In mechanical classification, particles are called equal-settling when their terminal settling velocities are identical. Under what conditions is this definition applied?

Introduction / Context:
Equal-settling concepts are central to the design and operation of classifiers (hydrocyclones, hydraulic classifiers, and settling tanks). The idea is used to predict how particles of different sizes and densities behave in a given fluid under a specified driving force, so that cut sizes and sharpness of separation can be estimated.

Given Data / Assumptions:

• The term “terminal settling velocity” means the steady velocity reached by a particle when gravitational (or centrifugal) forces balance drag.
• We are comparing particles in a specific, unchanged environment.
• “Field of force” refers to the driving force regime (usually gravitational settling; sometimes centrifugal fields).

Concept / Approach:
Particles are said to be equal-settling when they have the same terminal velocity in the same fluid and under the same field of force. This ensures that differences in viscosity, density of the fluid, or magnitude of the driving acceleration are not confusing the comparison. Under this definition, particles of different size–density combinations may follow the same trajectory and report to the same product stream in a classifier.

Step-by-Step Solution:
Fix the fluid: choose one fluid (e.g., water at given temperature) so its density and viscosity are defined.Fix the field: define gravity settling (g) or a specific centrifugal acceleration.Compare particles: identify pairs that have the same terminal velocity under those fixed conditions.Conclusion: equal-settling requires both same fluid and same field of force.

Verification / Alternative check:
Settling velocity charts (log–log plots) or Stokes/Allen–Oseen relations show iso-velocity lines. Particles lying on one iso-velocity line in a given fluid and field are equal-settling and will separate similarly in hydraulic classifiers.

Why Other Options Are Wrong:
Different fluids: viscosity and density change, so equal velocity in different fluids does not define equal-settling for classifier design.Same fluid only: incomplete; the driving acceleration must also be the same.Same field only: incomplete; the fluid properties must also be the same.

Common Pitfalls:
Confusing same-size particles with equal-settling particles; density differences can compensate for size differences. Also, forgetting temperature effects on viscosity leads to wrong equal-settling assumptions.

Final Answer:
both (b) and (c)