Power Number¶

A dimensionless rotating-fluid-device power coefficient, P/(ρn³D⁵), whose transfer across operating points requires matched geometry and flow conditions.

Version: v2 · 2026-10-03 · History
Domain-specific #: 13509
Domain group: Applied Sciences & Engineering
Origin domain: Engineering & Design (beyond software)
Subdomains: Rotating Machinery, Stirred Vessels → Engineering & Design (beyond software)
Aliases: Impeller Power Number, Newton Number Mixing

Core Idea¶

The power number normalizes the mechanical power of a specified rotating device acting on a fluid by a density–speed–diameter scale. With shaft or fluid-transfer power $P$, fluid density $\rho$, rotational frequency $n$ in revolutions per second, and rotor diameter $D$, its common form is

\[N_P=\frac{P}{\rho n^3D^5}.\]

Agitator literature commonly writes $N_P$; propeller literature often calls the same normalized quantity a power coefficient $C_P$. The expression is dimensionless because $\rho n^3D^5$ has units of mass × length² / time³, the same as power. The convention for which power is counted must still be stated: measured shaft input, power transferred to the fluid, and motor electrical consumption need not be identical.^[1]^[2]

The abstraction is a conditional comparison instrument, not a magic constant. A power number measured for one impeller geometry and flow regime can help infer power for a geometrically similar device at a matched operating state through $P=N_P\rho n^3D^5$. A baffled-tank chart commonly relates $N_P$ to impeller Reynolds number, while an aircraft propeller's $C_P$ can vary with advance ratio and blade pitch. Normalization removes dimensional size and speed factors; it does not erase geometry, free-surface, fluid-rheology or operating-point dependence.^[1]^[3]^[2]

Structural Signature¶

Sig role-phrases: specified rotating fluid device → mechanical power convention → fluid and rotational scales → normalized coefficient → operating similarity condition → power inference or comparison.

Specified rotating fluid device. An impeller in a defined vessel and a propeller in a defined flow both qualify, but their coefficient curves are not interchangeable. Baffles, blade design and spacing can change power draw; naming only “a rotor” is insufficient for transfer.^[3]^[4]^[2]
Mechanical power convention. $P$ is the power being normalized, commonly obtained from shaft torque $T$ as $P=2\pi nT$ when $n$ is revolutions per second. An electrical input that includes drivetrain losses must not silently replace shaft or fluid power.^[1]^[3]
Fluid and rotational scales. $\rho$, $n$ and $D$ must refer to the relevant fluid and rotor. The diameter is the impeller/propeller diameter, not automatically the vessel diameter. Using angular speed in radians per second as if it were $n$ introduces a factor of $(2\pi)^3$ in the normalization.^[1]^[2]
Normalized coefficient. The quotient $P/(\rho n^3D^5)$ packages measured power into a dimension-one quantity. It is not itself a power in watts, and two equal numerical values do not imply identical flow fields or engineering performance.^[1]^[2]
Operating similarity condition. Reuse a coefficient only with appropriate similarity. For baffled tanks, geometry and Reynolds regime matter; a free-surface vortex can add another influence. For a propeller, the NASA report explicitly plots coefficient against advance ratio $J$ and blade-pitch angle. The formula can always be evaluated, but predictive transfer requires this role.^[3]^[2]
Power inference or comparison. Under a chosen coefficient curve, invert the equation to estimate $P$ or compare normalized power demand. That use is optional to the definition; a measured point remains a power number even when no scale-up is attempted.^[1]

What It Is Not¶

It is not a universal numerical constant for all rotating devices. Columbia's tank example reads a coefficient from a geometry-specific $N_P$–Reynolds chart. NASA's propeller report plots $C_P$ against advance ratio and blade angle. A constant high-Re baffled-tank region is a conditional regime result, not an assertion that a propeller or a different vessel draws the same normalized power.^[1]^[2]

It is not a Reynolds number. Reynolds number locates an inertial–viscous regime using density, speed, diameter and viscosity; power number normalizes actual power. An $N_P$–Re chart uses the former to predict or organize the latter. Neither number substitutes for the other.^[1]

It is not mixing time, pumping number, thrust coefficient or propulsive efficiency. A mixing laboratory measures torque and tracer-response time separately; NASA reports propeller power, thrust and efficiency as distinct outputs. High or low $N_P$ by itself says what power is drawn relative to the chosen scale, not how well an engineering objective is met.^[3]^[2]

It is not simply “resistance force divided by inertial force.” One can motivate the scaling by inertial dimensions, but the defined observable is power divided by a power scale. Treating it as a force ratio without specifying rotation, torque and length conventions obscures the actual coefficient.

It is not a guarantee of safe motor sizing from one pilot point. The transfer assumes that the pilot and larger unit occupy matched relevant regimes and configurations. Different baffles, impeller counts or clearance can change power number even under turbulent conditions.^[3]^[4]

Scope of Application¶

In stirred-vessel engineering, $N_P$ expresses power consumption of turbines, paddles or other impellers. A fixed-geometry correlation versus impeller Reynolds number can separate low- and high-Re behavior. Columbia's mixing experiment uses a six-bladed turbine, baffles, measured torque and a coefficient chart to compare power across water and more viscous syrup, and proposes checking geometric similarity between differently sized tanks. The source distinguishes agitation power from a tracer-based assessment of mixing time.^[1]^[3]

In aeronautical propulsion, the same dimensional normalization appears as propeller power coefficient $C_P$. Barlow, Corsiglia and Katz's NASA report measured full-scale shaft thrust and torque for a 1.93-meter propeller in a wind tunnel. Its nomenclature defines $C_P=P/(\rho n^3D^5)$, and its results show coefficient curves as functions of advance ratio and blade angle. Those independent variables are not optional decoration: a propeller in forward motion samples a flow state unlike a baffled stirred tank.^[2]

The coefficient can be used in other rotating-fluid machines only after checking the community's numerator, diameter and speed conventions. A shared dimensionless expression does not imply a shared measured curve, and a non-Newtonian or aerated tank may require additional state descriptors. The sources here support the two named settings and the general dimensional relation, not a catalog of universal constants.

Clarity¶

The number resolves a misleading raw-power comparison. If a pilot impeller draws less power than a larger production impeller, that alone says little: power scales strongly with speed and diameter. Converting both to $N_P$ asks whether they have similar normalized draw under comparable geometry and flow state. The inversion $P=N_P\rho n^3D^5$ makes explicit why even a modest diameter change can strongly change shaft-power demand.^[1]

It also forces a unit and convention audit. Columbia gives $n$ as revolutions per second, uses impeller diameter, and derives power from torque. NASA likewise uses rev/s and propeller diameter. If a formula is fed rpm without conversion or tank diameter instead of rotor diameter, the resulting coefficient is not comparable despite being dimensionless on paper.^[1]^[2]

Finally, it resolves the ambiguity in “same coefficient.” Equality at matched Reynolds number for geometrically similar baffled tanks is a different claim from equality between different rotor families. Multiple impellers, altered bottom clearance, or a varying propeller advance ratio can shift the coefficient. Normalization is a prerequisite for comparison, not proof of similarity.^[3]^[4]^[2]

Manages Complexity¶

Many dimensional inputs influence rotary power: fluid density, frequency, rotor size, viscosity and geometry. $N_P$ collapses the first three major dimensional scales into one coefficient; Reynolds number and further dimensionless descriptors organize how that coefficient varies. In Columbia's example, a chart and a few measured inputs yield a torque prediction for a baffled turbine. Without normalization, every combination of tank and speed would require its own raw-power table.^[1]

The compression is deliberately incomplete. A laboratory chart must identify its impeller, vessel ratios and baffles; the NASA propeller curves must identify blade pitch and $J$. Armenante and Chang measured individual and total power numbers for multi-disk turbines and found changes with count, off-bottom clearance and spacing. Treating all those arrangements as one number would save notation at the expense of accuracy.^[3]^[2]^[4]

The coefficient also keeps a power budget separate from performance. A lower normalized power requirement may look attractive, but mixing time, dispersion, thrust and propulsive efficiency require their own observations. A smaller denominator-normalized input is not automatically a better process or vehicle.^[3]^[2]

Abstract Reasoning¶

For a specified rotating device, obtain the relevant power $P$—or shaft torque and frequency, then $P=2\pi nT$—and record $\rho$, $n$ in rev/s, and rotor $D$. Compute $N_P$ (or $C_P$) from the quotient. To infer power elsewhere, choose a coefficient from data at matched geometry and relevant dimensionless operating conditions, then multiply by $\rho n^3D^5$. This is an inference conditioned on the curve, not a direct consequence of dimensional algebra alone.^[1]^[2]

For a stirred tank, compare Reynolds numbers and baffle/impeller geometry before reading a pilot curve. In a sufficiently high-Re baffled regime the coefficient may be approximately stable over a range, simplifying scale-up, but the source's low-Re linear-torque and high-Re quadratic-torque behavior show why crossing regimes changes the inference.^[1]

For a propeller, ask for the advance ratio $J=V_\infty/(nD)$ and blade-angle setting before borrowing $C_P$. The NASA report's Figure 5 varies both and shows separate curves. A single $C_P$ carried across flight conditions would erase measured operating dependence. After predicting power, evaluate thrust or efficiency separately if those are the actual design objectives.^[2]

Knowledge Transfer¶

The literal transfer is within a family of geometrically and dynamically comparable devices. Columbia's two-tank proposal uses the same turbine type and matching Reynolds number to test whether a measured power number predicts power or torque in a second tank. The physics of normalized power is portable, but the coefficient's calibrated value is not portable without similarity.^[1]

The propeller case transfers the normalization form into a genuinely different fluid-device setting. It does not transfer a Rushton-turbine curve to an aircraft propeller. The NASA report uses the same $\rho n^3D^5$ denominator but indexes the measured coefficient by forward-speed ratio and blade setting. The general parent Dimensionless Quantity travels more broadly as a physical quantity of dimension one; Power Number remains the specific rotating-power normalization.^[2]

Examples¶

Baffled six-bladed turbine in water. A Columbia worked calculation gives water density $1000\,\mathrm{kg/m^3}$, impeller diameter $0.1016\,\mathrm{m}$, speed $5\,\mathrm{rev/s}$ and impeller Reynolds number $57{,}347$. A chart for this baffled six-bladed turbine supplies $N_P=6$. Reversing the definition gives approximately $8.1\,\mathrm{W}$ of agitation power; $P=2\pi nT$ gives about $0.258\,\mathrm{N\,m}$ of torque, matching the source's worked torque. The result belongs to that geometry and regime, not to an arbitrary stirrer.^[1]

Mapped back: specified rotating fluid device = six-bladed turbine in baffled tank; mechanical power convention = torque-derived fluid-agitation power; fluid/rotational scales = water density, 5 rev/s, 0.1016-m impeller; normalized coefficient = chart value 6; operating similarity = matched baffled geometry and Re 57,347; power inference = about 8.1 W and 0.258 N m.

Full-scale aircraft propeller. In a NASA wind-tunnel investigation, a 1.93-m propeller mounted on a wing panel was run at different tunnel speeds and blade-angle settings. Researchers measured shaft thrust and torque and presented $C_P=P/(\rho n^3D^5)$ against advance ratio $J$ and blade pitch. Figure 5 contains curves, not one invariant number. The coefficient permits normalized comparison across its measured operating points while retaining the flow-state variables required to interpret it.^[2]

Mapped back: specified rotating fluid device = full-scale 1.93-m aircraft propeller; mechanical power convention = shaft power from measured torque and speed; fluid/rotational scales = air density, rev/s frequency and propeller diameter; normalized coefficient = reported $C_P$; operating similarity = indexed $J$ and blade angle; power inference = comparison and model check across the reported points, with no unreported numeric coefficient asserted.

Boundary negative. A torque reading alone is not $N_P$. Without rotational frequency, fluid density, rotor diameter and the chosen power convention, it cannot be normalized to the defined coefficient.

Structural Tensions¶

Compact coefficient versus geometry-specific fidelity. Normalization makes different sizes and speeds comparable, but reusing one curve across changed baffles, impellers, clearance or propeller pitch suppresses physically relevant differences. Retaining every detail loses the economy of a similarity coefficient; dropping them all makes predictions unreliable. Diagnostic: Which geometric ratios and flow-state variables actually match before this measured $N_P$ or $C_P$ is transferred?^[3]^[4]^[2]

Power budget versus performance objective. Lower shaft-power draw can save energy, but does not by itself prove shorter mixing time, adequate suspension, more thrust or higher propulsive efficiency. Designing only for minimum power can underperform the task; designing only for maximum performance can overspend power. Diagnostic: What is the target outcome, and what measurement besides the power number verifies it?^[3]^[2]

Flat high-Re agitator curve versus variable propeller operation. A baffled agitator may show approximately constant $N_P$ in a high-Re regime, which makes prediction easy. A propeller at changed forward speed or blade pitch has visibly different $C_P$ in the NASA report. Extending the flat-curve convenience to the wrong apparatus loses a major operating dependency. Diagnostic: Is the measured curve flat over the actual planned operating range, or must $J$, pitch and geometry be retained?^[1]^[2]

Structural–Framed Character¶

The entry is predominantly structural within fluid engineering, but convention- and apparatus-framed. Evaluative weight: the coefficient is descriptive normalized power, not a value judgment that high or low power is good. Human-practice dependence: engineers decide which shaft, rotor diameter, power boundary and operating curve to report; the underlying dimensional relation does not depend on their preference, but its useful application does. Institutional origin: experimental and engineering conventions standardized names such as $N_P$ and $C_P$; no law or institution creates the physical power consumption itself.^[1]^[2]

Vocabulary travel: the form $P/(\rho n^3D^5)$ travels between mixing and propulsion, yet coefficient values and secondary parameters do not. Import versus recognition: recognizing this normalized relation in a different rotating-fluid device is legitimate if its power convention and flow scaling are checked; calling any “power ratio” a power number would merely import vocabulary. The portable skeleton belongs to live Dimensionless Quantity, and this named coefficient remains bound to rotating-fluid dynamics. Its character: a structural dimensionless engineering measure with meaningful apparatus and convention boundaries, not a general prime abstraction.

Structural Core vs. Domain Accent¶

The structural core is a unit-invariant physical ratio: a measured quantity divided by a same-dimension characteristic scale. The live Dimensionless Quantity parent supplies exactly that broad relation. Power Number fixes the numerator to mechanical power and the scale to $\rho n^3D^5$ for rotating flow, then requires interpretation against a specified geometry and operating regime.^[1]^[2]

Those domain details are constitutive, not detachable accents. Replace power with thrust and one obtains a different coefficient; replace the rotor diameter or rev/s convention and the reported number changes; move from a baffled tank to an advancing propeller and the relevant condition chart changes. A generic dimensionless-number prime should therefore not inherit the specific $N_P$–Re or $C_P$–$J$ curves. The named entry does not meet the prime bar merely because the same algebraic normalization appears in two fluid-device subfields.

This entry is a kind of Dimensionless Quantity. A power number is a dimensionless physical quantity specialized to rotating-device power.

Relationships to Other Abstractions¶

Current abstraction Power Number Domain-specific

Parents (1) — more general patterns this builds on

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 (87^th 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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Raw power: watts are dimensional; $N_P$ is dimensionless. Compare power only after declaring density, speed, diameter and operating context.

Reynolds number: measures a flow regime involving viscosity. It can help select $N_P$, but is not the normalized power itself.^[1]

Pumping or thrust coefficient: concerns fluid volume flow or thrust rather than mechanical input power. NASA's separate $C_T$ and $C_P$ nomenclature makes the distinction explicit.^[2]

Propulsive efficiency or mixing effectiveness: both require an output or outcome in addition to power. A lower $C_P$ does not by itself imply more useful thrust, and a lower impeller $N_P$ does not by itself imply faster tracer homogenization.^[3]^[2]

One universal turbulent value: a constant plateau is limited to a specific geometry and regime. Multi-impeller clearance and spacing change power draw, while propeller $C_P$ varies across advance ratio and pitch.^[4]^[2]

References¶

[1] Columbia University chemical-engineering laboratory, “Torque and Power” theory and sample calculations, sections E and Comments 3–8. Original course-authored instructional analysis, including Np-versus-Re correlation, torque relation and worked six-blade turbine values. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t

[2] 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, Summary, Results and Figure 5. Original measurements and coefficient convention. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y

[3] Columbia University chemical-engineering laboratory, “Power Consumption and Efficiency in Liquid Mixing”, Overview and Theory. Original course experiment distinguishing power from tracer mixing time and specifying geometric/baffle conditions. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m

[4] Piero M. Armenante and Gwo Ming Chang, “Power Consumption in Agitated Vessels Provided with Multiple-Disk Turbines”, Industrial & Engineering Chemistry Research 37 (1998), 284–291, original article abstract in author-institution repository. Full article not inspected; cited only for abstract-reported effects of impeller number, clearance and spacing. registry ↩a ↩b ↩c ↩d ↩e ↩f