Particle sizing in process engineering: The “equivalent diameter” of an irregular particle is defined as the diameter of a perfect sphere that has the same physical measure as the particle. Which measure is being equated in the usual volume-equivalent definition?

Introduction / Context:
In particle technology, engineers compare irregular particles to “equivalent” spheres to simplify calculations in settling, filtration, and heat/mass transfer. Several equivalent diameters exist (volume-equivalent, surface-equivalent, sieve-equivalent). Knowing which physical measure is matched in each definition avoids confusion in design correlations.

Given Data / Assumptions:

• We are discussing the most common “equivalent diameter” used in many correlations for sedimentation and fluidization.
• The alternative measures include surface area and various ratios; however, the standard volume-equivalent diameter equates particle volume to that of a sphere.
• Particle shape is irregular but nonporous for the purpose of definition.

Concept / Approach:
The volume-equivalent diameter d_v is defined by equating the volume of the irregular particle to that of a sphere: (pi/6) * d_v^3 = V_particle. This makes d_v useful in mass-based and buoyancy-driven relations, because both mass and buoyancy depend directly on volume.

Step-by-Step Solution:
Let V_p be the actual particle volume.Define d_v such that sphere volume equals V_p.Hence (pi/6) * d_v^3 = V_p ⇒ d_v = (6 * V_p / pi)^(1/3).Therefore, the correct “measure” being equated is volume, not a ratio like surface/volume.

Verification / Alternative check:
Compare with sphericity-based definitions: sphericity uses surface-area equivalence for the comparison but still relies on an independent diameter definition such as d_v. Handbooks consistently define equivalent diameter (unless otherwise specified) as the sphere of equal volume.

Why Other Options Are Wrong:

• Ratio of surface/volume or volume/surface: these are not the standard equivalence criteria for d_v.
• Surface area equal to that of the particle describes a different measure (surface-equivalent diameter), not the usual equivalent diameter asked here.
• “None of these” is incorrect because the correct choice is explicitly listed.

Common Pitfalls:
Confusing volume-equivalent diameter with sieve diameter (from screen analysis) or surface-equivalent diameter (used with sphericity). Always check the intended definition in a given correlation.

Final Answer:
Volume equal to that of the particle