The $P{\Phi}$-Compromise Function as a criterion of merit to optimize irreversible thermal engines

L. A. Arias-Hernandez; S. Levario-Medina

REVIEW 4 major objections 5 minor 1 cited by

The $P{\Phi}$-Compromise Function as a criterion of merit to optimize irreversible thermal engines

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The PΦ-compromise function, used directly as an objective function, reproduces the optima of three generalized heat-engine criteria.

desk verdict The conceptual point is modest and plausible, but the paper's own closed-form optima contradict its endoreversible limit, leaving the central equivalence unverified. read the letter →

arxiv 1908.11861 v1 pith:7Z2QD3JC submitted 2019-08-30 cond-mat.stat-mech physics.app-ph

S. Levario-Medina , L. A. Arias-Hernandez This is my paper · ORCID

classification cond-mat.stat-mechphysics.app-ph

keywords thermodynamicoptimizationirreversibleheatenginescompromisefunctionecologicalomegaefficientpowerleaknon-endoreversiblemodel

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 argues that the PΦ-compromise function—the difference between power and dissipation, each normalized by its value at the maximum-power regime—can be used directly as an objective function for an irreversible heat engine. The authors show that maximizing this single function over the high reduced temperature (the ratio of the working fluid's hot-side temperature to the hot reservoir temperature) gives exactly the same operating point as maximizing the generalized ecological function, the generalized $\Omega$ function, and the k-efficient power, when each generalization parameter is selected by the same compromise function. If true, the result means these four optimization criteria select the same power, dissipation, and efficiency, despite their different algebraic forms. The equivalence is demonstrated for two heat-engine models with internal irreversibility and heat leakage, and it reduces to the classic endoreversible result in appropriate limits. Because the compromise function requires no second optimization step to tune a parameter, it offers a simpler route to the same trade-off regime.

What carries the argument

The load-bearing object is the PΦ-compromise function, C_{PΦ}(a_h)=P(a_h)/$P^{{MP}}$−Φ(a_h)/$Φ^{{MP}}$, a dimensionless scalar that rewards power output and penalizes dissipation, each measured relative to the maximum-power regime. It works because the power and dissipation curves in the two models have shapes that give this normalized difference a unique interior maximum, and because the same normalized comparison is what fixes the generalization parameters ε, λ, and k in the ecological, $\Omega$, and k-efficient power families. The internal irreversibility is carried by a single scalar in each model: the non-endoreversibility factor R in one, and the ratio r=σ_i/α in the other, both assumed to act through the heat rejected to the cold reservoir. In the zero-heat-leak and zero-irreversibility limits, all expressions reduce to the endoreversible two-reservoir engine, preserving the known 75–25 result.

What would settle it

Add to either model an internal entropy production term that does not share this scaling, for instance σ_i=(1−R)Q_2/T_{2w}+c(Q_1/T_{1w})^2, or make R depend on temperature, and recompute the maxima. The paper's claim predicts that the maximizing high reduced temperature of C_{PΦ} still equals the maximizing high reduced temperatures of E_G, Ω_G, and Pη_k with compromise-selected parameters; any parameter set where these argmax differ would show the equivalence is an artifact of the two one-parameter irreversibility models.

Watch

Extended reading notes

Core claim

The central claim is that C_{PΦ}(a_h)=P(a_h)/$P^{{MP}}$−Φ(a_h)/$Φ^{{MP}}$, evaluated as a function of the high reduced temperature a_h, has a maximum that coincides with the optima obtained from the generalized ecological function E_G, the generalized $\Omega$ function Ω_G, and the k-efficient power Pη_k after their parameters ε, λ, and k are fixed by the compromise function. The paper proves this by deriving the maximizing high reduced temperature $a_h^{{M C_{PΦ}}$} for two irreversible models—a non-endoreversible model with heat leak and an irreversible model based on the uncompensated heat concept—and showing algebraically that the same a_h emerges from each generalized function when its parameter is chosen through the same compromise procedure. Consequently, at this optimum the engine's power, dissipation, efficiency, and entropy production are identical in all four regimes. The paper also shows that the 75–25 balance between power and dissipation, previously tied to the ecological function, is a property of the compromise function itself in the endoreversible limit.

Load-bearing premise

The argument rests on assuming that all internal irreversibilities can be captured by a single scalar that scales with the heat rejected to the cold reservoir; if real engine losses grow differently with operating conditions, the four optima need not coincide.

Editorial extensions

If this is right

Thermodynamic optimization of these engines can be performed in a single step: maximize C_{PΦ} over a_h, with no second optimization to tune a generalization parameter.
The generalized ecological, Omega, and k-efficient power regimes are the same operating point in these models, so choosing among them is not choosing among different power–dissipation trade-offs.
The 75–25 benchmark is a feature of the compromise function itself in the endoreversible limit, not a special property of the ecological criterion.
The equivalence holds for both a non-endoreversible model with heat leak and an irreversible model with uncompensated heat, strengthening the case that it is a property of the optimization procedure rather than of one particular irreversibility model.

Reading between the lines

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

Editorial inference: because C_{PΦ} is a normalized scalarization, the same one-function shortcut may reproduce the optima of other one-parameter families of objective functions in any model where P(a_h) and Φ(a_h) have the same convex–concave structure.
Editorial inference: a natural test is to replace the Newtonian heat-transfer law with a radiative or other non-linear law and check numerically whether the argmax coincidence survives; the algebra would change even if the equivalence does not.
Editorial inference: within these models, the debate about which optimization criterion is 'best' is effectively moot, since the criteria studied all land on the same operating point; the interesting choice is the trade-off ratio embodied in the normalization itself.
Editorial inference: the function also suggests a practical control strategy: measure power and dissipation at the maximum-power operating point, then run the engine at the a_h that maximizes the normalized difference, without needing to know the generalization parameters.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the idea is a natural one — take the compromise function C_{PΦ} as the objective directly, instead of using it only to pick parameters of the generalized ecological, omega, and efficient-power functions. If it worked, it would simplify a small corner of finite-time thermodynamics. The paper, though, doesn't demonstrate that it does. The closed-form optima in Eq. (32) and Eq. (43) fail the stated endoreversible limit, and since the appendices lean on exactly those formulas, the main claim is currently unsupported.

To its credit, the paper is honest about what it is doing: it extends the authors' earlier work on generalized objective functions. The 75-25 corollary is clearly stated. The two models (NEHL and IHL) are standard, and the idea that all three generalized functions collapse to one operating regime is a conceptually pleasing statement. The paper is not a repackaging of a known result; using C_{PΦ} directly as an objective function is a real, if small, simplification.

The load-bearing problem is internal. For the IHL model, Eq. (43) with r=0 and δ=0 gives a_h=1, while Table 1 promises a_h=(γ+τ^{1/4})/(1+γ); for γ=3, τ=0.5 that's 1 versus 0.960. For the NEHL model, Eq. (32) with R=1 and δ=0 leaves an α dependence even though P and Φ each scale linearly with α in that limit, so the optimum should be α-independent. No derivation of these formulas is supplied, only closed-form statements, so this is not a peripheral typo. Because the appendices establish the claimed equivalence by substituting these exact a_h formulas, the error propagates into the central result.

There is also a whiff of circularity: the generalization parameters are selected by maximizing the same compromise function, so it is not surprising that maximizing C directly gives the same regime. That does not kill the paper's contribution, but it lowers its significance. The manuscript is also badly copy-edited — Eq. (30) is garbled, and Appendix B has multiple typos — which makes verification harder. These issues are fixable, but they are not harmless.

Who is this for? Specialists in finite-time thermodynamics who care about objective functions for Curzon-Ahlborn-type engines. If the algebra is repaired, the paper is a modest but useful technical note. In its current form, it should not be accepted, but it deserves a serious referee. I would send it to peer review with an explicit request to verify Eqs. (32) and (43), including their endoreversible limits, and to supply the missing derivations.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes the P-Phi Compromise Function, C_{P Phi}(a_h) = P(a_h)/P^{MP} - Phi(a_h)/Phi^{MP}, as a direct objective function for optimizing irreversible thermal engines. For two Curzon-Ahlborn-type models with internal irreversibility and heat leakage (the NEHL and IHL models), the authors claim that maximizing C_{P Phi} reproduces the optimal regimes of the generalized ecological function, the generalized Omega function, and the k-efficient power, all with the same optimal high reduced temperature and with the 75/25 power-dissipation trade-off. The appendices purport to show that the same high reduced temperature results when the generalization parameters are selected through the Compromise Function. The central claim is therefore a unification of several finite-time-thermodynamics objective functions under one simpler criterion.

Significance. If the claimed equivalence were established rigorously, the paper would provide a genuine conceptual simplification: instead of optimizing a family of generalized objective functions and then selecting a family parameter through a second variational principle, one could optimize the single compromise function directly. The 75/25 trade-off and the unification across ecological, Omega, and efficient-power criteria are attractive and physically interpretable. The paper also formulates the claim in two nontrivial irreversible models, not just the endoreversible case, which is a reasonable scope for the journal. However, the manuscript as submitted does not provide verifiable derivations of its central closed-form optima: several displayed formulas are inconsistent with their own stated endoreversible limits, and no reproducible symbolic or numerical check is supplied. The strength of the contribution therefore depends on corrections that have not yet been made.

major comments (4)

[Section 3, Eq. (43)] Equation (43), the claimed maximizing high reduced temperature for the IHL model, does not reduce to the stated endoreversible no-leak limit. Setting r=0 and delta=0 in Eq. (43) gives a_h=(gamma+1)/(gamma+1)=1, whereas Table 1 and the text state the endoreversible result a_h=(gamma+tau^{1/4})/(1+gamma). For gamma=3 and tau=0.5 these values are 1 and 0.960, respectively. Because Eq. (43) is the defining result used in Section 3 and in Appendix B to identify the optimal regime, this inconsistency is load-bearing and must be resolved by rederiving the maximization.
[Section 2, Eq. (32)] Equation (32) fails the corresponding endoreversible limit R=1, delta=0. Direct substitution gives a_h = gamma/(1+gamma) + alpha sqrt(tau)(1-sqrt(tau))/(1+gamma), which depends on the thermal conductance alpha. In this limit both P and Phi are proportional to alpha, so their ratios in C_{P Phi}=P/P^{MP}-Phi/Phi^{MP} are alpha-independent and the optimizer cannot depend on alpha. The expression also disagrees with the Table 1 limit (gamma+tau^{1/4})/(1+gamma). Since Eq. (32) is the formula substituted into the generalized-regime calculations in Appendix A, the claimed equivalence of the three generalized objective functions is not supported until this formula is corrected.
[Sections 2 and 3, Eqs. (29)-(33) and (41)-(43)] The closed-form maximizers are asserted without derivation. The manuscript does not show the first-order condition dC_{P Phi}/da_h=0, nor does it provide any reproducible numerical or symbolic verification, and the displayed polynomial expressions contain visible defects (for example Eq. (30) begins with '= . [' and is not a complete expression). Given that two of the three explicit limit checks above fail, the reader cannot distinguish a transcription error from a genuine mistake in the optimization. The authors should supply a complete derivation or release a machine-checkable computation for the maximizers.
[Appendices A and B] The equivalence argument is only asserted through substitutions such as 'when these values are substituted ... the high reduced temperature which arise is the same,' without displaying the algebra or the first-order conditions of the generalized functions. Because the generalization parameters are themselves selected by maximizing the Compromise Function, the claimed coincidence of regimes could in part be enforced by construction; the manuscript needs an explicit statement showing that the same a_h satisfies the stationarity conditions of C_{P Phi}, E_G, Omega_G, and P_eta_k, rather than a statement that their numerically evaluated maxima coincide for the selected parameters.

minor comments (5)

[Equation (30)] Equation (30) is syntactically incomplete: it reads '= . [' and the following expression has unbalanced parentheses; this is not merely cosmetic because C_n1 and C_n0 enter Eq. (29).
[Table 1] Table 1 is typeset as an unreadable fragment rather than a proper table; it should give clear definitions of C_{P Phi}, a_h^{M C P Phi}, and the endoreversible-limit values with labeled columns.
[Notation] The argument lists of Eqs. (32) and (33) include a_h on the left-hand side's variable list even though the right-hand sides are supposed to define the maximizing a_h; this notation should be cleaned up, and T1 appears in the argument list of Eq. (32) although the expression does not depend on it.
[Figures 4 and 5] The captions of Figures 4 and 5 refer to 'the same value of their generalization parameters (epsilon=lambda=k=2)' and to compromise functions evaluated at the selected parameters, but the axes and curves are not described; the figures should be self-contained.
[Language and references] The manuscript contains many typos and grammatical errors (e.g., 'whit kappa =', 'fucntion', 'Eifucntion is the energy'), and several central algebra steps are delegated to Refs. [14] and [15], which are master's theses; the needed intermediate results should be restated in the paper for verifiability.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed equivalence of the generalized ecological, Omega, and efficient-power optima with the CPΦ optimum is built into the parameter-selection rule: those parameters are chosen by maximizing the very same CPΦ function, so substituting them back restates the selection criterion rather than providing an independent derivation.

self definitional [Appendix A (NEHL) and Appendix B (IHL), around Eqs. (66)-(68), (73), and (102); with Eq. (7) in Section 1.]
"To chose a value of each of the generalized parameters, in 2001 [12], 2006 [14] and 2016 [15] the compromise function (equation (3)) was used... values of ǫ, λ and k which maximize the compromise function in each case... When these values are substituted in their corresponding ah, the high reduced temperature which arise is the same... Which is just the same high reduced temperature that is get when the compromise function is used directly like PΦ–Compromise Function (equation (43))."

The compromise function of Eq. (3) has exactly the same functional form as the new objective CPΦ of Eq. (7), namely P/P^MP − Φ/Φ^MP, the only difference being that Eq. (3) is evaluated at the maximum of a generalized function. The parameters ǫ_MC, λ_MC, and k_MC are defined by maximizing this compromise function after inserting the corresponding generalized-function maximum. Thus the selection rule already optimizes CPΦ along the parametric curve a(parameter). Substituting the selected parameter back into a(parameter) and observing that the result equals Eq. (32) or Eq. (43) is therefore a formal restatement of the selection rule, not an independent proof that the three generalized regimes coincide with the CPΦ regime.

full rationale

The paper defines CPΦ(a_h)=P(a_h)/P^MP−Φ(a_h)/Φ^MP as a direct objective and then shows that the maxima of the generalized ecological, Omega, and efficient-power functions, after their parameters are chosen by the compromise function, have the same a_h. That demonstration is circular in a specific, quotable sense: the parameters are selected by maximizing the very same compromise function that CPΦ generalizes, so the subsequent substitution (e.g., Eq. (66) into the MEG optimum) returns the maximizer of CPΦ along the parameterized curve. The equality of the three generalized regimes is thus a consequence of applying the same selection rule to three reparameterizations of the same CPΦ objective, rather than an independent discovery. There is some independent content: the paper verifies by algebra that the substitution yields the claimed closed forms, and the 75-25 property is a separate corollary. However, the load-bearing claim that the direct CPΦ optimum is a special case of the generalized optima reduces by construction. A separate algebraic concern—Eq. (43) fails to reduce to the endoreversible no-leak limit—is a correctness issue, not a circularity, and does not affect this score. Overall, partial circularity, scored 6.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central claim rests on the linear heat-transfer model, the phenomenological irreversibility parameterizations, and the postulated compromise function. The generalization parameters epsilon, lambda, k are free in their families and are fixed by the compromise-function selection rule, which is the mechanism that makes the regimes coincide.

free parameters (3)

Generalization parameter epsilon (ecological) = Eq. (66) NEHL; Eq. (96) IHL
Chosen by maximizing the compromise function C; not measured. The equivalence of the ecological regime with C's maximum depends on this selection.
Generalization parameter lambda (omega) = Eq. (67) NEHL; Eq. (97) IHL
Chosen by maximizing C; the omega regime coincides with C's maximum only at this value.
Generalization parameter k (efficient power) = Eq. (68) NEHL; Eq. (98) IHL
Chosen by maximizing C; the efficient-power regime coincides with C's maximum only at this value.

assumptions (5)

domain assumption Newtonian heat transfer law for all heat fluxes (Eqs. 13-15): Q1=alpha(T1-T1w), Q2=beta(T2w-T2), Qhl=delta(T1-T2)
The explicit forms of P and Phi depend on this linear law. If heat transfer were e.g. radiative, the optimization would change.
domain assumption Non-endoreversibility hypothesis: sigma_i = (1-R)Q2/T2w (Eq. 20)
Phenomenological single-parameter description of internal irreversibilities; R=1 recovers the endoreversible limit.
domain assumption Uncompensated heat relation: ac = 1 + gamma(1-r) - gamma/a_h (Eq. 34) for the IHL model
Defines the relation between reduced temperatures using r = sigma_i/alpha; another phenomenological single-parameter model.
domain assumption Dissipation is defined as Phi = T2 * sigma_T (Eq. 16)
This Angulo-Brown definition is adopted without justification; alternative dissipation measures would change C.
ad hoc to paper The compromise function C is a valid criterion of merit
The paper postulates C as the objective function; its desirability is argued via the 75-25 trade-off, not derived from a more fundamental principle.

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Pith. "Pith review of The $P{\Phi}$-Compromise Function as a criterion of merit to optimize irreversible thermal engines." pith.science (2026). https://pith.science/paper/7Z2QD3JC

@misc{pith2026190811861,
  author       = {Pith},
  title        = {Pith review of: The $P\Phi$-Compromise Function as a criterion of merit to optimize irreversible thermal engines},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Z2QD3JC}},
  note         = {Machine review of arXiv:1908.11861}
}

abstract

Several authors have proposed out of equilibrium thermal engines models, allowing optimization processes involving a trade off between the power output of the engine and its dissipation. These operating regimes are achieved by using objective functions such as the ecological function ($EF$). In order to measure the quality of the balance between these characteristic functions, it was proposed a relationship where power output and dissipation are evaluated in the above mentioned $EF$-regime and they are compared with respect to its values at the regime of maximum power output. We called this relationship "Compromise Function" and only depends of a parameter that measures the quality of the compromise. Thereafter this function was used to select a value of the mentioned parameter to obtain the generalization of some different objective functions (generalizations of ecological function, omega function and efficient power), by demanding that these generalization parameters maximize the above mentioned functions. In this work we demonstrate that this function can be used directly as an objective function: the "$P{\Phi}$-Compromise Function" ($C_{P\Phi}$), also that the operation modes corresponding to the maximum Generalized Ecological Function, maximum Generalized Omega Function and maximum Efficient power output, are special cases of the operation mode of maximum $C_{P\Phi}$, having the same optimum high reduced temperature, then the characteristic functions will be the same in any of the above three working regimes, independent of the algebraic complexity of each generalized function. These results are presented for two different models of an irreversible energy converter: a non-endoreversible and a totally irreversible, both with heat leakage.

Figures

Figures reproduced from arXiv: 1908.11861 by the authors.

**Figure 2.** (a) PΦ–Compromise function under a non–endoreversible model as a function of the high reduced temperature, (b) Comparison of the power output and dissipation at maximum PΦ– Compromise Function with respect to the same process variables at maximum power output regime and (c) Efficiency at maximum efficiency regime ( ηMη), efficiency at maximum power output (ηMP ) and efficiency at maximum PΦ – Compromise Function (Da… view at source ↗

**Figure 3.** (a) PΦ – Compromise Function for an irreversible model with heat leak. (b) Comparison of the power output and dissipation at maximum compromise function with respect to the same process variables at maximum power output regime. (c) Efficiency at maximum efficiency regime (ηMη), efficiency at maximum power output regime (ηMP ) and efficiency at MCF–regime (dashed line, ηMCP Φ ). (d) Parametric curves of power output … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: (a) Generalization of the objective functions at t [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: (a) Generalization of the objective functions at t [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Efficiency at the maximum power of the power law dissipative Carnot-like Heat engines with non-adiabatic dissipation
cond-mat.stat-mech 2019-09 conditional novelty 4.0 of 10

Adding non-adiabatic, power-law dissipative friction in the adiabatic branches of a Carnot-like engine leaves the universal minimum and maximum efficiency-at-maximum-power bounds unchanged.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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