Download Finite-time thermodynamic analysis of an irreversible vacuum

Parametric optimum designs of a vacuum thermionic solar cell
Tianjun Liao
Abstract：In this paper, a vacuum thermionic solar cells consisting of a series of vacuum thermionic power
generator (VTPG), a heat sink, and a concentrator is proposed. The arithmetic expressions for the power output
and overall efficiency of the VTPG solar cell are derived. The Parametric optimum designs are presented and
discussed. The effects of the concentrating ratio and work function of the anode on the
performance characteristics are investigated. The obtained results can provide important guiding significance
for optimization of the actual VTPG solar cell and performance improvement.
Key words：Parametric optimum designs; Vacuum thermionic power generator; Solar cell
1. Introduction
2. Model description
The VTPG solar cell composed of a concentrator, a collector, a heat sink and VTPGS connected in series
exchange heat with the collector and heat sink, as shown in Fig. 1 (a). The collector acts as the
high-temperature heat reservoir of the VTPG solar cell for a further production of electric energy. The wasted
heat flow is transferred from the heat sink to the environment.
Fig. 1 (b) shows a VTPG consisting of two electrodes placed near one another in a vacuum gap, one
electrode is normally called the cathode, and the other is called the anode. Ordinarily, electrons in the cathode
are prevented from escaping from the surface by a potential-energy barrier. We call the minimum energy
required to allow an electron to be liberated from material as its work function. When the cathode at high
temperature, the numbers of electron have sufficient thermal energy to overcome the materials work function
and escape into vacuum to produce thermionic current. After collected by the anode, these electrons return to
the cathode through the load resistance RL . TC and TA are the temperatures of the cathode and the anode,
TL and T0 are the temperatures of heat sink and ambient, in this model, TC  TA  TL  T0 holds, C is the
concentrator ratio, G is the solar irradiation,  0 and  1 are the emissivity factor of the collector and
electrodes, 0 is the collector optical efficiency, Ar is the effective surface area of the cathode and anode,
U
is the heat transfer coefficient per unit area between the TPGM and the heat sink, 
is the
Stefan-Boltzmann constant, CGAr0 and  0 Ar TC4  T04  are the heat flow into the each VTPG and radiation
loss from the surface of the cathode to the environment, 1 TC4  TA4  is thermal radiant energy from the
cathode to the anode per unit area, J C and J A are the current densities emitted from the cathode and anode,
UAr TA  TL  is the heat flow from the anode to the heat sink.
Fig.1. (a) A schematic diagram of the VTPG solar cell and (b) structural schematic diagram of the VTPG.
3. Parametric optimum designs
In order to investigate the performance of the VTPG, some assumptions are given by: (i) the heat transfer
between the collector and the cathode isn’t taken into account. It is assumed that the temperature of the
collector is equal to the cathode. (ii) It is assumed that no space charge or other phenomena limit the current
density and the current density from the cathode to be in the positive direction. (iii) It is assumed that the
cathode and anode of the TPG have the same surface area Ar .
Considering the radiation loss on the surface of the collector and the heat transfer between the VTPG and
heat sink, the rate of heat Qin entering the VPTG and the rate of heat Qout leaving from the heat sink are
given by, respectively [2]:
Qin  CGAc0   0 Ac TC4  T04  ,
(1)
Qout  UAc TA  TL  .
(2)
Thermal efficiency of the solar collector is given by
s 
 0 TC4  T04 
Qin
 1
,
CGAc0
CG0
(3)
The Richardson-Dushman equation describes the current density emitted from a heated metal surface. To
derive the equation, a metal in thermal equilibrium is considered. The spectral density of the electrons is given
by the Fermi-Dirac distribution:
f FD  v  

me
4
3 3
me
4 3
3
1
 1

exp  me v 2   

 2

 kT   1

,
(4)
 1


exp    me v 2    kT 
2

 

where v is the velocity, k the Boltzmann constant, me the mass of a electron.  ,  and T are,
respectively, the electrochemical potential, work function and temperature of the metal materials
To calculate the current density J  nqv flowing away from the surface, the contributions from all electrons
with a positive velocity normal to the metal surface have to be integrated. If the x-direction is chosen as the
direction of the current, it follows:



0


J   dvx  dvy  dvz vx f FD  v   A0T 2e
where q the charge of an electron, A0 


kT
,
(5)
qme k 2
the Richardson constant.
2 2 3
According to the following Richardson-Dushman equations [1, 16], the emitted net current density
J  J C  J A determined by the two electrodes’ work functions and temperatures can be expressed as
J C  A0TC2 exp  C kTC  ,
(6)
J A  A0TA2 exp  A kTA  ,
(7)
where C and A denote the work function of the cathode and anode.
Internally, there exist two kinds of heat flows, one is the heat transport between the electrodes due to the
emission of electrons and another is the radiation loss of the two electrodes. According to the first law of
thermodynamics, the first kind of heat flow, the rate of heat Q1 leaving the cathode and the rate of heat Q2
entering the anode are given by [2]
Q1  Ar  J C   C  2kTC   J A   C  2kTA   q ,
(8)
Q2  Ar  J C   A  2kTC   J A   A  2kTA   q .
(9)
It is assumed that the two electrodes are perfect thermal radiators, thus, the second kind of heat flow, the
thermal radiation from the anode to the cathode can be given by [2]
QR  1 Ar TC4  TA4  ,
(10)
where  1 is the effective emissivity of the electrodes. For practical TPGM,  1 usually lies in the range of
0.1～0.2 [24].
According to the theory of non-equilibrium thermodynamics, combining Eqs. (7)-(9), one can obtain the rate
( QC ) of total leaving the cathode and the rate ( QA ) of total heat flow entering the anode [2]


(11)


(12)
QC =Q1  QR =Ar  J C  C  2kTC   J A  C  2kTA  q +1 TC4  TA4  ,
QA =Q2  QR =Ar  J C  A  2kTC   J A  A  2kTA  q +1 TC4  TA4  .
By using the law of conservation of energy, we can obtain
Qin  ns QC ,
(13)
Qout  ns QA ,
(14)
where ns is the number of VTPGS connected in series.
Combining Eqs. (1), (2), (10)-(13), the thermal balance equations can be derived:
Ac CG0   0 TC4  T04   ns Ar    J C  C  2kTC   J A  C  2kTA  q +1 TC4  TA4  ,
(15)
Ac U TA  TL   ns Ar    ns Ar    J C   A  2kTC   J A   A  2kTA  q +1 TC4  TA4  ,
(16)
The power output PVTPG and efficiency VTPG of TPGM are given by:
PVTPG  Qin  Qout  VJns Ar ,
VTPG 
PVTPG
VJns Ar

,
Qin
CGAc0   0 Ac TC4  T04 
(17)
(18)
where V   C  A  q is operating voltage of a per TPG unit.
The over thermal efficiency is product of the thermal efficiency of the collector and the thermal efficiency of
the VPTG solar cell
  sVTPG 
VJns Ar
CGAc0
(19)
Theoretically, combining Eq. (15) with Eq. (16), the TC and TA can be solved out under different voltage V
and Ac  ns Ar  . Then, substituting the values of TC and TA into Eqs. (17) and (19), the PVTPG and  can
be determined.
4. Conclusion
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