Power Number¶
A dimensionless rotating-fluid-device power coefficient, P/(ρn³D⁵), whose transfer across operating points requires matched geometry and flow conditions.
Core Idea¶
The power number compares mechanical power \(P\) drawn by a specified rotor in a fluid with the characteristic scale \(\rho n^3D^5\), where \(\rho\) is fluid density, \(n\) is rotational frequency in revolutions per second and \(D\) is rotor diameter: \(N_P=P/(\rho n^3D^5)\). The quotient is dimensionless. Mixing engineers commonly use \(N_P\); propeller reports often call the same normalization a power coefficient \(C_P\). The power boundary—shaft power, fluid-transfer power or another convention—must be declared.[ref-1b2806bb9aa8][ref-e4bf295803e7]
Scope of Application¶
In a stirred tank, a geometry-specific \(N_P\)–Reynolds chart helps estimate power from density, speed and impeller size. In a Columbia worked baffled six-blade turbine case, \(D=0.1016\) m, \(n=5\) rev/s, \(\rho=1000\) kg/m³ and chart \(N_P=6\) imply about 8.1 W and 0.258 N m torque. In a different setting, a NASA wind-tunnel report uses the same denominator for a 1.93-m aircraft propeller but plots \(C_P\) against advance ratio and blade pitch. The two settings share normalization, not a universal coefficient curve.[ref-1b2806bb9aa8][ref-e4bf295803e7]
Clarity¶
Raw watts cannot be compared across different rotor sizes and speeds without accounting for their strong scaling. The power number removes \(\rho n^3D^5\) but does not erase baffles, blade design, clearance or flow regime. Reynolds number helps index a mixing correlation; advance ratio helps index a propeller correlation. Confusing tank diameter with rotor diameter or rpm with rev/s changes the reported value substantially.[ref-1b2806bb9aa8][ref-3929c4d60a22][^ref-e4bf295803e7]
Manages Complexity¶
The coefficient condenses density, rotational speed, diameter and measured power into one dimension-one number, and a matched empirical curve organizes how that number changes with flow state. This makes a pilot-to-larger-device comparison possible when geometry and relevant dimensionless conditions match. It does not replace the second outcome measurement: mixing time, thrust and propulsive efficiency are not determined by \(N_P\) alone. Multi-impeller experiments also show that clearance and spacing affect normalized power.[ref-1b2806bb9aa8][ref-3929c4d60a22][ref-57dc9dd1ea2b][ref-e4bf295803e7]
Abstract Reasoning¶
Measure power, or derive it from shaft torque as \(P=2\pi nT\); record density, rev/s frequency and rotor diameter; then calculate \(N_P\). For prediction, choose a coefficient from a curve at the relevant geometry and operating state and invert to \(P=N_P\rho n^3D^5\). A high-Re baffled-tank plateau can simplify that step only within its validated regime. For a propeller, check \(J\) and blade pitch rather than importing a tank coefficient or assuming a constant \(C_P\).[ref-1b2806bb9aa8][ref-e4bf295803e7]
Knowledge Transfer¶
The literal transfer within mixing is between geometrically similar tanks at matched relevant dimensionless conditions, as the Columbia experiment proposes. The propeller case transfers only the normalization form: NASA's operating curves differ from a stirred-tank \(N_P\)–Reynolds chart. Live Dimensionless Quantity is the proposed strict DAG parent because \(P\) and its denominator have the same dimensions; the named power number remains a fluid-device-specific coefficient.[ref-1b2806bb9aa8][ref-3929c4d60a22][^ref-e4bf295803e7]
[^ref-1b2806bb9aa8]: Columbia University chemical-engineering laboratory, “Torque and Power” theory and sample calculations, sections E and Comments 3–8. [^ref-3929c4d60a22]: Columbia University chemical-engineering laboratory, “Power Consumption and Efficiency in Liquid Mixing”, Overview and Theory. [^ref-e4bf295803e7]: Philip R. Barlow, Victor R. Corsiglia and Joseph Katz, “Full-Scale Aerodynamic Characteristics of a Propeller Installed on a Small Twin-Engine Aircraft Wing Panel”, NASA Ames technical report (1981), Nomenclature, Results and Figure 5. [^ref-57dc9dd1ea2b]: Piero M. Armenante and Gwo Ming Chang, “Power Consumption in Agitated Vessels Provided with Multiple-Disk Turbines”, Industrial & Engineering Chemistry Research 37 (1998), original article abstract only.
Relationships to Other Abstractions¶
Current abstraction Power Number Domain-specific
Parents (1) — more general patterns this builds on
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Power Number is a kind of Dimensionless Quantity Domain-specific
A power number is a dimensionless physical quantity specialized to rotating-device power.
Hierarchy path (1) — routes to 1 parentless root
- Power Number → Dimensionless Quantity → Physical quantity → Measurement
Neighborhood in Abstraction Space¶
Power Number sits in a sparse region of the domain-specific corpus (87th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Thermodynamic Cycles & Engineering Measures (8 abstractions)
Nearest neighbors
- Energy Conversion Efficiency — 0.84
- Stokes's law — 0.81
- Potential Energy — 0.81
- Rotating Unbalance — 0.80
- Hartmann Number — 0.80
Computed from structural-signature embeddings · 2026-10-08