For mixing in the laminar regime (very low Reynolds number), how does the power number Po scale with the Reynolds number Re for a stirred tank (Po = P / (ρ N^3 D^5), Re = ρ N D^2 / μ)?
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APo ∝ Re^-1.0
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BPo ∝ Re^0.0
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CPo ∝ Re^0.5
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DPo ∝ Re^1.0
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EPo ∝ Re^-0.5
Answer
Correct Answer: Po ∝ Re^-1.0
Explanation
Introduction / Context:In mixing, dimensional analysis yields power number–Reynolds number correlations. In laminar flow, viscous forces dominate and inertial effects vanish from the power requirement.
Given Data / Assumptions:
- Definitions: Po = P/(ρ N^3 D^5), Re = ρ N D^2/μ.
- Low Re (creeping flow) limit.
Concept / Approach:For laminar mixing, P ∝ μ N^2 D^3, so substituting into Po gives Po ∝ (μ N^2 D^3) / (ρ N^3 D^5) = (μ / ρ N D^2) = 1/Re. Therefore, Po scales inversely with Re in the viscous regime.
Step-by-Step Solution:Use laminar mixing power correlation P ∝ μ N^2 D^3.Form Po and simplify.Obtain Po ∝ Re^-1.
Verification / Alternative check:Standard impeller correlations display a Po·Re = constant plateau in laminar flow, equivalent to Po ∝ 1/Re.
Why Other Options Are Wrong:Zero/positive exponents imply inertia influence, contrary to laminar limit.
Common Pitfalls:Extending turbulent correlations into laminar ranges.
Final Answer:Po ∝ Re^-1.0