Gravity discharge of granular solids: the mass flow rate M through a circular orifice of diameter D follows the Beverloo-type dependence. Which proportionality best represents this relation?

Difficulty: Medium

Correct Answer: M ∝ D^2.5

Explanation:


Introduction / Context:
Gravity discharge of granular materials through hoppers and orifices is a classic problem in bulk solids handling. Empirical correlations (e.g., Beverloo equation) capture how the flow rate scales with opening size, particle size, and bulk properties. Understanding the exponent on the orifice diameter is crucial for reliable scale-up.


Given Data / Assumptions:

  • Free discharge through a circular orifice.
  • Granular, non-cohesive bulk solid (no arching or rat-holing).
  • Orifice diameter D is much larger than particle diameter but not infinite.


Concept / Approach:
The Beverloo law gives M = C * ρ_b * √g * (D − k·d_p)^(5/2). For D ≫ d_p, this reduces to M ∝ D^(5/2). The 2.5 exponent arises from momentum and discharge geometry considerations and is validated across many materials, provided flow is mass flow and not hindered by cohesion.


Step-by-Step Solution:
Recall Beverloo relation: M ∝ (D − k·d_p)^(5/2).For D ≫ d_p, simplify to M ∝ D^(5/2).Select the closest proportionality: M ∝ D^2.5.


Verification / Alternative check:
Hopper discharge experiments consistently fit the 2.5 exponent better than 2 or 3 across a broad range of materials and orifice sizes, barring cohesive flow.


Why Other Options Are Wrong:
Linear D or quadratic D^2: underpredict flow increase with diameter.Cubic D^3: overpredicts and is inconsistent with observed data.


Common Pitfalls:
Ignoring particle-size correction (D − k·d_p); for small orifices relative to particle size, this correction is essential.


Final Answer:
M ∝ D^2.5

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