Proposed Lorentz Center workshop · Target: spring 2027
Towards independent assessment of a proposed general method for reversible-cycle benchmarking.
We have submitted a proposal for a workshop at the Lorentz Center at Leiden University to investigate whether the established Clausius reversible-cycle equality can be operationalized into a general, explicitly qualified method for benchmarking thermodynamic cycles with both isothermal and non-isothermal heat exchange.
The intended programme brings together thermodynamic science and engineering with relevant perspectives from physics and astronomy, to examine the proposed method independently, compare it with existing treatments and establish its scope, limitations and possible implications.
Can the established foundations of cycle theory be made usable as one parent reversible-cycle benchmark framework, exposing the full range of admissible reversible solutions rather than only familiar efficiency and COP results? This is the central proposition to be assessed.
The CarnotX Initiative proposes complete operationalization of the Clausius reversible-cycle equality: combining it with energy balance, the prescribed data and the physical conditions of each cycle to determine which reversible solutions are possible and which requested quantities can be calculated. These governing thermodynamic relations are established; the claimed completeness, organization and added value of the operational method require independent assessment.
Efficiency and COP are familiar outputs of a reversible-cycle problem. They do not describe every admissible relationship between work, heat quantities, temperatures and other cycle variables. The proposed framework starts with what is known and what must hold, then determines the requested quantities from the resulting qualified reversible solution space—the set of solutions that satisfy those conditions.
This broader view applies within a cycle model as well as across different models. Carnot provides the exact two-isothermal case. Brayton, Otto, Diesel, Lorenz and Rankine models bring their own process assumptions and physical requirements; using a common framework does not remove those differences or make an idealized model automatically fully reversible. The ambition is a consistent way to construct and qualify the corresponding reversible problems, whether heat exchange is isothermal or non-isothermal.
Changing the admissible selection of knowns, unknowns and target quantities can yield further performance measures and other reversible relations. Where the supplied conditions do not determine a unique answer, the result must remain a solution set or an explicitly underdetermined problem. The detailed method is documented in the Formal Core Specification.
The two expressions below use heat-engine efficiency and heating or refrigeration COP as familiar examples of target quantities. The first refers to the Carnot reversible-isothermal cycle equality; the second refers to the Clausius reversible-cycle equality, which also covers non-isothermal heat exchange. These performance measures illustrate the proposed operational approach; they do not exhaust its scope.
Established two-isothermal case
For reversible cyclic operation with heat supplied and removed at two constant temperatures, this balance, together with energy balance, yields the familiar Carnot efficiency and COP relations.
Proposed general operationalization
The proposed method operationalizes the established Clausius reversible-cycle equality to determine a qualified, process-specific reversible efficiency or COP for specified heat-exchange trajectories. It includes non-isothermal configurations and retains the two-isothermal case as a special case.
Proposed parent reversible-cycle benchmark framework
Define the cycle conditions: heat-transfer Q–T trajectories, known inputs and applicable process assumptions.
For the reversible two-isothermal case, the classical Carnot heat-engine efficiency, heat-pump COP and refrigeration COP formulas describe particular performance relationships. They represent only part of the complete reversible two-isothermal cycle solution space. Operationalizing the Carnot equality—the two-isothermal form of the Clausius equality—together with energy balance and the applicable physical conditions allows other relationships between heat, work and temperature to be explored. The CarnotX Initiative proposes to make that full qualified solution space and its derived relations systematically accessible.
The same distinction applies to non-isothermal models such as Brayton. The classical ideal Brayton efficiency and reversed-Brayton heat-pump or refrigeration COP expressions are particular results under their stated assumptions. The proposed Clausius-based approach aims to expose the complete qualified reversible Brayton solution space, from which these familiar expressions and other admissible performance formulas and cycle relations can be derived. This requires the Brayton process conditions, working-medium properties and reversible heat-transfer conditions to be included.
The framework’s primary role is to construct qualified reversible references and compare reversible alternatives under declared conditions. Comparing an actual, irreversible cycle with a relevant reference is a subsequent analysis: it requires information about actual performance and losses, and a consistent basis of comparison. A gap to a reversible limit is not automatically a recoverable engineering gain.
One application is Thermal Matching, which investigates cycle optimization through better coordination of the thermal behaviour of the heat-releasing and heat-receiving bodies. It considers both heat absorption and heat rejection, including the shapes and levels of their temperature trajectories. Reversible references help interpret these design choices; component models, practical constraints and validation establish their consequences for real systems.
Can the proposed framework recover established results, correctly identify admissible reversible solutions and demonstrate useful additional capability across its declared cycle domain?
Potential significance · Subject to independent assessment
The proposed modernization can be understood in two markedly different ways. At its most modest, it is “merely” a further mathematical operationalization of the Clausius relation of 1865: making established thermodynamic knowledge explicit, systematic and practically applicable, without introducing a new law of nature.
At its most ambitious, if the claimed scope and added value withstand independent scrutiny, the same operationalization could bring a profound modernization of how thermodynamic cycle theory is organized, taught and applied. The significance would lie in what a fully operational framework enables across cycle configurations, scientific disciplines and engineering applications.
The proposed operationalization aims to organize established reversible-cycle relations into a systematic framework for selecting known data, determining target quantities and constructing qualified, process-specific benchmarks. If independently validated, it could help researchers make assumptions explicit, compare cycle configurations and examine heat–work interactions within a common methodological structure.
Its potential scientific relevance extends beyond engineering applications to selected questions in materials science, physics, Earth sciences and astronomy. Applicability depends on whether a meaningful thermodynamic cycle can be defined and whether the system and processes satisfy the method’s scope and assumptions. Scientific assessment of that relevance provides the basis for any subsequent changes to teaching and professional practice.
Textbooks and technical references help shape lecture notes, course content and worked examples in universities and higher technical education. Through these materials, students learn which thermodynamic models to use, which assumptions to check and which benchmarks to select when assessing a cycle.
These choices carry into professional practice. The way cycle theory is taught influences how engineers formulate problems, interpret calculations and judge opportunities for improvement. Continuing education, engineering software and technical guidance can further transmit and reinforce those approaches.
If independently validated, the proposed modernization could strengthen this chain by giving the Carnot special case, non-isothermal cycles and process-specific reversible benchmarks a more explicit and systematic place in teaching. The educational value would lie in helping future engineers select and apply the appropriate benchmark for the actual question.
Benchmark selection helps determine what engineers regard as a performance limit, which losses they investigate and where they direct optimization efforts. This connects the treatment of cycle theory in education to the design, modelling and operation of combustion engines, power plants, heat pumps, refrigeration systems and other thermal installations.
For design decisions, this could help engineers assess whether to improve thermal matching, adjust heat-exchange trajectories or reconsider the cycle configuration. Within a defined comparison class, process-specific benchmarks could help distinguish configuration limits from potentially avoidable losses and guide which alternatives deserve detailed modelling or testing. The practical value would lie in directing optimization efforts towards physically justified opportunities and making the assumptions behind design choices explicit.
Where that leads to technically and economically feasible changes, the result may be lower fuel or electricity consumption for the required service. This is a pathway through which improved methods could support energy efficiency and sustainability; the achievable gains depend on the application, implementation and operating conditions and must be demonstrated in practice.
The route from scientific assessment to societal impact runs through textbooks and curricula, adoption in engineering tools, changed design decisions and implementation in real equipment. The figures below describe different steps along that conditional pathway.
These figures describe documented potential reach, selected engineering opportunities and conditional modeled outcomes—not demonstrated effects of CarnotX; realization depends on independent validation, adoption and implementation.
Publishing. The inventory identifies 724 evidence-supported textbook families across 20 selected publisher groups. Of a 143-title priority subset, 126 (88.1%) remain offered or digitally supported; seven have passage-level review and five form the first editorial-review group. Potential review relevance does not establish errors in every title or a need to revise all 724 families.
Higher education. The inventory documents 3,799 relevant providers across 42 countries, including 1,526 university-level engineering and 1,093 non-university technical providers. Explicit cycle content is documented in 10 of 18 reviewed disciplinary cases. These are measures of potential knowledge reach, not confirmed adoption or a complete global census.
Engineering. Eight of ten technology classes retain quantitative case evidence, with approximately 2–21% input-energy reductions under specific comparators. Nuclear steam and ORC/waste heat remain uncalibrated. The range is neither a sector-wide average nor a measured CarnotX effect.
| Scenario | Effective gain | Energy savings | Operational fossil CO₂ reductions |
|---|---|---|---|
| Low calibrated | 0.02% | 1.1 EJ | 31 Mt |
| Cautious calibrated | 0.12% | 6.6 EJ | 200 Mt |
| 1% broader scenario anchor | 1.0% | 20 EJ | 910 Mt |
| Broader realization | 1.3% | 41 EJ | 1,500 Mt |
These are distinct scenario-model outputs combining gain, diffusion and asset-turnover assumptions, not forecasts, probabilities or a statistical confidence interval. The totals must not be inferred from effective gain alone. No attributable CarnotX energy or emissions reduction has yet been established.
Implementation sensitivity. In the central diffusion scenario, modeled year-20 operating-asset penetration is 69% for heating heat pumps, 77% for refrigeration and AC, 65% for reciprocating engines and 46% for gas turbines and aviation. These are assumptions-based scenario outputs, not observed adoption. Excluding major-redesign pathways lowers the cautious effective gain from about 0.12% to about 0.09%.
Source: Beyond Carnot: Operationalizing Clausius — Metrics at a Glance, v2.9, September 2026. Scientific validation, knowledge transfer, tool adoption, changed engineering decisions and physical implementation are separate conditions along the impact pathway.
Evidence snapshot: Metrics at a Glance v2.9 · September 2026
The stakes extend well beyond a single formula. If the proposed operationalization withstands independent scrutiny, its implications could reach how thermodynamic cycles are explained, compared and taught across science and engineering.
Our educational and publishing evidence report identifies 724 evidence-supported textbook families within 20 selected publisher groups as potentially relevant to targeted review, should Formal Core be independently validated. The relevance extends from foundational textbooks to specialist engineering references. Coverage remains incomplete within the selected catalogues and does not encompass the full international publishing landscape.
That scale deserves a serious scientific response. If the method offers a substantive advance, teaching and reference literature should reflect it. If established approaches already provide an equivalent framework, that should be demonstrated just as clearly. Either outcome calls for rigorous examination of the derivations, existing knowledge and representative applications.
This is why the proposed workshop matters: to bring independent researchers, educators and publishers together around a question with potentially far-reaching consequences. The report establishes a case for review—not a finding that books are wrong or that the proposed method has already been validated.
The supporting report, Potential Educational and Publishing Relevance of Beyond Carnot: Operationalizing Clausius, is available on request through Research Documents.
Help examine these possibilities: contribute scientific expertise, an engineering or educational case, explore funding, or receive updates on significant milestones. Critical review is welcome; contributing does not require endorsing the proposed method.
Contribute or collaborateLeiden, the Netherlands
The Lorentz Center is an international scientific workshop centre based at Leiden University and structurally supported by both Leiden University and the Dutch Research Council (NWO). It brings researchers from around the world together to explore important open questions, examine emerging ideas and develop new research directions.
Its scope encompasses the natural, social and medical sciences, the humanities, and the interfaces between them. Lorentz Center describes this openness to all disciplines and their intersections as a globally distinctive feature: researchers can bring together precisely the combination of expertise that a scientific question requires. Institutional background
The centre combines high scientific standards with an explicit openness to daring initiatives. This creates room for ambitious, exploratory work whose outcomes remain uncertain, but whose potential scientific value merits serious investigation. Its mission emphasizes scientific quality, diverse perspectives and connections between academia, society and industry.
Lorentz Center pursues these aims through focused scientific workshops: international working meetings, typically lasting five days, built around intensive discussion, collaborative work and informal exchange. This sustained interaction gives participants time to examine assumptions, challenge interpretations and work through questions together.
Scientific coordinators and advisory boards help organizers shape the programme, while the centre provides practical and scientific support. The format brings complementary expertise into direct dialogue and helps participants identify concrete next steps for research. The workshop format
For Beyond Carnot: Operationalizing Clausius, we see this combination of scientific ambition, complementary expertise and sustained interaction as particularly valuable for independently assessing the proposed method, its limitations and its potential implications for thermodynamic cycle theory.
The workshop proposal has been submitted to the Lorentz Center for consideration in its programme. Assessment by its scientific team is expected over the coming months. We are further developing the scientific programme and organizing team and working on preparations for the proposed workshop in spring 2027. This remains the target period, subject to formal acceptance and confirmation of workshop dates.
Programme under development
Three tasks guide the proposed scientific assessment:
The proposed programme envisages five days with approximately 50 participants, organized around four conditional validation stages and two scientific test laboratories. These are proposed programme parameters; participation, acceptance and workshop dates remain to be confirmed.
01
Clarify the exact two-isothermal comparison domain, reconstruct its reversible-cycle relations and agree on known data, target quantities and qualification conditions. Establish a shared basis for testing the proposed generalization.
02
Reconstruct the proposed derivations for multiple and varying heat-exchange temperatures. Test correctness, completeness and closure, and compare with established entropy, exergy, Lorenz-cycle and finite-reservoir treatments to identify equivalent or additional capability.
03
Use two scientific test laboratories: four controlled benchmark families and four cross-disciplinary stress-test breakouts. Examine reproducibility, counterexamples, physical admissibility and failure boundaries, including cases where the method is redundant or underdetermined.
04
Proceed to application exploration only where the earlier scientific stages justify it. Assess supported implications for engineering, education and literature, and record restrictions, unresolved questions and further numerical or experimental work.
A disposition map recording the assessment of the claims; a benchmark and stress-test set; an equivalence/difference matrix comparing methods; a failure-boundary dossier; a cross-medium comparison; and an unresolved-question list.
The assessment may lead to confirmation, restriction, reorganization, a finding of redundancy or rejection. Bounded and negative results remain useful: they establish what the method does and does not add and define the next research questions.
Thermodynamics provides a common framework across engineering and several branches of the physical sciences. The proposed workshop connects two complementary areas of expertise.
Expertise in thermodynamic cycles, reversible benchmarks, non-isothermal heat exchange, energy and exergy analysis, and computational methods is central to assessing the proposal.
Relevant application areas include heat engines, power generation, combustion engines, heat pumps and refrigeration. Contributions may include established comparison methods, representative cycle cases, numerical reconstruction and critical examination of the proposed framework.
Contributions from physics and astronomy can strengthen the examination of thermodynamic assumptions, theoretical consistency and applicability to selected physical systems.
Relevance must be established for each case: whether a meaningful cycle can be defined, how heat and work interactions are specified, and whether the required thermodynamic conditions are satisfied. Applicability to a natural or astrophysical system is a question to investigate.
The organizing team and prospective participant group are in development. We welcome exploratory discussions with researchers who can contribute subject expertise, independent scrutiny, programme development or relevant comparison cases.
Meet the initiating researchers and contributors →
Potential contributions include reviewing the scientific question, helping shape the programme, identifying prior work, reconstructing derivations and proposing cases that test the method’s boundaries.
Industrial researchers and engineers may contribute application questions, modelling expertise and suitable comparison cases. Possible performance improvements remain application-dependent and require substantiation.
Educators, textbook authors and academic publishers may contribute perspectives on the presentation of reversible benchmarks, cycle theory and non-isothermal heat exchange. They can help secure access to the priority chapters identified in the evidence report and assess representative teaching cases. Editorial decisions and proposed changes to teaching or literature should follow independent scientific assessment.
Organizations interested in supporting the initiative are invited to discuss possible arrangements. The availability and form of sponsorship acknowledgement or other visibility would need to be agreed with the relevant parties. Scientific conclusions must remain independent of financial support.
The introductory presentation, From a Specific Formula to a General Method, uses the square–rectangle analogy to explain the proposed Carnot–Clausius relationship and the purpose of independent assessment.
Dutch and English PDF editions are available for direct download. The submitted workshop proposal describes the intended scientific scope and programme and can be requested separately. Formal Core review access has its own application route.
Tell us about your expertise, your interest in the scientific question and the contribution you would like to discuss. An expression of interest starts a conversation; participation and other arrangements will be confirmed separately.
To receive the proposal, use the workshop-proposal request form. You can describe your interest there; a second contact message is not needed.
Scientific review, including critical examination and counterexamples, industrial cases, educational contributions and sponsorship interests can be registered through our central stakeholder form. Contributing does not require endorsing the proposed method.
Contribute or collaborateFor a scoped technical study or software evaluation, explore Early Access & Technical Collaboration.