Filtration theory units: in Darcy’s law modeling, the clean filter medium resistance Rm carries what unit in cgs terms?

Introduction / Context:
Filter design and cycle-time prediction use Darcy’s law, which relates pressure drop to flow rate through resistances contributed by the clean medium and by the cake. Getting units right ensures correct scale-up and parameter estimation from test data.

Given Data / Assumptions:

• Darcy’s law form: ΔP = μ * (Rm + α * (w/A)) * (V̇/A).
• μ is viscosity, V̇ is volumetric flow, A is area, Rm is medium resistance.
• Using cgs units for clarity (cm, g, s).

Concept / Approach:
Rearranging dimensions shows Rm must have units of inverse length to keep ΔP dimensions consistent. Specifically, μ has units (g/(cm·s)), V̇/A has (cm/s), and ΔP has (g/(cm·s^2)). Thus Rm must be cm^-1 so that μ * Rm * (V̇/A) yields pressure units.

Step-by-Step Solution:
Write ΔP term from medium: ΔP_m = μ * Rm * (V̇/A).Substitute units: μ = g/(cm·s), V̇/A = cm/s.Therefore μ * Rm * (cm/s) must equal g/(cm·s^2) → Rm = 1/cm.

Verification / Alternative check:
Dimensional analysis of the cake term α*(w/A) similarly ensures α has units of cm/kg (SI: m/kg), consistent with common references.

Why Other Options Are Wrong:
gm/cm-1, cm/gm-1, gm-1: these do not reconcile units in Darcy’s equation for ΔP.

Common Pitfalls:
Mixing SI and cgs during calculations; always convert consistently to avoid magnitude errors.

Final Answer:
cm-1