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This author is not a philosopher nor a historian of science, but an engineering thermodynamicist. In that regard, and in addition to various philosophical “why and how” treatises and existing historical analyses, the physical and logical “what it is” reflections, as sequential Key Points, where a key Sadi Carnot reasoning infers the next one, along with novel contributions and original generalizations, are presented. We need to keep in mind that in Sadi Carnot’s time (early 1800s), steam engines were inefficient (below 5%, so the heat in and out was comparable within experimental uncertainty, as if caloric were conserved), the conservation of caloric flourished (might be a fortunate misconception leading to the critical analogy with the waterwheel), and many critical thermal concepts, including the conservation of energy (The First Law), were not even established. If Clausius and Kelvin earned the title “Fathers of thermodynamics”, then Sadi Carnot was ‘the ingenious’ “Forefather of thermodynamics-to-become”.

Cyclical heat engines are a paradigm of classical thermodynamics, but are impractical for miniaturization because they rely on moving parts. A more recent concept is particle-exchange (PE) heat engines, which uses energy filtering to control a thermally driven particle flow between two heat reservoirs1,2. As they do not require moving parts and can be realized in solid-state materials, they are suitable for low-power applications and miniaturization. It was predicted that PE engines could reach the same thermodynamically ideal efficiency limits as those accessible to cyclical engines3–6, but this prediction has not been verified experimentally. Here, we demonstrate a PE heat engine based on a quantum dot (QD) embedded into a semiconductor nanowire. We directly measure the engine’s steady-state electric power output and combine it with the calculated electronic heat flow to determine the electronic efficiency η. We find that at the maximum power conditions, η is in agreement with the Curzon–Ahlborn efficiency6–9 and that the overall maximum η is in excess of 70% of the Carnot efficiency while maintaining a finite power output. Our results demonstrate that thermoelectric power conversion can, in principle, be achieved close to the thermodynamic limits, with direct relevance for future hot-carrier photovoltaics10, on-chip coolers or energy harvesters for quantum technologies.

We study how to achieve the ultimate power in the simplest, yet non-trivial, model of a thermal machine, namely a two-level quantum system coupled to two thermal baths. Without making any prior assumption on the protocol, via optimal control we show that, regardless of the microscopic details and of the operating mode of the thermal machine, the maximum power is universally achieved by a fast Otto-cycle like structure in which the controls are rapidly switched between two extremal values. A closed formula for the maximum power is derived, and finite-speed effects are discussed. We also analyze the associated efficiency at maximum power showing that, contrary to universal results derived in the slow-driving regime, it can approach Carnot's efficiency, no other universal bounds being allowed.

In this study, a novel irreversible cyclic model of a capacitive mixing blue heat engine mainly consisting of super capacitors, charging and discharging circuits, a heat source, as well as two water sources with given salt concentrations is established for harvesting salinity gradient energy and waste heat. Additionally, the effects of the charging voltage and ratio of the minimum to maximum surface electric charge density on the thermodynamic efficiency and power output of the cycle are discussed. The maximum power output of the cycle is calculated. The optimized ranges of efficiency and power output as well as the temperatures of two isothermal processes are determined. It is established that during the isoelectric quantity process, there is not only an increase in thermal voltage owing to the temperature difference, but also an increase in concentration voltage owing to the salinity gradient. Consequently, the blue heat engine can obtain higher energy conversion efficiency than a conventional heat engine. When the temperature ratio of the heat source to the heat sink is 1.233, the maximum efficiency can reach approximately 36%. The results obtained can promote the application of capacitive mixing technology in real life, reducing the consumption of fossil fuels.

We undertake a theoretical study of a finite-time quantum Otto engine cycle driven by inter-particle interactions in a weakly interacting one-dimensional (1D) Bose gas in the quasicondensate regime. Utilizing a c -field approach, we simulate the entire Otto cycle, i.e. the two work strokes and the two equilibration strokes. More specifically, the interaction-induced work strokes are modelled by treating the working fluid as an isolated quantum many-body system undergoing unitary evolution. The equilibration strokes, on the other hand, are modelled by treating the working fluid as an open quantum system tunnel-coupled to another quasicondensate which acts as either the hot or cold reservoir, albeit of finite size. We find that, unlike a uniform 1D Bose gas, a harmonically trapped quasicondensate cannot operate purely as a heat engine; instead, the engine operation is enabled by additional chemical work performed on the working fluid, facilitated by the inflow of particles from the hot reservoir. The microscopic treatment of dynamics during equilibration strokes enables us to evaluate the characteristic operational time scales of this Otto thermochemical engine, crucial for characterizing its power output, without any ad hoc assumptions about typical thermalization timescales. We analyse the performance and quantify the figures of merit of the proposed Otto thermochemical engine, finding that it offers a favourable trade-off between efficiency and power output, particularly when the interaction-induced work strokes are implemented via a sudden quench. We further demonstrate that in the sudden quench regime, the engine operates with an efficiency close to the near-adiabatic (near maximum efficiency) limit, while concurrently achieving maximum power output.

Compared with endoreversible heat engine with pure heat transfer and endoreversible isothermal chemical engine with pure mass transfer, endoreversible non-isothermal chemical engine (ENICE) is a more reasonable model of practical mass exchanger, solid device and chemo-electric systems. There exists heat and mass transfer (HMT) simultaneously between working fluid and chemical potential reservoir in ENICE. There is coupled HMT effect that in ENICE should be considered. There are two ways to consider this coupled effect. One is based on Onsager equations, and another is based on Lewis analogy. For the mathematical and physical description of the above HMT process, the model using Onsager equations are more appropriate in the linear HMT region not far from the equilibrium state, while that based on Lewis analogy is more appropriate in nonlinear HMT region far from the equilibrium state. Different from the previous research on the power optimization of ENICEs with Onsager equations, this paper optimizes power and efficiency of ENICE based on Lewis analogy. HMT processes are assumed to obey Newtonian heat transfer law ( q ∝ Δ T , and T is temperature) and Fick’s diffusive mass transfer law ( g ∝ Δ c , and c is concentration), respectively. Analytical results of power output and corresponding vector efficiency ( η T and η μ ) of ENICE are obtained, which provide important parallel results with those based on Onsager equations. They include special cases for endoreversible Carnot heat engine with q ∝ Δ T and endoreversible isothermal chemical engine with g ∝ Δ c . Adopting Lewis analogy in the modelling of ENICEs with simultaneous HMT is an important work. It provides important analytical and numerical results different from those with Onsager equations obtained previously and enriches the research contents of FTT. The research results in this paper have a certain guiding significance for the optimal designs of single irreversible NICEs, multistage NICE systems, practical mass exchangers, solid devices, chemo-electric systems, and so on.

Relativistic quantum systems exhibit unique features not present at lower energies, such as the existence of both particles and antiparticles, and restrictions placed on the system dynamics due to the light cone. In order to understand what impact these relativistic phenomena have on the performance of quantum thermal machines we analyze a quantum Otto engine with a working medium of a relativistic particle in an oscillator potential evolving under Dirac or Klein–Gordon dynamics. We examine both the low-temperature, non-relativistic and high-temperature, relativistic limits of the dynamics and find that the relativistic engine operates with higher work output, but an effectively reduced compression ratio, leading to significantly smaller efficiency than its non-relativistic counterpart. Using the framework of endoreversible thermodynamics we determine the efficiency at maximum power of the relativistic engine, and find it to be equivalent to the Curzon–Ahlborn efficiency.

Heat engines used to output useful work have important practical significance, which, in general, operate between heat baths of infinite size and constant temperature. In this paper, we study the efficiency of a heat engine operating between two finite-size heat sources with initial temperature difference. The total output work of such heat engine is limited due to the finite heat capacity of the sources. We firstly investigate the effects of different heat capacity characteristics of the sources on the heat engine’s efficiency at maximum work (EMW) in the quasi-static limit. Moreover, it is found that the efficiency of the engine operating in finite-time with maximum power of each cycle is achieved follows a simple universality as η=ηC/4+OηC2, where ηC is the Carnot efficiency determined by the initial temperature of the sources. Remarkably, when the heat capacity of the heat source is negative, such as the black holes, we show that the heat engine efficiency during the operation can surpass the Carnot efficiency determined by the initial temperature of the heat sources. It is further argued that the heat engine between two black holes with vanishing initial temperature difference can be driven by the energy fluctuation. The corresponding EMW is proved to be ηMW=2−2.

Isothermal transformations are minimally dissipative but slow processes, as the system needs to remain close to thermal equilibrium along the protocol. Here, we show that smoothly modifying the system-bath interaction can significantly speed up such transformations. In particular, we construct protocols where the overall dissipationWdissdecays with the total timeτtotof the protocol asWdiss∝τtot−2α−1, where each valueα>0can be obtained by a suitable modification of the interaction, whereasα=0corresponds to a standard isothermal process where the system-bath interaction remains constant. Considering heat engines based on such speed-ups, we show that the corresponding efficiency at maximum power interpolates between the Curzon-Ahlborn efficiency forα=0and the Carnot efficiency forα→∞. Analogous enhancements are obtained for the coefficient of performance of refrigerators. We confirm our analytical results with two numerical examples whereα=1/2, namely the time-dependent Caldeira-Leggett and resonant-level models, with strong system-environment correlations taken fully into account. We highlight the possibility of implementing our proposed speed-ups with ultracold atomic impurities and mesoscopic electronic devices.

Curzon and Ahlborn’s 1975 paper, a pioneering work that inspired the birth of the field of finite-time thermodynamics, unveiled the efficiency at maximum power (EMP) of the endoreversible Carnot heat engine, now commonly referred to as the Curzon–Ahlborn (CA) engine. Historically, despite the significance of the CA engine, similar findings had emerged at an earlier time, such as the Yvon engine proposed by J. Yvon in 1955 that shares the exact same EMP, that is, the CA efficiency ηCA. However, the special setup of the Yvon engine has circumscribed its broader influence. This paper extends the Yvon engine model to achieve a level of generality comparable to that of the CA engine. With the power expression of the extended Yvon engine, we directly explain the universality that ηCA is independent of the heat transfer coefficients between the working substance and the heat reservoirs. A rigorous comparison reveals that the extended Yvon engine and CA engine represent the steady-state and cyclic forms of the endoreversible Carnot heat engine, respectively, and are equivalent.