Laminar mixing correlation: for laminar flow around an impeller, how is power number Np correlated with impeller Reynolds number Rei?

Introduction / Context:
Power consumption in stirred vessels is often expressed using dimensionless groups: power number Np and Reynolds number Rei (based on impeller). In the laminar regime, viscous forces dominate and classical correlations show a simple inverse relationship between Np and Rei for a given geometry.

Given Data / Assumptions:

• Laminar regime (low Rei, typically Rei < 10–100 depending on geometry).
• Constant impeller and tank geometry.
• K denotes a geometry-dependent constant.

Concept / Approach:
By definition, Np = P / (ρ*N^3*Di^5) and Rei = ρ*N*Di^2/μ. In viscous-dominated (laminar) flow, P scales with μ, N, and Di such that Np ∝ 1/Rei. Thus, doubling Rei halves Np, all else equal, until transitional/turbulent regimes are reached where Np becomes nearly constant with Rei for baffled tanks.

Step-by-Step Solution:
1) Start from dimensional analysis and laminar scaling: P ∝ μ*N^2*Di^3 (for a given geometry).2) Substitute into Np and collect terms with Rei = ρ*N*Di^2/μ.3) Resulting relation shows Np = K*(Rei)^-1 for laminar mixing.

Verification / Alternative check:
Mixing handbooks list Np = K/Rei for many impeller types under laminar conditions, with K depending on blade shape and clearance.

Why Other Options Are Wrong:
Option B/D introduce extra dimensional terms; option C makes Np grow with Rei, opposite to laminar behavior; option E applies to turbulent, not laminar flow.

Common Pitfalls:
Applying turbulent, constant-Np assumptions to laminar design; ensure regime check via Rei before selecting correlations.

Final Answer:
Np = K*(Rei)^-1