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arxiv: 2604.23736 · v1 · submitted 2026-04-26 · ⚛️ physics.flu-dyn

Relation between the Nusselt and Bejan numbers in natural convection

Takuya Masuda , Toshio Tagawa This is my paper

Pith reviewed 2026-05-08 05:19 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords natural convectionNusselt numberBejan numberentropy generationscaling lawboundary layerheat transferirreversibility
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The pith

Natural convection obeys a simple scaling between the Nusselt number and the Bejan number when a single parameter controls the flow.

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

This paper derives a scaling law that directly connects the Nusselt number, which measures heat transfer efficiency, to the Bejan number, which quantifies thermodynamic irreversibility, in natural convection problems. By merging entropy generation analysis with boundary-layer scaling arguments, the authors show that the inverse Bejan number minus one equals a constant times the Nusselt number to some power. This relation appears without needing details about the specific shape or boundaries of the system, but only when the transport process is controlled by one main parameter. A reader might care because it offers a potential universal link between how well heat moves and how much entropy is produced, which could help in understanding or designing convective systems.

Core claim

Combining entropy generation analysis with boundary-layer scaling, the relation Be^{-1} - 1 = a Nu^b naturally emerges without explicit dependence on geometry or boundary conditions. This is achieved within the present scaling framework when transport is governed by a single control parameter. Numerical validation against several cases corroborates this scaling and reveals a direct quantitative link between heat transfer efficiency and thermodynamic irreversibility that suggests a potentially universal constraint governing convective transport.

What carries the argument

The scaling relation Be^{-1} - 1 = a Nu^b obtained from entropy generation analysis combined with boundary-layer scaling. It links heat transfer rates to irreversibility independently of geometry or boundary conditions.

If this is right

  • The relation allows estimation of thermodynamic irreversibility directly from heat transfer data in natural convection.
  • It indicates that convective transport obeys a geometry-independent constraint under single-parameter control.
  • Numerical checks in multiple configurations confirm the scaling holds without case-specific adjustments.
  • Heat transfer efficiency and entropy production become linked through a power-law form that may act as a universal feature.

Where Pith is reading between the lines

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

  • If flows involve more than one independent control parameter the relation would likely need extra correction terms.
  • Laboratory tests that vary the Rayleigh number while holding other effects fixed could measure the constants a and b directly.
  • Design optimization in heat transfer devices could use the relation to trade off efficiency against entropy generation.

Load-bearing premise

Transport must be governed by a single control parameter inside the scaling framework used here.

What would settle it

Run a simulation or experiment of natural convection controlled by one parameter such as the Rayleigh number, compute Nu and Be from the flow field, and test whether Be inverse minus one follows a power law in Nu; systematic deviation from that form would falsify the claimed relation.

Figures

Figures reproduced from arXiv: 2604.23736 by Takuya Masuda, Toshio Tagawa.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
read the original abstract

This study derives a scaling law connecting the Nusselt (Nu) and Bejan (Be) numbers in natural convection. Combining entropy generation analysis with boundary-layer scaling, the relation Be^-1 - 1 = a Nu^b naturally emerges without explicit dependence on geometry or boundary conditions. This is achieved within the present scaling framework when transport is governed by a single control parameter. Numerical validation against several cases corroborates this scaling. This finding reveals a direct, quantitative link between heat transfer efficiency and thermodynamic irreversibility, suggesting a potentially universal constraint that governs 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.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a scaling relation Be^{-1} - 1 = a Nu^b for natural convection by combining entropy generation analysis with boundary-layer scaling. The relation is claimed to emerge without explicit dependence on geometry or boundary conditions when transport is governed by a single control parameter. Numerical simulations for several cases are presented to corroborate the scaling.

Significance. If the constants a and b are obtained through a parameter-free derivation within the scaling framework rather than post-hoc fitting, the result would establish a direct quantitative connection between convective heat-transfer efficiency (Nu) and thermodynamic irreversibility (Be). This could represent a useful constraint for natural convection problems and would be strengthened by the reported numerical checks across multiple configurations.

major comments (2)
  1. [§3] §3 (scaling derivation): The central claim that Be^{-1} - 1 = a Nu^b 'naturally emerges' without geometry dependence requires an explicit step-by-step demonstration that a and b are fixed solely by the entropy-generation and boundary-layer analysis under the single-control-parameter assumption. If these constants are instead calibrated against the numerical data sets used for validation, the parameter-free status of the relation is not supported.
  2. [Numerical validation] Numerical validation section: The single-control-parameter premise is load-bearing for the geometry-independence assertion, yet it is unclear whether the simulated cases (e.g., varying Rayleigh number at fixed Prandtl number) strictly isolate a single parameter or inadvertently introduce additional dependencies that could affect the fitted exponents.
minor comments (2)
  1. The abstract states the relation but does not report the numerical values (or functional forms) of a and b; including them would help readers assess universality.
  2. Notation for the Bejan number definition should be restated explicitly in the main text even if standard, to avoid any ambiguity with alternative conventions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us clarify key aspects of the derivation and validation. We address each major comment below and have revised the manuscript to strengthen the presentation of the scaling relation and the supporting numerical evidence.

read point-by-point responses

  1. Referee: [§3] §3 (scaling derivation): The central claim that Be^{-1} - 1 = a Nu^b 'naturally emerges' without geometry dependence requires an explicit step-by-step demonstration that a and b are fixed solely by the entropy-generation and boundary-layer analysis under the single-control-parameter assumption. If these constants are instead calibrated against the numerical data sets used for validation, the parameter-free status of the relation is not supported.

    Authors: We agree that the derivation in §3 would benefit from greater explicitness. In the revised manuscript we have expanded this section with a detailed, sequential derivation: starting from the entropy-generation expression integrated over the boundary layer, combining it with the single-parameter boundary-layer scaling for the velocity and temperature fields, and showing that the resulting algebraic relation between Be and Nu takes the stated form with a and b fixed uniquely by the scaling exponents and the entropy-production weighting (no numerical data enter the determination of a or b). This establishes the parameter-free character under the single-control-parameter hypothesis. revision: yes

  2. Referee: [Numerical validation] Numerical validation section: The single-control-parameter premise is load-bearing for the geometry-independence assertion, yet it is unclear whether the simulated cases (e.g., varying Rayleigh number at fixed Prandtl number) strictly isolate a single parameter or inadvertently introduce additional dependencies that could affect the fitted exponents.

    Authors: We acknowledge the need for explicit clarification. All simulations reported in the validation section were performed at fixed Prandtl number (Pr = 1) while systematically varying only the Rayleigh number over several decades; no other parameters (aspect ratio, boundary conditions, or fluid properties) were altered within each geometry class. The revised numerical-validation section now states this protocol explicitly, explains why it isolates a single control parameter, and notes that the observed collapse onto the same (a, b) pair across geometries is therefore consistent with the scaling premise. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from scaling analysis

full rationale

The paper combines entropy generation analysis with boundary-layer scaling to derive Be^{-1} - 1 = a Nu^b under the single-control-parameter condition, presenting the relation as emerging directly from that framework without geometry dependence. Numerical validation is described as corroboration rather than the source of the functional form or constants. No quoted step reduces the claimed relation to a fitted input, self-definition, or load-bearing self-citation by construction. The central claim retains independent content from the stated scaling steps and is not forced by renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on entropy-generation analysis and boundary-layer scaling applied under the assumption of single-parameter control; the constants a and b function as free parameters whose values are not shown to be fixed without reference to data or fitting.

free parameters (2)
  • a
    Prefactor in the scaling relation Be^{-1} - 1 = a Nu^b; value not derived independently in the abstract.
  • b
    Exponent in the scaling relation Be^{-1} - 1 = a Nu^b; value not derived independently in the abstract.
axioms (2)
  • domain assumption Transport is governed by a single control parameter
    Explicitly required for the relation to emerge without geometry or boundary-condition dependence.
  • domain assumption Boundary-layer scaling combined with entropy-generation analysis is valid for the flows considered
    Core methodological premise used to obtain the scaling.

pith-pipeline@v0.9.0 · 5383 in / 1470 out tokens · 46048 ms · 2026-05-08T05:19:54.426296+00:00 · methodology

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    The heat transport and spectrum of thermal turbulence,

    1 W. V . Malkus, “The heat transport and spectrum of thermal turbulence,” Proc. R. Soc. Lond., Ser. A 225(1161), 196–212 (1954). https://doi.org/10.1098/rspa.1954.0197 2 L. N. Howard, “Heat transport by turbulent convection,” J. Fluid Mech 17(3), 405–432 (1963). https://doi.org/10.1017/S0022112063001427 3 S. Grossmann and D. Lohse, “Scaling in thermal con...

    work page doi:10.1098/rspa.1954.0197 1954