Brownian micro-engines and refrigerators in a spatially periodic temperature field: Heat flow and performances

Physics Letters A 352 (2006) 286–290 www.elsevier.com/locate/pla

Brownian micro-engines and refrigerators in a spatially periodic temperature field: Heat flow and performances Bao-Quan Ai a,∗ , Liqiu Wang b , Liang-Gang Liu c a School of Physics and Telecommunication Engineering, South China Normal University, GuangZhou, China b Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong c Department of Physics, ZhongShan University, GuangZhou, China

Received 18 October 2005; received in revised form 2 December 2005; accepted 2 December 2005 Available online 9 December 2005 Communicated by R. Wu

Abstract We study the thermodynamic features of a thermal motor driven by temperature differences, which consists of a Brownian particle moving in a sawtooth potential with an external load. The motor can work as a heat engine or a refrigerator under different conditions. The heat flow driven by both potential and kinetic energy is considered. The former is reversible when the engine works quasistatically and the latter is always irreversible. The efficiency of the heat engine (Coefficient Of Performance (COP) of a refrigerator) can never approach Carnot efficiency (COP).  2005 Elsevier B.V. All rights reserved. PACS: 05.40.-a; 05.70.-a; 87.10.+e Keywords: Efficiency; Heat flow; Thermal motor

1. Introduction Brownian ratchets (motors) have recently attracted considerable attention due to their importance in developing molecular motors [1,2]. A Brownian ratchet appeared first in Feynman’s famous textbook as a thermal ratchet [3]. It is a machine using the rectify thermal fluctuation to produce a directed current. This comes from the desire of understanding molecular motors [4], nanoscale friction [5], surface smoothening [6], coupled Josephson junctions [7], optical ratchets and directed motion of laser cooled atoms [8], and mass separation and trapping schemes at the microscale [9]. Brownian ratchets are spatially asymmetric but periodic structure in which transport of Brownian particles is induced by some nonequilibrium processes [10–15]. Typical examples are external modulation of underling potential, a nonequilibrium chemical reaction coupled to a change of the potential and * Corresponding author.

E-mail address: [email protected] (B.-Q. Ai). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.010

contact with reservoirs at different temperatures. The most studies have been on the velocity of the transport particle. Another important aspect is efficiency of energy conversion. Sekimoto has recently unambiguously defined the concept of the heat at mesoscopic scales in terms of Langevin equation [16]. He refers to the formalism providing this definition as stochastic energetics. The essential point of this formalism is that the heat transferred to the system is nothing but the microscopic work done by both friction force and the random force in the Langevin equation. Stochastic energetics has been applied to the study of thermodynamic processes. Derenyi and Astumian [17] have studied the efficiency of onedimensional thermally driven Brownian engines. They identify and compare the three basic setups characterized by the type of the connection between the Brownian particle and the two reservoirs: (1) simultaneous [3], (2) alternating in time [18], and (3) position dependent [19]. Parrondo and Cisneros [15] has reviewed the literature the energetics of Brownian motors, distinguishing between forced ratchet, chemical motorsdriven out of equilibrium by difference of chemical potential, and thermal motors-driven by temperature differences. In this

B.-Q. Ai et al. / Physics Letters A 352 (2006) 286–290

Letter we give a detailed study of the last motors-thermal motors. Thermodynamic features of the thermal motors-driven by temperature differences are investigated by some previous works [20,21]. Lindner and Schimansky-Geier [22] studied the transport of overdamped Brownian particles in periodic potentials subject to a spatially modulated Gaussian white noise. Asfaw et al. [20] have explored the performance of the motors under various conditions of practical interest such as maximum power and optimized efficiency. They found that the efficiency can approach the Carnot efficiency under the quasistatic limit. The same results are also obtained in Matsuo and Sasa’s work [21] by stochastic energetics theory. The previous works are limited to case of heat flow via potential. The present work extends the study to case of heat flow driven by both potential and kinetic energy. The motor can work as a heat engine or a refrigerator under various conditions. The efficiency of heat engine (COP of refrigerator) is different from the results of previous works and can never approach the Carnot efficiency (COP). The heat flow driven by potential is reversible when the engine works at quasistatic limit. The heat flow driven by the kinetic energy is always irreversible in nature. 2. Thermal motor works as a heat engine Consider a Brownian particle moving in a sawtooth potential with a external load force F . The medium is periodically in contact with two heat reservoirs along the space coordinate (shown in Fig. 1). There are two different driving factors in the motor. The first one is noise-induced transport. The second one is that temperature difference makes the particle move from high temperature reservoir to low temperature reservoir. Let E be the barrier height of the potential. L1 , L2 are the length of the left part and the right part of the ratchet, respectively. E + F L1 is the energy needed for a forward jump, while E − F L2 is the energy required for a backward jump. The left part of a period ratchet is at temperature TH (hot reservoir) and the right one is at temperature TC (cold reservoir). We can assume that the rates of both forward and backward jumps are proportional to the corresponding Arrhenius’ factor [23], such that 
E + F L1 1 , N˙ + = exp − (1) t k B TH 
E − F L2 1 ˙ N− = exp − (2) , t k B TC are the number of forward and backward jumps per unit time, respectively, kB the Boltzmann constant, t a proportionality constant with dimensions of time. If N˙ + > N˙ − , the ratchet works as a two-reservoir heat engine shown in Fig. 2. The rate of heat flow from hot reservoir to the heat engine driven by potential is given by pot Q˙ H = (N˙ + − N˙ − )(E + F L1 ).

(3)

287

Fig. 1. Schematic illustration of the motor: a Brownian particle moves in a sawtooth potential with an external load where the medium is periodically in contact with two reservoirs along the space coordinate. The period of the potential is L = L1 + L2 . The temperature profiles are also shown.

Fig. 2. The engine acts as a heat engine. Heat flows via both potential energy and kinetic energy in a thermal motor in contact with two thermal baths at temperatures TH > TC , W is power output, heat flow via potential energy is reversible, heat flow via kinetic energy is irreversible.

The rate of heat flow from the engine to the cold reservoir driven by potential is ˙ pot = (N˙ + − N˙ − )(E − F L2 ). Q C

(4)

The heat flow driven by the kinetic energy of the particle is much more complicated to determined [15]. Whenever a particle stays at a hot segment (temperature TH ) it absorbs 12 kB TH energy on average from the hot reservoir. It can pick up 12 kB TC energy on average from the cold reservoir when the particle stays at a cold segment. When a particle leaves from a hot segment to a cold segment and then returns to the hot segment, the hot reservoir loses 12 (kB TH − kB TC ) energy on average, the lost energy is released to the colder reservoir and never returns to the hot reservoir or to the particle’s potential energy, which indicates the inherently irreversible nature of this heat flow. On the other hand, if a particle goes out from a hot segment to a cold segment and never returns to the hot segment, the hot reservoir loses 12 kB TH energy on average. Therefore the rate of net heat flow driven by kinetic energy from the hot reservoir to the cold reservoir is given by ˙ kin = 1 N˙ + kB TH − 1 N˙ − kB TC . Q 2 2

(5)

Therefore, the rate of total heat transferred from the hot reservoir is given by ˙ pot + Q ˙ kin ˙H =Q Q H 1 1 = (N˙ + − N˙ − )(E + F L1 ) + N˙ + kB TH − N˙ − kB TC . 2 2 (6)

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The rate of total heat transferred to the cold reservoir is

Therefore, it is easy to obtain

˙ pot + Q˙ kin ˙C =Q Q C

ηpot =

1 1 = (N˙ + − N˙ − )(E − F L2 ) + N˙ + kB TH − N˙ − kB TC . 2 2 (7) The rate of power output is ˙H −Q ˙ C = (N˙ + − N˙ − )F L. W˙ = Q

(8)

It is easy to obtain the efficiency of the heat engine η=

W˙ . ˙H Q

(9)

If the heat flow via the kinetic energy is not considered, the efficiency is given by ηpot =

W˙ . ˙ pot Q

(10)

H

In order to discuss simply, we can rewrite Eqs. (5)–(10) in a dimensionless form, then we get ˙Ht Q qH = k B TH 
 1 = e−/τ  + + µf e0 −µf 2    1 −  + τ + µf e(1−µ)f/τ , 2
  1 qC = e−/τ  + + µf − f e0 −µf 2    1 (1−µ)f/τ , −  + τ + µf − f e 2   Q˙ kin t 1 q kin = = e−/τ e0 −µf − τ e(1−µ)f/τ , k B TH 2   W˙ t w= = e−/τ e0 −µf − e(1−µ)f/τ f, k B TH [e0 −µf − e(1−µ)f/τ ]f , η= ( + µf + 12 )e0 −µf − ( + µf + 12 τ )e(1−µ)f/τ f , ηpot =  + µf

(11)

(12) (13) (14)

fm f TC  =1−τ =1− = ηC ,  + µf  + µfm TH

(19)

where ηC is Carnot efficiency. When f = fm , ηpot can approach the Carnot efficiency, namely, the motor works quasistatically. By letting ∂w ∂f = 0, we have the optimum f = fc for the maximum wm ,   1 − µ (1−µ)fc /τ e (1 − µ)e0 −µfc − 1 + (20) = 0. τ By letting

∂η ∂f

= 0, similarly, we have

( + 0.5)e2(0 −µfc ) 
 1−µ fc + 0.5(τ − 1) τ

 1−µ − (2 + 2µfc + 0.5τ + 0.5) e0 +( τ )fc + ( + 2µfc + 0.5τ )e2(1−µ)fc /τ = 0,

(21)

which leads to the optimum f = fc for the maximum ηm . Fig. 3 shows the heat flow qH , qC and power output w as a function of the load f . When f = 0, namely, the engine runs without a load, qH is equal to qC , which indicates that the heat that absorbs from the hot reservoir releases to the cold reservoir entirely and no power output is obtained. When f increases, qH and qC decreases. When f → 0, no power output is obtained (w = 0). When f → fm , the motor works quasistatically, the ratchet cannot give any power output. So the power output w has a maximum value at certain value (wm = 0.1778) of fc (0.1713) which depends on τ ,  and µ. Fig. 4 shows the variation of the efficiency ηC , ηpot , η with the load f . If the heat flow driven by kinetic energy is ignored, the efficiency ηpot increases with the load f , it approaches the Carnot efficiency ηC at quasistatic condition (f = fm ). The result is also presented in Asfaw’s work [20]. When the heat flow driven by kinetic energy is considered, the curve of the

(15) (16)

where FL E TC L1 , , = , τ= , µ= k B TH k B TH TH L (1 − τ ) 0 = (17) . τ From the above equations, one has 
0, 0  τ  1, 0  µ  1. Since the motor acts as a heat engine, the power output cannot be negative, namely, w
0. One can also obtain the value range of f , 0  f  fm , where f=

fm =

(1 − τ ) . (τ − 1)µ + 1

(18)

Fig. 3. Dimensionless heat flow qH , qC and power output w vs. the load f at τ = 0.1,  = 2.0, µ = 0.1. fc = 1.7134, wm = 0.1778, fm = 1.998.

B.-Q. Ai et al. / Physics Letters A 352 (2006) 286–290

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Fig. 5. The engine acts as a refrigerator. Heat flows via both potential energy and kinetic energy in a thermal motor in contact with two thermal baths at temperatures TH > TC , W is power input, heat flow via potential energy is reversible, heat flow via kinetic energy is irreversible.

The above equations can also be rewritten in a dimensionless form with Eq. (17) ˙Ht Q k B TH
  1 = e−/τ  + τ + µf e(1−µ)f/τ 2    1 −  + + µf e0 −µf , 2 ˙ QC t qC = k B TH
  1 = e−/τ  + τ + µf − f e(1−µ)f/τ 2    1 −  + + µf − f e0 −µf , 2 ˙   Wt w= = e−/τ e(1−µ)f/τ − e0 −µf f. k B TH qH =

Fig. 4. ηC , ηpot and η vs. the load f at τ = 0.1,  = 2.0, µ = 0.1. fc = 1.8170, ηm = 0.6452, fm = 1.998.

efficiency η is observed to be bell-shaped, a feature of resonance. When the load f = fc (1.8100), the efficiency attains its maximum value (ηm = 0.6452). The efficiency η is always less than ηpot and never approaches Carnot efficiency ηC . It is obvious that the heat flow driven by kinetic energy is always irreversible and energy leakage is inevitable, so the efficiency is less than ηpot and cannot approach Carnot efficiency. When the load f is less than the maximum load fm , the motor works as a heat engine. The power output is a peaked function of the load f . The efficiency η is less than the efficiency ηpot and can never approach the Carnot efficiency ηC . The heat flow driven by kinetic energy causes the energy leakage unavoidably and reduces the efficiency of the heat engine. 3. Thermal motor works as a refrigerator If the load is large enough along with appropriately chosen other quantities, the motor will run in the reverse direction N˙ +  N˙ − . The motor will absorb the heat from the cold reservoir and release it to the hot reservoir. Under this condition, the thermal motor acts as a refrigerator shown in Fig. 5. Similarly, the rate of total heat transferred to the hot reservoir is given by ˙ pot + Q ˙ kin Q˙ H = Q H 1 = (N˙ − − N˙ + )(E + F L1 ) − kB (N˙ + TH − N˙ − TC ). 2

(22)

The rate of total heat transferred from the cold reservoir to the heat engine is given by ˙ pot + Q ˙ kin Q˙ C = Q C 1 = (N˙ − − N˙ + )(E − F L2 ) − kB (N˙ + TH − N˙ − TC ). 2

(23)

The rate of power input is W˙ = (N˙ − − N˙ + )F L.

(24)

(25)

(26) (27)

For a refrigerator, N˙ +  N˙ − , one can get f
fm . The heat flow from cold reservoir must not be negative (qC
0), which is an important feature of a refrigerator. As for a refrigerator, its generalized COP is most important, which is given by P=

qC w 

 1 =  + τ + µf − f e(1−µ)f/τ 2    1 0 −µf −  + + µf − f e 2  (1−µ)f/τ  −1 × e − e0 −µf f .

(28)

If the heat flow driven by the kinetic energy is not considered, the COP is given by P pot =

  τ +µ−1 = PC , +µ−1= f fm 1−τ

(29)

where P C is COP of a Carnot refrigerator. When Ppot attains the PC , namely, f = fm , the power input is zero. Fig. 6 gives the plot of P C , P pot and P versus the input force f . When the heat flow driven by kinetic energy is ignored, the COP of the refrigerator P pot decreases with increasing of f and it approaches the Carnot COP at f = fm . When the heat flow via kinetic energy is considered, the curve of COP P is

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When the external load is so large that the motor moves reversely, the thermal motor can work as a refrigerator. When the heat flow driven by kinetic energy is ignored, the COP P pot decreases with increasing of the input force f and can approach the Carnot COP P C . When the heat flow driven by both potential and kinetic energy is considered, the COP P is less than P pot when fm < f < fC , larger than P pot when f > fC and equal to P pot at f = fC . However, P can never approach the Carnot COP P C . The heat flow driven by kinetic energy reduces the performance of the refrigerator. References

Fig. 6. PC , Ppot and P vs. input force f at τ = 0.7,  = 2.0, µ = 0.5. fm = 0.7129, fc = 1.0132.

bell-shaped and it can never approach the Carnot COP P C . The COP P with the heat flow driven by kinetic energy is less than P pot when fm < f < fC , larger than P pot when f > fC and equal to P pot at f = fC . However, P can never approach the P C . When f is so large that the motor runs reversibly, f becomes the input force, so the motor works as a refrigerator. The COP P is a peaked function of input force f which indicates the the COP cannot obtain a maximum value at quasistatic limit. The COP of refrigerator can never approach its Carnot COP because of the heat flow driven by kinetic energy. 4. Concluding remarks In present work, we study the thermodynamic features of a thermal motor which consists of a Brownian particle moving a sawtooth potential with an external load where the viscous medium is alternately in contact with hot and cold reservoirs along the space coordinate. The thermal motor can work as both a heat engine and a refrigerator under different conditions. We make a clear distinction between the heat flow driven by the kinetic and the potential energy of the particle, and show that the former is always irreversible and the latter may be reversible when the engine works quasistatically. When the external load is not enough, the thermal motor can work as a heat engine. The power output has a maximum value at certain value of the load f which depends on the other parameters. If only the heat flow driven by potential is considered, the efficiency ηpot increases with the load f and approaches the Carnot efficiency ηC under quasistatic condition, which agrees with the previous works [20,21]. When the heat flow driven by kinetic energy is also considered, the efficiency η is less than ηpot and can never approach the Carnot efficiency ηC . The heat flow driven by potential is reversible, while the heat flow driven by kinetic energy is irreversible. The heat flow driven by kinetic energy reduces the heat engine efficiency.

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