Efficiency at maximum power of Feynman’s ratchet as a heat engine

The maximum power of Feynman’s ratchet as a heat engine and the corresponding efficiency (η∗) are investigated by optimizing both the internal parameter and the external load. When a perfect ratchet device (no heat exchange between the ratchet and the paw via kinetic energy) works between two thermal baths at temperatures T1 > T2, its efficiency at maximum power is found to be η∗ = η 2 C /[ηC − (1 − ηC) ln(1 − ηC)], where ηC ≡ 1 − T2/T1. This efficiency is slightly higher than the value 1 − √ T2/T1 obtained by Curzon and Ahlborn [Am. J. Phys. 43 (1975) 22] for macroscopic heat engines. It is also slightly larger than the result ηSS ≡ 2ηC/(4 − ηC) obtained by Schmiedl and Seifert [EPL 81 (2008) 20003] for stochastic heat engines working at small temperature difference, while the evident deviation between η∗ and ηSS appears at large temperature difference. For an imperfect ratchet device in which the heat exchange between the ratchet and the paw via kinetic energy is non-vanishing, the efficiency at maximum power decreases with increasing the heat conductivity. PACS numbers: 05.70.Ln J. Phys. A: Math. Theor. 41, 312003 (2008) FAST TRACK COMMUNICATION 2 As is well known, the Carnot efficiency ηC ≡ 1− T2/T1 gives the upper bound for heat engines working between two thermal baths at temperatures T1 > T2. However, the engines at the Carnot efficiency cannot produce output power because the Carnot cycle requires an infinitely slow process. The cycle should be speeded up to obtain a finite power. Curzon and Ahlborn [1] derived the efficiency, ηCA ≡ 1− √ T2/T1, at maximum power for macroscopic heat engines within the framework of finite-time thermodynamics. The same expression was also obtained from linear irreversible thermodynamics for perfectly coupled systems [2–4]. It is pointed out that the efficiency at maximum power might be different from ηCA for imperfectly coupled systems [2–4]. In a recent paper [5], Schmiedl and Seifert investigated cyclic Brownian heat engines working in a time-dependent harmonic potential and obtained the efficiency at maximum power, ηSS ≡ 2ηC/(4− ηC), within the framework of stochastic thermodynamics [6–8]. To illustrate the second law of thermodynamics, Feynman introduced an imaginary microscopic ratchet device in his famous lectures [9]. Following Feynman’s spirit, many models were put forward, such as on-off ratchets [10], fluctuating potential ratchets [11, 12], temperature ratchets [13, 14], chiral ratchets [15–17], and so on, which have potential applications in biological motors. Thus it is significant to investigate the efficiency and power of these ratchets. Feynman’s ratchet, as a parental model, attracts researchers’ interests [18–21] in the nature of things. In particular, the efficiency at maximum power of Feynman’s ratchet was obtained by optimizing the external load for given internal parameter in Refs. [19][21]. However, there is still lack of a result by optimizing both the internal parameter and the external load of the ratchet device. In the present paper, we will analytically derive this result. Let us consider Feynman’s ratchet device as shown in figure 1. It consists of a ratchet, a paw and spring, vanes, two thermal baths at temperatures T1 > T2, an axle and wheel, and a load. For simplicity, we assume that the axle and wheel is a rigid and frictionless thermal insulator. Figure 1. Feynman’s ratchet device. Now we follow Feynman’s discussion [9]. In one-step forward motion, we must FAST TRACK COMMUNICATION 3 borrow an energy ǫ to overcome the elastic energy of spring and then lift the pawl. Assume that the wheel rotates an angle θ per step and the load induces a torque Z, thus we also need an additional energy Zθ. Then the total energy that we have to borrow is ǫ + Zθ. In fact, we can borrow it from the hot thermal bath in the form of heat. The rate to get this energy is RF = r0e −(ǫ+Zθ)/T1 , (1) where r0 is a constant with dimension s , and the Boltzmann factor is taken to 1. In this process, the ratchet absorbs heat ǫ + Zθ from the hot thermal bath. A part of this heat is transduced into work Zθ, and the other is transferred as heat ǫ to the cold thermal bath through the interaction between the ratchet and the paw. Now let us consider one-step backward motion. To make the wheel backwards, we have to accumulate the energy ǫ to lift the pawl high enough so that the ratchet can slip. Here the rate to get this energy is RB = r0e −ǫ/T2 . (2) In the backward process, the load does work Zθ. This energy and the accumulated energy ǫ are returned to the hot thermal bath in the form of heat. In an infinitesimal time interval ∆t, the net work done on the load by the system may be expressed as W = (RF −RB)Zθ∆t. (3) The net heat absorbed from the hot thermal bath via the potential energy [22,23] may be expressed as: Q 1 = (RF − RB)(ǫ+ Zθ)∆t. (4) Since the ratchet contacts simultaneously with two thermal baths at different temperatures, there may exist a heat conduction from the hot thermal bath to the cold one via the kinetic energy [22, 23]. In time interval ∆t, it can be expressed as Q 1 = σ(T1 − T2)∆t, (5) where σ is the heat conductivity due to the heat exchange between the ratchet and the paw via kinetic energy. The analysis by Parrondo and Español suggests that σ is inversely proportional to the masses of ratchet and paw [22]. We first consider an perfect ratchet device in which the masses of ratchet and paw are infinitely large relative to the gas molecules full in both thermal baths. In this case, σ and Q 1 are vanishing. Thus the efficiency can be defined as η = W/Q 1 = Zθ/(ǫ+ Zθ). (6) The power is defined as P = W/∆t = r0Zθ[e −(ǫ+Zθ)/T1 − e2 ]. (7) We find that P depends on the internal parameter ǫ and the external load Z. It is easy to tune the external load Z. In fact, ǫ can also be adjusted by changing the strength of FAST TRACK COMMUNICATION 4 the spring. We can optimize both ǫ and Z to achieve the maximum power. Before doing that, we introduce two dimensionless parameters ε = ǫ/T2 and z = Zθ/T1. Equations (6) and (7) are then respectively transformed into η = z ε(1− ηC) + z , (8) and P = r0T1z[e −ε(1−ηC )−z − e]. (9) Maximizing P with respect to ε and z, we have