationship depends on the porosity of the medium and is important for estimating residence time and mass-transfer coefficients.

Given Data / Assumptions:

• Packing density α = volume fraction of fibers; porosity ε = 1 − α.
• V0 is superficial velocity (flow rate divided by total area).
• V is interstitial velocity within the void space.

Concept / Approach:
Because only the void fraction carries flow, continuity implies that the actual fluid speed must be higher in the voids than the superficial average by a factor 1/ε. Therefore, V = V0 / ε = V0 / (1 − α). This correction is essential when predicting contact times, Reynolds number in pores, and collection efficiencies that depend on local velocities.

Step-by-Step Solution:
Define ε = 1 − α.Write conservation of volumetric flow: V0 * A_total = V * (ε * A_total).Solve for V: V = V0 / ε = V0 / (1 − α).Interpretation: as α increases (ε decreases), interstitial velocity rises.

Verification / Alternative check:
Dimensional analysis and simple control-volume arguments give the same relationship; empirical measurements of pressure drop also reflect changes with ε via correlations (e.g., Ergun-like forms).

Why Other Options Are Wrong:

• V = V0(1 − α): This would make velocity decrease with decreasing void space, contradicting continuity.
• Other algebraic forms do not satisfy conservation of volumetric flow through reduced area.

Common Pitfalls:
Confusing superficial and interstitial velocities when computing Reynolds number or residence times; always use the correct definition for the phenomenon being modeled.

Final Answer:
V = V0/(1-α)