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Efficiency at the maximum power of the power law dissipative Carnot-like Heat engines with non-adiabatic dissipation

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Pith's one-line read Adding non-adiabatic dissipation to a power-law dissipative Carnot-like heat engine leaves the universal bounds on efficiency at maximum power unchanged.

desk verdict Modest but plausible extension of the power-law dissipation universality bounds to engines with non-adiabatic friction; the main result survives the algebra, but the manuscript's equation typos and the unexamined equal-exponent assumption need work. read the letter →

arxiv 1909.02424 v2 pith:7M7N2W3S submitted 2019-09-04 cond-mat.stat-mech

M. Ponmurugan This is my paper · ORCID
classification cond-mat.stat-mech PACS 05.70.-a05.70.Ln
keywords efficiencyatmaximumpowerpower-lawdissipationnon-adiabaticCarnot-likeheatengineuniversalboundsfinite-timethermodynamicslow-dissipationmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether internal friction in the finite-time adiabatic branches of a Carnot-like heat engine changes the efficiency at maximum power. It extends the power-law dissipation model by giving the adiabatic branches the same power-law irreversible entropy production as the isothermal branches. The paper finds that these extra dissipative terms only renormalize a lumped ratio parameter and do not alter the universal bounds: $\eta_C/(\delta+1) \le \eta_P \le \eta_C/((\delta+1)-\delta\eta_C)$. A sympathetic reader would care because it says the standard efficiency limits for finite-time engines survive a realistic source of friction that earlier derivations ignored.

What carries the argument

The machinery is the power-law irreversible entropy production $\Delta S^{\mathrm{irr}}_i = \alpha_i(\sigma_i/t_i)^{1/\delta}$ imposed on every branch, hot, cold, and both adiabats. Because each branch shares the same exponent $\delta$, the optimal-time condition gives a ratio relation among branch times that forces all non-adiabatic contributions into the single ratio parameter $\zeta = 1 + \varsigma(T_c/T_h)^{\delta/(\delta+1)}$. This collapse is what makes the bounds independent of the adiabatic dissipation strengths.

What would settle it

Measure the irreversible entropy production of a driven finite-time adiabatic process as a function of its duration; if the exponent in its power-law scaling differs from the isothermal exponent $\delta$, then the ratio relation in Eq. (11) changes and the bounds in Eq. (16) are not guaranteed. Comparing such measurements for isothermal and adiabatic branches of the same engine would settle the claim.

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Extended reading notes

Core claim

The central claim is that non-adiabatic dissipation does not influence the universal minimum and maximum bounds on efficiency at maximum power in power-law dissipative Carnot-like heat engines. The derivation adds entropy production terms $\Delta S^{\mathrm{irr}}_j = \alpha_j(\sigma_j/t_j)^{1/\delta}$ for the two finite-time adiabatic branches and optimizes power with respect to all four branch times. The resulting efficiency at maximum power depends on the power-law exponent $\delta$ and a ratio parameter $\zeta$ that absorbs the cold-isotherm and adiabatic dissipation coefficients; in the asymmetric dissipation limits one recovers exactly the lower bound $\eta_C/(\delta+1)$ and the upper bound $\eta_C/((\delta+1)-\delta\eta_C)$, the same bounds as in the model without non-adiabatic dissipation.

Load-bearing premise

The load-bearing premise is that the extra entropy produced during the finite-time adiabatic branches follows the same power-law dependence on time, with the same exponent $\delta$, as the isothermal branches; if adiabatic friction scales differently, the ratio relation used to derive the bounds no longer holds.

Editorial extensions

If this is right

  • For engines whose adiabatic branches take finite time and dissipate internally, the same universal lower and upper bounds on efficiency at maximum power apply as for instantaneous adiabats.
  • The non-adiabatic dissipation coefficients enter only through the lumped parameter $\zeta$, so within the bounds the efficiency can be tuned by changing isothermal and adiabatic dissipation ratios.
  • In the completely symmetric case with $\delta = 1$, the efficiency at maximum power reduces to the known stochastic heat engine result under an appropriate tuning of the parameters.
  • The expansion of $\eta_P$ in powers of $\eta_C$ retains the same universal form, so the leading coefficient $1/(\delta+1)$ is unaffected by non-adiabatic dissipation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the adiabatic entropy production were instead to scale with a different exponent than the isothermal branches, the clean collapse into $\zeta$ would fail; testing this would require an experimental or numerical measurement of adiabatic friction scaling.
  • The same style of argument suggests that adding further dissipative branches that share the same power-law form would again only renormalize $\zeta$ and leave the bounds intact, but this is an extrapolation the paper does not make.
  • A practical consequence, not drawn by the paper, is that efforts to beat the Curzon-Ahlborn-type bounds should target the dissipation-law exponent or break the ratio symmetry rather than merely reduce adiabatic friction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies a finite-time Carnot-like heat engine with power-law dissipation, now including non-adiabatic dissipation during the adiabatic branches. Each irreversible entropy production term is assumed to scale as ΔS_irr_i = α_i(σ_i/t_i)^{1/δ}, with the same exponent δ for isothermal and adiabatic branches. The author derives an expression for the efficiency at maximum power, ηP, and argues that the non-adiabatic dissipation does not affect the universal bounds ηC/(δ+1) ≤ ηP ≤ ηC/((δ+1)−δηC), nor the universal expansion in powers of ηC, previously obtained without non-adiabatic dissipation. The paper also discusses symmetric and asymmetric limits and compares with observed power-plant efficiencies.

Significance. If correct, the result would extend the known universality of efficiency-at-maximum-power bounds to engines with finite-time adiabatic branches, a natural and worthwhile question given prior low-dissipation results. The paper provides an explicit analytical model rather than a numerical fit, and the combination ς in Eq. (14) is a clean object that controls the corrections. However, the printed derivation contains several algebraic errors that invalidate the equations as written, and the central conclusion rests on an unexamined assumption about the adiabatic exponent. With the corrected algebra the advertised bounds do follow, so the contribution is valuable after a careful revision.

major comments (4)
  1. [Section 2, Eq. (10)] The optimal times in Eq. (10) do not satisfy the first-order conditions of Eq. (9). For the dissipation terms x_i = α_i T_i(σ_i/t_i)^{1/δ}, stationarity gives t_i = [B_i S_i (1+1/δ)/A]^δ, with B_i = α_i T_i σ_i^{1/δ}, A = (T_h−T_c)ΔS, and S_i = 1 + Σ_{j≠i}(B_j/B_i)^{δ/(δ+1)}. Equation (10) instead places B_i in the numerator divided by A(1+1/δ)S_i, i.e., the factor (1+1/δ) and the bracket S_i are inverted relative to the correct expression. Consequently, the ratios t_j/t_h obtained from Eq. (10) do not reduce to Eq. (11); with the corrected times, the relation (t_j/t_h)^{1+1/δ} = (α_j T_c)/(α_h T_h)(σ_j/σ_h)^{1/δ} does hold.
  2. [Section 2, Eq. (12)] Equation (12) is not the efficiency at maximum power that follows from Eqs. (8), (10), and (11). The correct expression is ηP = [ηC/(δ+1)] / [1 − ηC/((1+1/δ)ζ)] = ηC/[(1+δ) − δηC/ζ], with ζ defined in Eq. (13). The printed formula has the numerator (1+1/δ)ηC instead of ηC/(δ+1). As a result, the printed Eq. (12) gives ηP → (1+1/δ)ηC when ς→∞ and ηP → (δ+1)^2ηC/[δ(δ+1−δηC)] when ς→0, rather than ηC/(δ+1) and ηC/(δ+1−δηC) as claimed in Section 3. The corrected formula does yield the bounds in Eq. (16), but the derivation as printed does not support them.
  3. [Section 2, Eq. (15)] The expansion in Eq. (15) is not the Taylor expansion of ηP because ζ in Eq. (13) depends on ηC through (1−ηC)^{δ/(δ+1)}. While the first two coefficients in Eq. (15) agree with the corrected expression to order ηC^2, the coefficient of ηC^3 is δ^2/[(δ+1)^3(1+ς)], not δ^2/[(δ+1)^3(1+ς)^2]. The printed series therefore overstates the universality of the expansion coefficients beyond leading order and should be recomputed from the correct ηP formula.
  4. [Section 2, assumption before Eq. (3)] The central modeling assumption that the non-adiabatic entropy production obeys the same power-law exponent δ as the isothermal branches, ΔS_irr_j = α_j(σ_j/t_j)^{1/δ} for j=a,b, is asserted without derivation or discussion. Reference [49], cited by the author, reports a 1/τ^2 scaling for a finite-time adiabatic process, which corresponds to δ=1/2, while the 1/τ scaling of the isothermal branches corresponds to δ=1. If the adiabatic exponent differs, the sum in Eq. (3) does not collapse to the single-ς structure that produces Eq. (16), and the advertised independence from non-adiabatic dissipation is not established.
minor comments (4)
  1. [Eq. (4)] The summation index is printed as 'j=,c,a,b' in Eq. (4); it should read 'j=c,a,b'.
  2. [Section 3] The word 'preent' in the sentence announcing the second main result should be 'present'.
  3. [Section 2, Eq. (8)] The intermediate algebra between Eqs. (7) and (8) is not displayed clearly; Eq. (8) is hard to parse and should be rewritten with explicit parentheses.
  4. [Section 3] The tuning condition α_j/α_h = (1/3)T_h/T_c that reduces Eq. (17) to the stochastic heat engine result for δ=1 is stated without derivation; a brief explanation or citation to the relevant step would improve reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the bounds in Eq. (16) follow from an explicit optimization of the stated model, not from fitting or from a self-citation chain.

full rationale

The paper's central result is a direct mathematical consequence of its stated model. The non-adiabatic dissipation terms are added to the cold-reservoir heat expression in Eq. (3), and the power is optimized explicitly: the first-order conditions yield the time ratios in Eq. (11), and substitution into Eq. (8) gives the closed-form efficiency at maximum power in Eq. (12). The universal bounds in Eq. (16) are obtained by taking the limits ς→∞ and ς→0, realized respectively by σ_h→0 and σ_h→∞ for fixed finite positive σ_j (j=c,a,b). This is a derivation from the model's definitions, not a fit or a renamed empirical pattern. The paper scans rather than fits the parameters ς and δ in Fig. 1, so no fitted input is relabeled as a prediction. The self-citations (notably ref. [43], the author's own prior work on the power-law model without non-adiabatic dissipation) supply the baseline result and motivation, but the new derivation does not assume Eq. (16); it reproduces it from the extended model. The main caveat—that adiabatic entropy production is assumed to obey the same power-law exponent δ as the isothermal branches—is an explicit modeling assumption stated in Section 2, and is a matter of model correctness rather than circularity. No uniqueness theorem is imported, and no load-bearing claim reduces to a self-citation by construction. Overall, the paper is self-contained for the stated model, with only minor non-load-bearing self-citation.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The model uses the author's earlier power-law dissipation framework with additional adiabatic friction terms. No new physical entities are introduced; the only additions are model parameters α_j and σ_j for the adiabatic branches, which are combined into ς.

free parameters (2)
  • δ = None (model parameter); Figure 1 scans 0.6 to 1.75
    Power-law dissipation exponent; not fitted in the derivation, but chosen in the figure to encompass observed power-plant efficiencies.
  • ς (dissipation ratio combination) = None (model parameter); Figure 1 scans 0 to 4
    Combination of dissipation coefficients and tuning parameters, Eq. (14); not fitted, but scanned in the figure to match observed efficiencies.
assumptions (3)
  • domain assumption Heat exchange formulas in Eqs. (1)-(3) with additive irreversible entropy production terms and zero net entropy change of the working substance per cycle.
    Invoked at the start of Section 2; the model is built on this low-dissipation ansatz.
  • ad hoc to paper Non-adiabatic entropy production obeys the same power-law time scaling as the isothermal branches, ΔS_irr_j = α_j(σ_j/t_j)^(1/δ), j=a,b.
    Introduced in Section 2 without physical derivation; this is the load-bearing extension of the prior model and directly controls whether the bounds survive.
  • standard math The power maximum is an interior maximum and the first-order conditions in the four process times are valid.
    Needed for Eq. (10); the paper does not prove the optimum is interior or that the stationary point is a maximum.

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Pith. "Pith review of Efficiency at the maximum power of the power law dissipative Carnot-like Heat engines with non-adiabatic dissipation." pith.science (2026). https://pith.science/paper/7M7N2W3S

@misc{pith2026190902424,
  author       = {Pith},
  title        = {Pith review of: Efficiency at the maximum power of the power law dissipative Carnot-like Heat engines with non-adiabatic dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7M7N2W3S}},
  note         = {Machine review of arXiv:1909.02424}
}
read the original abstract

We study the efficiency at maximum power of non-adiabatic dissipative (internally dissipative friction in finite time adiabatic processes) Carnot-like heat engines operate in finite time under the power law dissipation regime. We find that the non-adiabatic dissipation does not influence the universal minimum and maximum bounds on the efficiency at maximum power obtained in the generalized dissipative Carnot-like heat engines which does not take in to account the non-adiabatic dissipation.

Figures

Figures reproduced from arXiv: 1909.02424 by the authors.

Figure 1
Figure 1. Efficiency at maximum power ηP plotted as a function of ηC for different values of δ and ς. Top (dot-dashed line): for δ = 0.6, ς = 0, Middle (dotted line): δ = 1 and ς = 2 and bottom (dashed line): δ = 1.75, ς = 4. The observed efficiencies of the various thermal power plants are shown in circles [37, 39, 52]. Solid line represents ηCA. References [1] R. S. Berry, V. A. Kazakov, S. Sieniutyez, Z. Szwast and A. M. T… view at source ↗

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