Eur. Phys. J. B (2015) 88: 322
https://doi.org/10.1140/epjb/e2015-60782-0

Regular Article

The effect of temperature dependence of viscosity on a Brownian heat engine

¹ Department of Physics, California State University Dominguez Hills, Carson, CA 90747, USA
² National Center for Biotechnology Information, National Library of Medicine and National Institute of Health, 8600 Rockville Pike, Bethesda MD, 20894 USA

^a e-mail: taye@es.hokudai.ac.jp

Received: 1 October 2015
Received in final form: 13 October 2015
Published online: 7 December 2015

Abstract

We modeled a Brownian heat engine as a Brownian particle that hops in a periodic ratchet potential where the ratchet potential is coupled with a spatially varying temperature. The strength for the viscous friction γ(x) is considered to decrease exponentially when the temperature T(x) of the medium increases (γ(x) = Be^{− AT(x)}) as proposed originally by Reynolds [O. Reynolds, Phil. Trans. R. Soc. London 177, 157 (1886)]. Our result depicts that the velocity of the motor is considerably higher when the viscous friction is temperature dependent than that of the case where the viscous friction is temperature independent. The dependence of the efficiency η as well as the coefficient of performance of the refrigerator P_ref on model parameters is also explored. If the motor designed to achieve a high velocity against a frictional drag, in the absence of external load f, we show that Carnot efficiency or Carnot refrigerator is unattainable even at quasistatic limit as long as the viscous friction is temperature dependent A ≠ 0. On the contrary, in the limit A → 0 or in general in the presence of an external load (for any A) f ≠ 0, at quasistatic limit, Carnot efficiency or Carnot refrigerator is attainable as long as the heat exchange via kinetic energy is omitted. For all cases, far from quasistatic limit, the efficiency and the coefficient of performance of the refrigerator are higher for constant γ case than the case where γ is temperature dependent. On the other hand, if one includes the heat exchange at the boundary of the heat baths, Carnot efficiency or Carnot refrigerator is unattainable even at quasistatic limit. Moreover, the dependence for the optimized and maximum power efficiencies on the determinant model parameters is explored.