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Experimental Characterization of Autonomous Heat Engine Based On Experimental characterization of autonomous heat engine based on minimal dynamical-system model Shoichi Toyabea and Yuki Izumidab, * aDepartment of Applied Physics, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan bDepartment of Complexity Science and Engineering, Graduate School of Frontier Sciences, University of Tokyo, Kashiwa 277-8561, Japan *[email protected] The autonomous heat engine is a model system of autonomous nonequilibrium systems like biological cells, exploiting nonequi- librium flow for operations. As the Carnot engine has essentially contributed to the equilibrium thermodynamics, autonomous heat engine is expected to play a critical role in the challenge of constructing nonequilibrium thermodynamics. However, the high complexity of the engine involving an intricate coupling among heat, gas flow, and mechanics has prevented simple model- ing. Here, we experimentally characterized the nonequilibrium dynamics and thermodynamics of a low-temperature-differential Stirling engine, which is a model autonomous heat engine. Our experiments demonstrated that the core engine dynamics are quantitatively described by a minimal dynamical model with only two degrees of freedom. The model proposes a novel concept that illustrates the engine as a thermodynamic pendulum driven by a thermodynamic force. This work will open a new approach to explore the nonequilibrium thermodynamics of autonomous systems based on a simple dynamical system. Modern physics is challenging to characterize autonomous a 90° nonequilibrium systems like biological cells. These systems are typically complex, but there has been a long pursuit for T Ideal Stirling cycle deriving universal and simple relations governing them. For this purpose, the extension of thermodynamics would be a Stirling engine promising approach because thermodynamicsillustrates a uni- θ = 0° T > T + versal structure of the system behind the details. Thermody- Flywheel Δ Δ c 180° T < T + namics is originally formulated based on infinitely slow qua- Δ Δ c Power piston T sistatic processes1. The recent challenge in constructing the top finite-time thermodynamics tries to characterize the thermo- Displacer Tbtm dynamic quantities of nonequilibrium systems by incorporat- (switch of heat bath) 270° ing the finite-speed dynamics. The finite-time thermodynam- V ics was already successful in characterizing the efficiency at maximum power2–9, optimal control with the minimal energy cost10–12, the trade-off relations between the power and effi- ciency13–17, and stochastic heat engines18–20. A simple model is always the basis of scientific understand- b Target pattern ing. Carnot’s heat engine would be one of the most prominent examples in history and played an essential role in construct- Water block ing thermodynamics21. As a natural extension along with this Al plate approach, autonomous heat engines are expected to play es- Pressure sensor Peltier module sential roles in the development of finite-time thermodynam- Thermometer ics. The Carnot’s engine requires control by an external agent, whereas the autonomous heat engines implement autonomous Figure 1. Schematic of the experiments. a, Thermodynamic diagram of the regulation like biological cells. However, the high complexity low-temperature-differential Stirling engine. The heating from the bottom in- of the engine dynamics involving a mechanical motion, heat creases the internal pressure, pushes the power piston upward, and drives the flywheel rotation, which then pushes the displacer downward. The displacer flow, and gas flow has prevented simple modeling. serves to switch the heat baths. When the displacer moves downward, most arXiv:1911.02810v3 [cond-mat.stat-mech] 23 Apr 2020 Here, we experimentally characterize the nonequilibrium gas in the cylinder moves to the upper side and makes contact with the top dynamics of a low-temperature-differential Stirling engine plate at a lower temperature of Ttop. The cooling of gas results in the pres- sure decrease and pushes the power piston downward, and the cycle restores (LTD-SE), which is a model autocatalytic heat engine with to the initial state. The rotation can be inverted by an opposite temperature a minimal structure (Fig. 1)22. We especially focus on the difference. b, Experimental set up. The temperatures at the top and bottom bifurcation dynamics because the bifurcation behavior char- plates are controlled by Peltier modules. The pressure difference between the outside and inside of the cylinder is monitored by a differential pressure sen- acterizes the system’s universal properties behind the details. sor. The rotation of the flywheel is monitored at 100 Hz by a videoscopy of We deduce a simple two-variable model of this engine based the target pattern (three circles aligned in an isosceles triangle configuration) on the experimental results. Such simple modeling would en- attached to the crank screw. able us to build a theoretical framework of autonomous heat engines towards the establishment of the finite-time thermo- 1 dynamics. The Stirling engine is an autonomous and closed heat en- gine23–26. Given a temperature difference, the engine cycles the volume, temperature, and pressure inside a cylinder au- tonomously without external timing control and rotates a fly- wheel unidirectionally. Theoretically, an ideal Stirling engine, i.e., a Stirling cycle, achieves the Carnot efficiency. The LTD- SE22, 27 consists of a power piston, displacer, flywheel, two cranks, and two rods connecting the piston and displacer to the flywheel (Fig. 1a). The flywheel rotates when a suffi- ciently large temperature difference is given between the top and bottom plates of the cylinder. The flywheel rotation is syn- chronized with the oscillation of the internal displacer and the power piston. The displacer serves to switch the heat baths between the top and bottom plates. Thus, the gas tempera- ture and pressure oscillate and move the power piston up and down. This piston motion drives the flywheel rotation. The π/2 out of phase of the displacer and the power piston makes a cycle. The flywheel provides inertia necessary for a smooth rotation. When the opposite temperature difference is given, the flywheel rotates in the opposite direction with an inverted mechanism. Experiment An LTD-SE (N-92 type) was bought from Kontax (UK). We controlled the temperatures at the top and bottom plates, Ttop and Tbtm, of the cylinder (Fig. 1b) and monitored the angular position θ(t) and angular velocity ω(t) of the flywheel and the pressure p(t) inside the chamber. See the Materials and Methods for details. Figure 2. Steady rotation. a, The time-averaged angular velocity hωi of the Rotation. Without stimulation, the engine was settled at a limit cycle was plotted against the temperature differences ∆T. At |∆T|≥ 8K, stationary position θ ≃−38◦, where the pressure difference three experimental traces were averaged (circle). The error bar corresponds to across the power piston and the gravity force on the power S.D. At |∆T| < 8 K, twelve traces were superposed (solid lines, six for ∆T > 0 and six for ∆T < 0). b, The stability of the rotation state at ∆T = 12 K. The piston, displacer, crank screws, and rods are presumably bal- rotation state is stable against perturbations (indicated by arrows). Dashed anced. When an initial angular momentum with a sufficiently lines are fitting curves by exponential functions, of which time constant cor- large magnitude was given, the flywheel rotated steadily with responds to I/Γ. c, Pressure-volume curves for ∆T = 12 K (red) and -12 K (navy) obtained by experiments (solid) and theories (dashed). The cycling an angular velocity determined by ∆T = Tbtm − Ttop (Fig. 2a). direction was clockwise independent of the sign of ∆T. The average of ∆p The rotation direction changed depending on the sign of ∆T. for the theoretical curves was forced to zero. d, Bifurcation diagrams without When the engine in this steady state was perturbed by hand, (open) or with (closed) additional frictional load. The solid curves are numer- the angularvelocity was soon recoveredto the steady rate (Fig. ical simulations with Γ obtained by measuring the relaxation time I/Γ to the perturbation (b and Fig. S2). 2b), implying a stable limit cycle. The pressure-volume curve exhibited a circular diagram (Fig. 2c), demonstrating a heat engine. The cycling direc- the homoclinic bifurcation point. However, such the continu- tion in the PV diagram was the same independent of the sign ity is too steep to be observed in the experiments because ω is of ∆T , and the PV curves were nearly symmetric for the sign ± ± inversely proportional to −ln|∆T −∆Tc | for |∆T | > |∆Tc | in of ∆T. The area increased with |∆T | (Fig. S1). the vicinity of the bifurcation point28. Instead, discontinuous The time-averaged steady angular velocity hωi changed change in hωi was observed. nearly linearly with ∆T (Fig. 2a). |hωi| decreased with |∆T| We also induced additional frictional load by pressing a and vanished at a finite value of ∆T . The threshold value, ∆Tc, brush for Chinese calligraphy to the flywheel. The increase + + was slightly different for the sign of ∆T ; ∆Tc = 6.2 ± 0.3K in load suppressed the angular velocity and increased ∆Tc − and ∆Tc = −5.7 ± 0.2 K (mean ± standard deviation) , indi- (Fig. 2c). cating the asymmetry of the dynamics for the sign of ∆T . ± The stalling at ∆Tc was accompanied by a steep change Bifurcation analysis. We characterize the bifurcation dy- in hωi, implying a homoclinic bifurcation28. The homoclinic
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