Unit Converters
For design engineers, scale-up teams, and capacity planners, project power from a validated baseline, test exponent sensitivity, and compare modeled demand with rating and operating exposure. The calculation is credible only inside the physical or empirical regime supporting the exponent; distant extrapolation, changing efficiency, and regime transitions require additional evidence.
Power-law scaling
Demand and exponent sensitivity
Scale a baseline power requirement with a declared exponent, duty modifier, efficiency ratio, capacity rating, and operating duration. Use it for first-pass similarity studies where the power-law assumption has physical support.
Input evidence: document the measured baseline, physical scaling variable, exponent source, operating modifiers, and capacity-rating condition.
| Element | Value | Unit | Interpretation |
|---|
Exponent sensitivity
The sensitivity view compares nearby exponents because uncertainty in the scaling law can dominate ordinary rounding error.
| Scenario or component | Basis | Calculated value | Decision use |
|---|
How to use
Document the baseline measurement, physical similarity conditions, and evidence supporting the exponent before projecting. Keep geometric scaling separate from duty and efficiency modifiers so every assumption can be challenged independently.
Target characteristic divided by its baseline value.
Controls how quickly power changes with scale; cubic affinity behavior is one specific case.
Separates active loading from full-time baseline operation.
Represents change in required input per useful output.
Installed rating minus modeled demand at the target point.
Result interpretation
The scale ratio changes baseline power through the declared exponent. Duty and efficiency modifiers then adjust the operating estimate. Capacity utilization compares that estimate with installed rating, operating energy extends it through time, and the sensitivity span shows how much nearby exponent choices move demand.
A negative margin means the modeled steady demand exceeds entered rating, not that physical overload has already been measured. A ratio of one should reproduce baseline after modifiers, while a zero or negative baseline scale makes the ratio invalid. Wide sensitivity relative to margin means model choice controls the decision.
The page evaluates P₂ = P₁(S₂/S₁)ⁿ and then applies independent operating modifiers. The live curve makes exponent sensitivity visible.
Fan and pump laws require dynamically similar operation and may fail when control, static head, or system resistance changes.
Small errors in the ratio or exponent become large when the exponent is high or the target lies far from baseline.
A positive margin is only a rating comparison; transient, thermal, and redundancy requirements remain separate.
Law selection
Linear, square, and cubic relationships represent different mechanisms, and the exponent may change when controls, geometry, or flow regime changes. Selecting three because equipment resembles a fan or pump is not sufficient evidence.
Document the physical relationship or fitted dataset, variable definitions, intercept treatment, valid scale range, uncertainty, and goodness of fit. If several laws are plausible, compare them explicitly and use the sensitivity span as model-choice risk rather than hiding the choice inside extra decimal places.
Extrapolation
When the scale ratio moves far from one, small baseline, ratio, and exponent errors compound nonlinearly. The curve can appear smooth even after the real system reaches saturation, a control limit, a different regime, or an efficiency collapse.
Prefer interpolation within validated observations. For extrapolation, compare nearby exponents, check intermediate operating points, and state the distance beyond evidence. Obtain new measurements when sensitivity overlaps installed capacity or changes the economic decision.
Capacity intersection
Capacity margin compares one modeled steady demand with one entered rating. It does not test transient demand, thermal duty, cooling, voltage drop, torque-speed limits, redundancy, environment, control stability, or protection settings.
Use the margin as an early screen. Confirm manufacturer operating curves, duty class, ambient derating, startup and contingency requirements before approving equipment. A large positive margin can also indicate inefficient oversizing or poor part-load operation rather than a better design.
Visual reading guide
The primary curve traces modeled demand versus scale under the selected exponent and modifiers, marks the current target, and shows installed capacity. Crossing the rating line identifies a mathematical intersection, not a validated safe operating point.
The supporting view holds baseline, ratio, duty, and efficiency constant while changing only the exponent. Divergence grows away from baseline and reveals model-choice leverage. Neither curve shows confidence intervals, regime changes, thermal limits, or measured observations unless those are supplied separately.
r = S₂/S₁; f = rⁿ; P₂ = P₁ f d e; M = Prating − P₂
S is the scaling variable, n is the power-law exponent, d is duty modifier, e is efficiency ratio, M is capacity margin.
| r | scale ratio | dimensionless |
| n | declared exponent | dimensionless |
| P₂ | scaled demand | W |
| M | rating margin | W |
Reverse check:
Default-value audit
Defaults: 15 kW baseline at scale 1; target scale 1.35; exponent 3; duty 0.82; efficiency ratio 1.04; rating 40 kW; 1,200 hours.
| Symbol | Meaning | Unit | Calculation role |
|---|---|---|---|
| P1 | Baseline power | W | 15,000 |
| S1, S2 | Baseline and target scales | declared scale unit | 1 and 1.35 |
| r | Scale ratio | dimensionless | S2/S1 |
| n | Power-law exponent | dimensionless | 3 |
| f | Power-law factor | dimensionless | rⁿ |
| d, e | Duty and efficiency modifiers | dimensionless | 0.82 and 1.04 |
| P2 | Scaled demand | W | P1fde |
| Pr | Installed rating | W | 40,000 |
| M, E | Capacity margin and operating energy | W; kWh | Pr−P2; P2h/1,000 |
Reverse check: dividing P2 by P1, duty and efficiency modifiers returns rⁿ; nearby-exponent results isolate model sensitivity.
Scaling evidence
Retain baseline measurement and uncertainty, scale-variable definition, geometry, speed, flow, pressure, environment, control state, efficiency, fitted observations, exponent source, regression diagnostics, duty basis, and rating condition.
Confirm that baseline and target use the same boundary and that modifiers are not already embedded in the exponent or baseline. Reproduce known observations with the selected law and compare prediction residuals before extending the curve. Unverified similarity assumptions should remain explicit limitations.
Limits and exclusions
The model excludes nonzero intercepts, saturation, regime changes, control limits, changing system curves, transient load, thermal constraints, correlation between modifiers, baseline uncertainty, and uncertainty or covariance in fitted coefficients. It applies one exponent over the full displayed range.
Therefore, use it for transparent sensitivity and preliminary capacity screening, not certification or final design outside validated conditions. Replace it with measured curves, regression intervals, or a mechanistic model when the operating regime, efficiency map, or decision risk demands them.
Scaling glossary
Worked cases
Inputs: 15 kW baseline, speed ratio 1.35, validated cubic law, duty 0.82 and efficiency ratio 1.04.
Calculation: cube the ratio, apply modifiers and compare target demand with the 40 kW rating.
Decision: retain the result as a first-pass demand estimate and still check the system curve and motor conditions.
Inputs: 15 kW baseline, target ratio 1.8, duty 0.82, efficiency ratio 1.04, exponent 3 with ±0.5 sensitivity, and 40 kW rating.
Calculation: modeled demand is approximately 55.606 kW at n = 2.5, 74.603 kW at n = 3, and 100.090 kW at n = 3.5. All cases exceed rating, and the 44.485 kW span is larger than the rating itself.
Decision: reject the 40 kW capacity screen and do not select equipment from the nominal exponent alone. Obtain intermediate measurements or fit a validated regime-specific model.
Do not use an assumed cubic law for a system whose static head, control strategy, efficiency curve, or geometry changes materially.
Frequently asked questions
The exponent acts on the scale ratio before other modifiers, so its effect grows rapidly when the ratio is far from one. Small exponent changes can therefore dominate rating margin and operating energy.
No. Cubic affinity behavior requires dynamically similar operation and compatible system conditions. Static head, control valves, variable efficiency, geometry changes, and operating limits can make actual demand depart materially from a cubic law.
It is a separate operating factor applied after geometric or throughput scaling. Use it for a justified load or active-duty adjustment and avoid double-counting behavior already included in baseline measurements or the fitted exponent.
Only with validated physics or data and an explicit sensitivity or uncertainty review. Seek intermediate measurements when the target is far outside evidence, especially when model variation crosses capacity or changes the decision.
The ratio is undefined because target scale cannot be divided by zero. Choose a nonzero physical reference state within the same operating regime; do not replace zero with an arbitrary small number.
Nearby exponents expose model-choice risk while holding other assumptions constant. If their demand span is large relative to rating margin, improving numeric precision will not solve the underlying uncertainty about the relationship.
No. It shows only that modeled steady demand is below entered rating. Final adequacy also requires transient, thermal, environmental, redundancy, control, and protection checks against manufacturer and code requirements.
Yes. Efficiency may vary with speed, load, flow, temperature, and control state. Use an efficiency curve or state model when that change is material instead of compressing it into one constant ratio.
Fit a regression when several validated observations exist and no accepted physical law fully describes them. Review residuals, domain, uncertainty, and possible intercepts rather than selecting an exponent solely for a visually smooth curve.