Universal Energy Extraction Limit in Open-Channel Flow

Abstract. A universal energy extraction limit for ideal open-channel flow is derived analytically. The extractable energy exhibits a subcritical maximum at the extremal Froude number Fr ≈ 0.3094, corresponding to a maximum energy fraction η ≈ 34.52%. The result is independent of friction laws and plays a role analogous to the Betz limit for wind turbines.

Core Result

The extractable energy of an ideal incompressible open-channel flow exhibits a subcritical extremum at the universal Froude number:

Fr_ext ≈ 0.3094

This extremum represents a fundamental energetic limit that does not depend on friction, channel roughness, or empirical resistance laws (Manning, Chezy, etc.).

Dimensionless Energy Function

The extractable energy is described by a dimensionless function Φ(Fr). Its maximum defines the universal limit for open-channel flow:

Φ_max = Φ(Fr ≈ 0.3094) ≈ 0.3452

Energy Limit (Betz-Type Analogy)

In wind energy theory, the Betz limit establishes a universal upper bound on extractable kinetic energy:

η_Betz = 16/27 ≈ 59.3%

For open-channel free-surface flow, an analogous universal constraint arises from energy balance and flow continuity:

η_OCF ≈ 34.52% at Fr ≈ 0.3094

Thus, Fr ≈ 0.3094 plays the same theoretical role for open-channel flow as the Betz limit does for wind turbines.

Analogy with the Betz Limit

System	Control parameter	Maximum efficiency	Nature of limit
Wind turbine	Axial induction factor	η = 16/27 ≈ 59.3%	Momentum & energy balance
Open-channel flow	Froude number	η ≈ 34.52%	Free-surface energy balance

Energy Hump and Subcritical Extremum

The figure below illustrates the dimensionless extractable energy Φ(Fr) as a function of the Froude number. A pronounced energy hump appears in the subcritical regime, with a universal maximum at Fr ≈ 0.3094.

Energy hump and universal subcritical extremum at Fr ≈ 0.3094 — Dimensionless energy Φ(Fr) showing a universal subcritical maximum at Fr ≈ 0.3094.

Domain of Validity

Incompressible fluid
Free-surface (open-channel) flow
Ideal flow without friction or dissipative losses
Subcritical regime (Fr < 1)

Why This Result Matters

Provides a theoretical upper bound for free-flow hydrokinetic devices
Separates geometric energy limits from technological efficiency
Clarifies the fundamental role of the Froude number in energy extraction

Primary Reference

Treshchalov, G. V. (2026).
A Universal Energy Extraction Limit in Open-Channel Flow.
Zenodo. DOI:
https://doi.org/10.5281/zenodo.18321384

Citation

If you refer to this result, please cite:

Treshchalov, G. V. (2026).
Universal Energy Extraction Limit in Open-Channel Flow (Fr ≈ 0.3094).
Available at:
https://althydroenergy.github.io/ERGroup/

Discussion

Comments, questions, and independent checks are welcome.

Notes

This page serves as the primary public reference for Fr ≈ 0.3094.
The limit is exact within the framework of ideal fluid mechanics.
The result is conceptually analogous to the Betz limit but applies to free-surface flows.