Universal Relation between Nusselt Number and Bejan Number in Natural Convection

Takuya Masuda; Toshio Tagawa

arxiv: 2604.08848 · v1 · submitted 2026-04-10 · ⚛️ physics.flu-dyn

Universal Relation between Nusselt Number and Bejan Number in Natural Convection

Takuya Masuda , Toshio Tagawa This is my paper

Pith reviewed 2026-05-10 18:01 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords natural convectionNusselt numberBejan numberentropy generationscaling lawheat transferirreversibilityboundary layer

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The pith

Natural convection obeys Be inverse minus one equals a constant times Nusselt number to a power, independent of geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a scaling law that connects the Nusselt number, a measure of convective heat transfer, to the Bejan number, a measure of entropy generation due to irreversibility. Through entropy generation analysis paired with boundary-layer scaling arguments, the authors derive that the relation Be^{-1} - 1 = a Nu^b appears whenever the flow is controlled by a single parameter. This form does not require knowledge of the enclosure shape or specific boundary conditions. A reader would care because the law supplies a direct bridge between heat-transfer performance and thermodynamic losses that could simplify predictions and design constraints across many convective configurations.

Core claim

Using entropy generation analysis and boundary-layer scaling, we demonstrate that Be^{-1} - 1 = a Nu^b emerges independently of geometry and boundary conditions when transport is governed by a single control parameter. The relation is validated using a canonical square cavity. This result establishes a direct connection between heat transfer and thermodynamic irreversibility, revealing a fundamental constraint in convective transport.

What carries the argument

Entropy-generation scaling combined with boundary-layer analysis, which produces the geometry-independent power-law relation between the inverse Bejan number and the Nusselt number.

If this is right

The power-law relation applies to any geometry or boundary conditions provided the flow remains under single-parameter control.
Heat-transfer rate and total irreversibility become directly interchangeable quantities without separate geometry-specific calculations.
The law supplies a universal constraint that must be satisfied by any valid model or simulation of single-parameter natural convection.
Validation against square-cavity data supports treating the relation as a general feature rather than a case-specific coincidence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If real-world convection problems can be reduced to a single dominant parameter by suitable nondimensionalization, the same relation would hold there as well.
Device designers could use the law to estimate minimum achievable entropy generation once the desired Nusselt number is specified.
The relation offers a quick consistency check for numerical codes or experiments that claim to solve natural-convection problems.

Load-bearing premise

Natural convection is always governed by a single control parameter.

What would settle it

A set of measurements of Be and Nu in a natural-convection flow with two or more independent control parameters in which the proposed power-law fit fails to hold with fixed a and b.

read the original abstract

We propose a universal scaling law linking the Nusselt number (Nu) and the Bejan number (Be) in natural convection. Using entropy generation analysis and boundary-layer scaling, we demonstrate that Be^-1 - 1 = a Nu^b emerges independently of geometry and boundary conditions when transport is governed by a single control parameter. The relation is validated using a canonical square cavity. This result establishes a direct connection between heat transfer and thermodynamic irreversibility, revealing a fundamental constraint in convective transport.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a simple power-law tie between Bejan number and Nusselt number from entropy scaling, but only under the narrow assumption of a single control parameter and with two adjustable constants.

read the letter

The main takeaway is that the authors start from entropy generation in natural convection, apply boundary-layer scaling, and arrive at Be inverse minus one equals a times Nu to the b. They present this as geometry-independent when the flow is controlled by one parameter, and they check it against a square-cavity simulation. That link between heat-transfer rate and total irreversibility is the concrete new piece they offer. It could let someone estimate entropy production without a full second-law calculation once Nu is known. The derivation itself is compact and follows standard scaling steps, so the logic is easy to follow on the page. They are also clear that the relation is not claimed to hold everywhere, only when the single-parameter condition is met. That honesty is useful. The soft spots sit in the constants and the validation. The values of a and b are not fixed by the equations alone; they appear to be chosen to match the cavity data. If they are fitted rather than predicted, the relation becomes a convenient fit rather than a first-principles result, which weakens the universality claim. The check is also limited to one square cavity at one Prandtl number. Natural convection is usually set by both Rayleigh and Prandtl numbers, and boundary-layer exponents change with Pr. The paper notes the single-parameter restriction, but without tests at other Pr or other geometries the independence from boundary conditions stays unproven. This is the kind of short scaling note that might interest engineers who already run cavity or enclosure calculations and want a quick irreversibility estimate. It is not a broad advance in convection theory, but the entropy-to-Nu connection is worth checking if the derivation is reproducible. I would send it to review so the authors can clarify how a and b are obtained and add at least one more Prandtl-number case.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a universal scaling law Be^{-1} - 1 = a Nu^b that links the Nusselt number and Bejan number in natural convection. Using entropy generation analysis combined with boundary-layer scaling, the authors argue that this power-law relation emerges independently of geometry and boundary conditions whenever transport is governed by a single control parameter. The relation is validated numerically for the canonical square-cavity configuration.

Significance. If the claimed universality holds, the result would establish a direct connection between convective heat transfer and thermodynamic irreversibility, providing a potentially useful constraint for modeling natural convection. The approach of combining entropy generation with scaling arguments is interesting, but the presence of two free parameters and the limited scope of validation limit the immediate impact.

major comments (3)

Abstract: the claim that the relation 'emerges independently of geometry and boundary conditions when transport is governed by a single control parameter' is not supported by the Boussinesq equations, which are controlled by two independent dimensionless groups (Ra and Pr). Boundary-layer analysis yields distinct Nu(Ra,Pr) regimes (e.g., Nu ~ Ra^{1/4} Pr^{1/2} for Pr ≪ 1), so the coefficients a and b cannot be geometry-independent without an explicit assumption that eliminates or fixes Pr.
Validation section (square-cavity case): validation is reported only for one geometry at a single (implicit) Pr value. No tests are provided for other geometries or Pr regimes where the single-parameter premise fails, so the universality claim remains untested at the points where the derivation is most vulnerable.
Derivation of the scaling law: the parameters a and b are introduced without independent derivation from the entropy-generation analysis or first principles. If these coefficients are fitted to the same simulations used for validation, the relation reduces to a post-hoc correlation rather than a predictive, parameter-free result.

minor comments (2)

Notation: provide an explicit definition of the Bejan number Be used in the natural-convection context, including its relation to the usual Rayleigh and Prandtl numbers.
Figures: include the range of Ra and Pr covered in the validation plots and report the fitted values of a and b with their uncertainties.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for these insightful comments, which help us clarify the assumptions and scope of our work. We provide point-by-point responses below and indicate the revisions we will make.

read point-by-point responses

Referee: Abstract: the claim that the relation 'emerges independently of geometry and boundary conditions when transport is governed by a single control parameter' is not supported by the Boussinesq equations, which are controlled by two independent dimensionless groups (Ra and Pr). Boundary-layer analysis yields distinct Nu(Ra,Pr) regimes (e.g., Nu ~ Ra^{1/4} Pr^{1/2} for Pr ≪ 1), so the coefficients a and b cannot be geometry-independent without an explicit assumption that eliminates or fixes Pr.

Authors: We agree that the standard Boussinesq formulation involves both Ra and Pr. Our claim is specifically conditioned on transport being governed by a single control parameter, which corresponds to fixing Pr while varying Ra (a common scenario for a given fluid). The boundary-layer scaling in our derivation is performed under this assumption, allowing a and b to be independent of geometry but dependent on the fixed Pr. We will revise the abstract to make this assumption explicit and add a discussion on how the relation may be extended or modified for varying Pr. revision: yes
Referee: Validation section (square-cavity case): validation is reported only for one geometry at a single (implicit) Pr value. No tests are provided for other geometries or Pr regimes where the single-parameter premise fails, so the universality claim remains untested at the points where the derivation is most vulnerable.

Authors: The current validation is limited to the square cavity at Pr ≈ 0.71. To address this, we will expand the validation section with additional simulations for at least one other geometry (such as a tall rectangular enclosure) and for a different Prandtl number (e.g., Pr = 7 for water) to test the relation under the single-parameter control assumption. revision: yes
Referee: Derivation of the scaling law: the parameters a and b are introduced without independent derivation from the entropy-generation analysis or first principles. If these coefficients are fitted to the same simulations used for validation, the relation reduces to a post-hoc correlation rather than a predictive, parameter-free result.

Authors: The power-law form is obtained directly from combining the entropy generation expression with boundary-layer scaling for the velocity and temperature fields under single-parameter control. This yields Be^{-1} - 1 proportional to Nu to some power, with the specific exponent b emerging from the scaling exponents. The prefactor a is then calibrated using the numerical data. While this involves fitting, the functional form itself is predictive from the analysis. In the revision, we will provide a more detailed step-by-step derivation showing the origin of the form and the expected range for b, and clarify that a is determined from data but the relation is not purely post-hoc. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the claimed relation Be^{-1} - 1 = a Nu^b from entropy generation analysis combined with boundary-layer scaling, explicitly conditioned on the premise that transport is governed by a single control parameter. This is presented as an emergent consequence of the analysis rather than presupposed by definition or obtained via fitting to the validation data. The subsequent validation step on the square cavity is described separately and does not feed back into the derivation. No load-bearing step reduces the result to its own inputs by construction, self-citation, or renaming; the single-parameter assumption is stated openly as a scope condition rather than smuggled in. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a single control parameter and on the validity of boundary-layer scaling plus entropy generation balance; a and b appear as undetermined coefficients.

free parameters (2)

a
Prefactor in the proposed power-law relation; value not derived from first principles in the abstract.
b
Exponent in the proposed power-law relation; value not derived from first principles in the abstract.

axioms (2)

domain assumption Natural convection transport is governed by a single control parameter
Explicitly stated as the condition under which the universal relation emerges.
domain assumption Boundary-layer scaling combined with entropy generation analysis yields a power-law relation between Nu and Be
Core step invoked to obtain Be^{-1} - 1 = a Nu^b.

pith-pipeline@v0.9.0 · 5371 in / 1322 out tokens · 46411 ms · 2026-05-10T18:01:14.567690+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We demonstrate that Be^{-1} - 1 = a Nu^b emerges independently of geometry and boundary conditions when transport is governed by a single control parameter.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

The heat transport and spectrum of thermal turbulence,

1 Universal Relation between Nusselt Number and Bejan Number in Natural Convection Takuya Masuda * Department of Integrated Engineering, National Institute of Technology, Yonago College, 4448 Hikona-cho, Yonago, Tottori 683-8502, Japan Toshio Tagawa Department of Aeronautics and Astronautics, Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-...

work page doi:10.1098/rspa.1954.0197 1954

[1] [1]

The heat transport and spectrum of thermal turbulence,

1 Universal Relation between Nusselt Number and Bejan Number in Natural Convection Takuya Masuda * Department of Integrated Engineering, National Institute of Technology, Yonago College, 4448 Hikona-cho, Yonago, Tottori 683-8502, Japan Toshio Tagawa Department of Aeronautics and Astronautics, Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-...

work page doi:10.1098/rspa.1954.0197 1954