Power number–Reynolds number relation: for a turbine-impeller, baffled tank operating at low Reynolds number (laminar regime), how does the power number Np vary with NRe?

Introduction / Context:
The power number Np characterizes impeller power draw: P = Np * ρ * N^3 * D^5. Its dependence on the impeller Reynolds number NRe = ρ * N * D^2 / μ reveals the mixing regime. At low NRe (laminar), viscous effects dominate; at high NRe (turbulent), Np approaches a constant that depends on geometry.

Given Data / Assumptions:

• Turbine impeller in a baffled tank.
• Low Reynolds number (laminar flow).
• Newtonian liquid.

Concept / Approach:
In the laminar regime, dimensional analysis and experiments show P ∝ μ * N^2 * D^3 (for a given geometry). Rearranging the general power correlation gives Np = P / (ρ * N^3 * D^5). Substituting the laminar expression for P yields Np ∝ (μ / (ρ * N * D^2)) = 1 / NRe. Thus, as NRe increases (approaching transition), Np decreases inversely until it levels off in turbulence.

Step-by-Step Solution:
Start with P_laminar ∝ μ * N^2 * D^3.Compute Np = P / (ρ * N^3 * D^5) ∝ (μ / (ρ * N * D^2)) = 1/NRe.Therefore, Np ∝ 1/NRe in laminar mixing.

Verification / Alternative check:
Power curves in mixing handbooks plot Np vs NRe and show a straight inverse relationship at low NRe, transitioning to a constant plateau at high NRe.

Why Other Options Are Wrong:
Np ∝ NRe or constant contradicts laminar scaling; a constant Np applies only in the turbulent asymptote.

Common Pitfalls:
Using turbulent power numbers for viscous scale-up; always check Reynolds number range before applying Np values.

Final Answer:
Np ∝ 1/NRe