Optimal single-particle-exchange heat engines and
refrigerators
Yanchao Zhang, Jincan Chen
1. Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of
China
We propose a mesoscopic heat engine and refrigerator model consisting of single
particle exchange system embedded between two reservoirs. The two reservoirs are
assumed
to
be
particle
reservoir
which
the
constituent
particles
obey
Maxwell-Boltzman (M-B), Fermi-Dirac (F-D), or Bose-Einstein (B-E) distributions,
respectively. Based on the detailed balance condition, the optimize performance of the
particle exchange heat engine and refrigerator are calculated, and compared with that
of the classical Carnot-type heat engine and refrigerator.
I. Introduction
In classical thermodynamics, a reversible Carnot heat engine produces the maximum
possible work for a given temperature of the hot and cold reservoirs but generates
zero power because it is an infinitely slow operation. The efficiency ( C  1  Tc Th )
of Carnot cycle is the upper bound on the efficiency which real heat engine is
unrealistically high. In order to seek a more realistic upper bound on the efficiency of
a heat engine, Curzon and Ablborn proposed an endoreversible model of Carnot cycle
and calculated its efficiency at maximum power (EMP), i.e. so-called CA efficiency
CA  1  Tc Th .[1]
Yan and Chen [2] first reported an optimization study taking as the target function
Q c to the particular case of an endoreversible Carnot-type refrigerator
The Q c is the cooling power of the refrigerator and   Qc W is the usual
coefficient of performance (COP) for refrigerators.
W is denotes the work input.
linear (Newtonian) finite heat transfer laws.
They provides the counterpart of Curzon-Ahlborn coefficient of performance:
 CA 
1
 1.
1  Tc Th
(1)
II. GENERIC MODEL AND THEORY
A quantum dot refrigerator model is illustrated in fig.1.
Fig. 1. The schematic diagram of a single particle exchange heat engines and refrigerators.
The evolution of the occupation probability of particle exchange system state is
described by the master equation [6,22,23]:
 p0    01
 
 p1    01
 10   p0 
  .
 10   p1 
Where  i  j is the transition rate from state i to state j
 01 



 h ,c
0 1
,  10 



 h ,c
1 0
(2)
i, j  0,1 ,
and
respectively.
The model was operated in steady state situation, i.e. dpi dt  0 . According to Eq.
(2) and p0  p1  1 . Then the occupation probability are
p0 
 10
,
 01   10
(9)
p1 
 01
.
 01   10
(10)
In the case of steady state, the probability current from the hot/cold particle reservoirs
to enter the system are given by
I h c   0hc1 p0   1hc0 p1 ,
(11)
Consider first the case of a classical (M-B distribution) heat reservoir, and let  be
the bare tunneling rates between system and each of the reservoirs.
The transition rates of a particle out of reservoir  to the system and into reservoir
 from the system are given by
MB
0MB
  h f hMB
1   c f c
1MB
0   c   h
(10)
and
 0FD1   c f cFD   h f hFd
FD
1FD
   h 1  fhFD 
0   c 1  f c
(11)
and
0BE1   c f cBE   h f hBE
BE
1BE
   h 1  fhBE 
0   c 1  f c
(12)
BE




f cFD
Where f cMB
h  1 exp  xc h  ,
h  1  exp  xc h   1 and f c h  1  exp  xc h   1 are
the Maxwell-Boltzman, Fermi-Dirac and Bose-Einstein distributions, respectively.
Fig. 3. Three-dimension projection graph of the power output with the
III. Optimization analysis of heat engine
In order to obtain the optimal performance of the engine at maximum power for given
temperatures Th and Tc , we determine the values of xh and xc through the
extremal conditions P xh  0 and P xc  0 . The dimensionless energy barriers
xh and xc at maximum power as a function of the Carnot efficiency C are plotted
by numerical calculation, as shown in Fig. 2.
Fig. 2. The curves of the dimensionless energy barriers xc , xh at maximum power varying with
c for different particle reservoirs.
According to Eq. (13), the curves of the dimensionless maximum power 20P Th h
versus the Carnot efficiency is plotted, as shown in Fig. 5. It is seen that the maximum
power is a monotonically increasing function of  C
FD
BE
MB
Pmax
 Pmax
 Pmax
Fig. 3. The curves of the dimensionless maximum power P Th   varying with c for different
particle reservoirs.
FD
MB
BE
mp
 mp
 mp
Fig. 4. Difference of efficiencies at maximum power between the single particle exchange heat
engines and classical CA heat engine as a function of c .
A. Refrigerators
Fig. 4. Three-dimension projection graph of the cooling power with the
IV. OPTIMIZATION ANALYSIS AND COMPARISON
A. Heat engines
B. Refrigerators
V. CONCLUSIONS
ACKNOWLEDGMENTS
This paper is supported by
References
[1] Curzon F L and Ahlborn B 1975 Am. J. Phys. 43 22
[2] Z. Yan and J. Chen, J. Phys. D: Appl. Phys. 23, 136 (1990).
[3]